1
Controllable Thermal Conductivity in Twisted Homogeneous
Interfaces of Graphene and Hexagonal Boron Nitride
Wengen Ouyang1 Huasong Qin2 Michael Urbakh1 and Oded Hod1
1Department of Physical Chemistry School of Chemistry The Raymond and Beverly
Sackler Faculty of Exact Sciences and The Sackler Center for Computational
Molecular and Materials Science Tel Aviv University Tel Aviv 6997801 Israel
2State Key Laboratory for Strength and Vibration of Mechanical Structures School of
Aerospace Xirsquoan Jiaotong University Xirsquoan 710049 China
ABSTRACT
Thermal conductivity of homogeneous twisted stacks of graphite is found to strongly
depend on the misfit angle The underlying mechanism relies on the angle dependence
of phonon-phonon couplings across the twisted interface Excellent agreement between
the calculated thermal conductivity of narrow graphitic stacks and corresponding
experimental results indicates the validity of the predictions This is attributed to the
accuracy of interlayer interactions descriptions obtained by the dedicated registry-
dependent interlayer potential used Similar results for h-BN stacks indicate overall
higher conductivity and reduced misfit angle variation This opens the way for the
design of tunable heterogeneous junctions with controllable heat-transport properties
ranging from substrate-isolation to efficient heat evacuation
Keywords Interfacial thermal conductivity graphite h-BN twisted interface misfit
angle phonon-phonon coupling registry-dependent interlayer potential
2
Graphene is considered to be one of the most promising heat dissipating materials in nanoelectronics
[1] due to its ultrahigh in-plane room-temperature thermal conductivity of ~3000-5000 Wm-1K-1 [23]
This however can be hindered by graphene-substrate interactions that may lead to a substantial
reduction of the heat-transport due to phonon leakage across the graphene-substrate interface and
strong interfacial scattering of flexural phonon modes [4] Such undesirable substrate effects can be
reduced by considering multilayer graphene stacks These are expected to effectively isolate the top
graphene layers from the substrate due to the considerably lower cross-plane thermal conductivity
(~68 Wm-1K-1) [5] while exhibiting high in-plane conductivity that can be tuned via the stack
thickness [6-13] Anisotropic thermal conductivity is also observed for bulk hexagonal boron nitride
(h-BN) with the in-plane and cross-plane thermal conductivity in the range of 390-420 Wm-1K-1 and
25-48 Wm-1K-1 respectively [1415]
Efficient is-situ tuning of the thermal conductivity of such graphitic structures can be achieved by
controlling the twist angle between adjacent layers within the stack This has been recently
computationally demonstrated for finite-sized nanoscale few-layer graphene junctions [1617] Two
factors however limit the applicability of these results (i) the simulations were performed using
simplistic isotropic interlayer potentials that are known to be inaccurate for simulating the interlayer
interactions in layered materials [18-21] and (ii) the relevance of the results for large-scale interfaces
is questionable due to significant edge scattering effects inherent to the small finite-sized model
systems studied
To address these issues we investigate the interlayer thermal conductivity of graphene and h-BN
stacks of varying thicknesses and twist angles This allows us to gain fundamental understanding of
the heat transport mechanisms in layered materials stacks and identify feasible means to control it
Our model system consists of two contacting identical AB (AArsquo)-stacked graphite (h-BN) slabs
whose interfacing graphene (h-BN) layers are twisted with respect to each other to create a stacking
fault of misfit angle θ (see Figure 1) Recent experiments demonstrated fine control over the misfit
angle in such setups [2223] The thickness of the entire construction is varied between 27 nm-35 nm
(8-104 layers) and periodic boundary conditions are applied in all directions Heat transport
simulations are performed using state-of-the-art anisotropic interlayer potentials [18-21] applied to
the twisted stacks These potentials were shown to capture well the structural dynamic heat
dissipation and phonon spectrum of graphitic and h-BN layered systems [24-27] A thermal bias is
induced by applying Langevin thermostats with different temperatures to two layers residing on
opposite sides away from the twisted interface (see Sec 1 of the Supporting Information (SI) for
further details)
3
Figure 1 Schematic representation of the simulation setup Two identical AB-stacked graphite slabs
(gray and orange respectively) are twisted with respect to each other to create a stacking fault of misfit
angle θ A thermal bias is induced by applying Langevin thermostats to the two layers marked by
dashed red (Thot) and green (Tcold) rectangles The arrows indicate the direction of the vertical heat
flux Since periodic boundary conditions are applied also in the vertical direction two twisted
interfaces are shown across which heat flows in opposite directions
We start by studying the effect of the misfit angle on the cross-plane thermal conductivity of the
twisted graphite and h-BN stacks Figure 2 presents the dependence of the cross-plane thermal
conductivity of the entire stack on the misfit angle for model systems consisting of 8 (red circles) and
16 (black triangles) layers for (a) graphite and (b) h-BN A pronounced dependence of the cross-plane
thermal conductivity (120581120581CP) of the entire graphitic stack is clearly evident which above a misfit angle
of sim 5deg 120581120581CP drops by a factor of 3-4 with respect to the value obtained for the aligned contact
Similar misfit-angle dependence of 120581120581CP is obtained for twisted bilayer graphene (tBLG) using the
transient MD simulation approach (see Sec 2 of the SI) We note that this sharp drop for graphite is
steeper and that the overall reduction is higher than those previously obtained using Lennard-Jones
interlayer potentials in finite model systems [1617] The corresponding cross-plane thermal
conductivity of the commensurate h-BN stack is found to be approximately double that of graphite
for the same number of layers Notably it reduces more gradually with the twist angle and saturates
at sim 15deg with an overall two-three fold reduction
The thermal conductivity of both graphite and h-BN stacks is found to increase when doubling their
thickness To identify the source of this thickness dependence we plot in Figure 2(c-d) the interfacial
thermal resistance (ITR) (see Sec 12 of the SI for the definition) associated with the twisted junction
formed between the contacting graphene or h-BN layers of the two optimally-stacked slabs Note that
unlike 120581120581CP which measures the conductivity of the entire stack the ITR corresponds to the heat
transport resistance of the two adjacent layers forming the twisted interface Two important
4
observations can be made (i) the ITR is weakly dependent on the stack thickness indicating that the
thickness dependence arises from the conductivity of the optimally-stacked interfacing slabs
Specifically in the thickness range considered the heat conductivity grows with slab thickness due to
reduction of phonon-phonon interactions and increased contribution of long wave-length phonons
below the mean-free path [28-30] (ii) the ITR strongly depends on the twist angle demonstrating a
~10-fold (4-fold) increase when the twist angle at the graphene (h-BN) interface is varied from 0deg
to 15deg This clearly indicates that the twist angle can be utilized to control the cross-plane thermal
conductivity of hexagonal two-dimensional (2D) materials and to effectively thermally isolate the top
layers from the underlying substrate
Figure 2 Twist-angle dependence of the cross-plane thermal conductivity of the entire stack (a b)
and the interfacial thermal resistance (c d) of the twisted contact formed between the optimally-
stacked slabs of graphite (a c) and bulk h-BN (b d) respectively Red circles and black triangles
correspond to the results obtained using 8 and 16 layer models respectively
The strong dependence of the cross-plane thermal conductivity of graphene and h-BN on the stacking
fault twist angle is related to the degree of coupling between the phonon modes of the two contacting
layers at the twisted interface Note that the term ldquocouplingrdquo used herein is not related to the standard
notion of phonon-phonon couplings due to anharmonic effects Instead we regard to the off-diagonal
terms of the Hessian when represented in the basis of the harmonic phonon modes of the isolated
5
layers To demonstrate this we write the dynamical matrix (the mass-reduced Fourier transform of
the force constant matrix) in block form as follows
120625120625(119954119954) = 12062512062511(119954119954) 12062512062512(119954119954)12062512062521(119954119954) 12062512062522(119954119954) (1)
where 12062512062511(119954119954) and 12062512062522(119954119954) are the block matrices relating to the first and second layer and 12062512062512(119954119954)
and 12062512062521(119954119954) = 12062512062512dagger (119954119954) all evaluated at wave-vector 119954119954 The interlayer phonon-phonon couplings are
obtained by diagonalizing separately 12062512062511(119954119954) and 12062512062522(119954119954) such that 12062512062511(119954119954) =
1199321199321dagger(119954119954)12062512062511(119954119954)1199321199321(119954119954) and 12062512062522(119954119954) = 1199321199322
dagger(119954119954)12062512062522(119954119954)1199321199322(119954119954) are diagonal matrices containing the
frequencies (120596120596119894119894) of the phonon modes of the two layers and 1199321199321(119954119954) and 1199321199322(119954119954) are unitary
matrices of the corresponding eigenvectors We now construct a global block diagonal transformation
matrix of the form
119932119932(119954119954) = 1199321199321(119954119954) 120782120782120782120782 1199321199322(119954119954) (2)
and transform the full dynamical matrix as follows
119932119932dagger(119954119954)120625120625(119954119954)119932119932(119954119954) = 12062512062511(119954119954) 12062512062512(119954119954)12062512062521(119954119954) 12062512062522(119954119954)
(3)
where 12062512062512(119954119954) = 1199321199321dagger(119954119954)12062512062512(119954119954)1199321199322(119954119954) and 12062512062521(119954119954) = 12062512062512
dagger (119954119954) are the interlayer phonon-phonon
coupling blocks Naturally when the two layers are infinitely separated these coupling blocks vanish
and the diagonal blocks converge to those of the isolated layers
The overall coupling between the two layers can be obtained from the individual phonon-phonon
coupling matrix elements via Fermirsquos golden rule [31] which reads as (see Sec 4 of SI for a detailed
derivation)
Γtot = 120587120587ℏ3
2sum 119890119890minus120573120573119864119864119954119954120582120582
119885119885
120588120588119864119864119954119954120582120582119881119881120582120582120582120582+31199031199032(119954119954)
2
1198641198641199541199541205821205822119954119954120582120582 (4)
where 119885119885 = sum 119890119890minus120573120573119864119864119954119954120582120582119954119954120582120582 is the partition function 119864119864119954119954120582120582 is the energy of phonons at branch 120582120582 with
wave number 119954119954 120588120588119864119864119954119954120582120582 is the density-of-states (DOS) at 119864119864119954119954120582120582 and 119881119881120582120582120582120582+31199031199032(119954119954)
2 is the coupling
matrix element between branches of phonons of similar energy in the two layers whose number of
atoms in one unit cell is 119903119903
Using Eq (4) we can rationalize the misfit angle dependence of the heat flux across the twisted
interface from the calculated inter-phonon coupling To that end we performed room temperature
(300K) simulations (technical details can be found in Sec 3 of SI) for tBLG with different misfit
angles using the Greenrsquos function molecular dynamics (GFMD) developed by Kong et al [32] as
6
implemented in LAMMPS [33] The simulations allow us to evaluate the dynamical matrix from
which the phonon-phonon couplings can be extracted (see details in Sec 3 of SI) and the overall heat
transfer rate calculated Figure 3 shows the resulting heat transfer rate (normalized to is value for the
aligned contact (120579120579 = 0) ) as a function of the misfit angle compared to the interfacial thermal
conductivity defined as the inverse of the ITR presented in Figure 2(c) ITC equiv 1ITR The
remarkable agreement between the calculated interfacial thermal conductivity and Fermirsquos golden
rule results indicate that the dependence of the interlayer phonon-phonon couplings on the misfit
angle is responsible for the strong angle dependence of the interfacial conductivity Notably the sharp
heat conductivity drop at misfit angles in the range of 0deg-5deg as well as the small conductivity for larger
misfit angles are well captured by Fermirsquos golden rule
Figure 3 Comparison between Fermirsquos golden rule results (open blue squares) for the interfacial heat-
transfer rate of a tBLG and the calculated interfacial thermal conductivity at various misfit angles
ITC simulation results are presented for both 8 layers (open red circles) and 16 layers (open black
triangles) showing similar behavior For comparison purposes all data sets are normalized to their
value obtained for the aligned contact
To correlate our results with experimentally measured thermal conductivities that are often obtained
for thick samples we repeated our calculations for increasing stack thicknesses at fixed misfit angles
Figure 4 presents results for the calculated heat conductivity of (a) graphite and (b) h-BN stacks either
aligned (open red circles) or twisted by 120579120579 = 3016deg (open black diamond symbols) as a function of
number of layers in the stack As discussed above for both systems the misoriented stack exhibits
lower heat conductivity compared to the aligned system however its thickness dependence is
considerably stronger This can be attributed to the significantly higher interface resistance of the
twisted interface that when plugged in Eq (S2) of the SI for the overall conductivity induces stronger
7
thickness dependence
Comparing our calculated heat conductivities for the aligned contact (open red circles) to available
experimental data for ~35 nm thick graphite slabs [34] (dashed green line) we find that at the thickest
model system considered of 104 graphene layers (~34 nm thick) the calculated value of 085plusmn005
W m sdot Kfrasl is in remarkable agreement with the measured value of ~07 W m sdot Kfrasl Furthermore
experimental values for bulk graphite [5] indicate that the thermal conductivity continues to grow up
to ~68 W m sdot Kfrasl (black dash-dotted line) which is consistent with the general trend of the calculated
heat conductivity that does not saturate for the thickest model system considered These results
strongly enforce the validity of our force-field and model systems to model the heat conductivity of
twisted layered materials interfaces Available experimental results for the heat conductivity of bulk
h-BN are marked by the dashed-dotted black and dashed-green lines in Figure 4(b) In line with our
findings for the graphitic interface our calculated finite slab heat conductivities for the aligned
interface (open red circles) continue to grow with the number of layers and are consistently below the
bulk value
Figure 4 Thickness dependence of the thermal conductivity 120581120581CP of aligned (open red circles) and
twisted by 3016deg (open black diamond symbols) graphite (a) and h-BN (b) stacks Blue squares
represent results obtained using the isotropic Lennard-Jones potential for the aligned contacts The
green dashed and black dash-dotted lines represent experimental results measured for graphite (a)
(Refs [534]) and bulk h-BN (b) (Refs [1415]) Note that both axis scales are logarithmic Error bar
estimation procedure is discussed in Sec 1 of the SI
Another important factor that may affect the interlayer thermal transport properties of 2D material
stacks is the average temperature of the system which was taken to be ~300K in all abovementioned
simulations To evaluate the sensitivity of our results towards this parameter we repeated the heat
conductivity and interfacial resistance calculations of optimally stacked graphite and h-BN stacks for
an average temperature of 400 K The results presented in see Sec 4 of the SI indicate that 120581120581CP and
8
ITR weakly depend on the average temperature over the entire thickness range considered This
conclusion is consistent with recent experimental findings [11] We may attribute this to the fact that
the thickness of the graphite and h-BN stack model considered is much smaller than their phonon
mean-free path (~200 nm for graphite [1011133536] and ~100 nm for bulk h-BN [15]) such that
the phonon transport is dominated by phonon-boundary scattering which weakly depends on
temperature in the range considered Therefore temperature dependent Umklapp processes have only
marginal contribution to our results [11]
Finally we note that previous calculations of the heat conductivity of twisted graphitic interfaces
relied on Lennard-Jones (LJ) potentials describing the interlayer interactions [8-1013] To
demonstrate the importance of using registry-dependent interlayer potentials we have repeated our
calculations of the heat conductivity of graphitic slabs with the REBO intralayer potential augmented
by LJ interlayer interactions [37] (120576120576 = 284 meV120590120590 = 34 Å ) We find that the calculated heat
conductivities obtained using the LJ interlayer potential are consistently higher than those obtained
by our ILP and that the difference between them grows with the model system thickness Notably the
heat conductivity obtained using the LJ potential for a graphitic slab of thickness ~34 nm is 154
W m sdot Kfrasl overestimating the experimental value by more than a factor of 2
The excellent agreement of our ILP calculations with experimental data of nanoscale graphitic stacks
therefore demonstrates the reliability of our predictions for the strong interfacial misfit angle
dependence of cross-layer thermal conductivity in graphite and h-BN The observed sharp
conductivity decrease of twisted graphitic interfaces at misfit angles lt 5deg opens the way to control
the thermal evacuation rate and thermal isolation of active layers in graphene-based electronic and
mechanical devices The revealed underlying mechanism suggests that design rules can be obtained
by carefully tailoring the phonon-phonon couplings across the twisted interface While the misfit
angle dependence of h-BN is found to be weaker than that of graphite the overall thermal
conductivity of the former is found to be higher This may be utilized to achieve higher conductivity
and controllability in twisted heterogeneous junctions of layered materials
ASSOCIATED CONTENT
Supporting Information
The supporting Information section includes a description of the methodology thermal conductivity
of twisted bilayer graphene theory for calculating the phonon coupling of twisted bilayer graphene
9
derivation of Fermirsquos golden rule and temperature dependence of the cross-plane thermal
conductivity
AUTHOR INFORMATION
Corresponding Author
E-mail urbakhtauextauacil
Notes
The authors declare no competing financial interest
ACKNOWLEDGMENTS
The authors would like to thank Prof Abraham Nitzan Dr Guy Cohen and Dr Yiming Pan for
helpful discussions WO acknowledges the financial support from a fellowship program for
outstanding postdoctoral researchers from China and India in Israeli Universities and the support from
the National Natural Science Foundation of China (Nos 11890673 and 11890674) HQ
acknowledges the financial support from the National Natural Science Foundation of China (No
11890674) MU acknowledges the financial support of the Israel Science Foundation Grant
No 114118 and the ISF-NSFC joint grant 319119 OH is grateful for the generous financial
support of the Israel Science Foundation under grant no 158617 and the Naomi Foundation for
generous financial support via the 2017 Kadar Award This work is supported in part by COST Action
MP1303
10
References
[1] A A Balandin Thermal properties of graphene and nanostructured carbon materials Nat Mater 10 569 (2011) [2] S Ghosh I Calizo D Teweldebrhan E P Pokatilov D L Nika A A Balandin W Bao F Miao and C N Lau Extremely high thermal conductivity of graphene Prospects for thermal management applications in nanoelectronic circuits Appl Phys Lett 92 151911 (2008) [3] A A Balandin S Ghosh W Bao I Calizo D Teweldebrhan F Miao and C N Lau Superior thermal conductivity of single-layer graphene Nano Lett 8 902 (2008) [4] J H Seol et al Two-Dimensional Phonon Transport in Supported Graphene Science 328 213 (2010) [5] R Taylor The thermal conductivity of pyrolytic graphite J Philosophical Magazine 13 157 (1966) [6] S Ghosh W Bao D L Nika S Subrina E P Pokatilov C N Lau and A A Balandin Dimensional crossover of thermal transport in few-layer graphene Nat Mater 9 555 (2010) [7] Z Wang R Xie C T Bui D Liu X Ni B Li and J T L Thong Thermal Transport in Suspended and Supported Few-Layer Graphene Nano Lett 11 113 (2011) [8] Z Wei Z Ni K Bi M Chen and Y Chen Interfacial thermal resistance in multilayer graphene structures Phys Lett A 375 1195 (2011) [9] Y Ni Y Chalopin and S Volz Significant thickness dependence of the thermal resistance between few-layer graphenes Appl Phys Lett 103 061906 (2013) [10] J Chen J H Walther and P Koumoutsakos Strain Engineering of Kapitza Resistance in Few-Layer Graphene Nano Lett 14 819 (2014) [11] Q Fu J Yang Y Chen D Li and D Xu Experimental evidence of very long intrinsic phonon mean free path along the c-axis of graphite Appl Phys Lett 106 031905 (2015) [12] G Fugallo A Cepellotti L Paulatto M Lazzeri N Marzari and F Mauri Thermal Conductivity of Graphene and Graphite Collective Excitations and Mean Free Paths Nano Lett 14 6109 (2014) [13] Z Wei J Yang W Chen K Bi D Li and Y Chen Phonon mean free path of graphite along the c-axis Appl Phys Lett 104 081903 (2014) [14] E K Sichel R E Miller M S Abrahams and C J Buiocchi Heat capacity and thermal conductivity of hexagonal pyrolytic boron nitride Phys Rev B 13 4607 (1976) [15] P Jiang X Qian R Yang and L Lindsay Anisotropic thermal transport in bulk hexagonal boron nitride Phys Rev Mater 2 064005 (2018) [16] M-H Wang Y-E Xie and Y-P Chen Thermal transport in twisted few-layer graphene Chinese Physics B 26 116503 (2017) [17] X Nie L Zhao S Deng Y Zhang and Z Du How interlayer twist angles affect in-plane and cross-plane thermal conduction of multilayer graphene A non-equilibrium molecular dynamics study Int J Heat Mass Tran 137 161 (2019) [18] A N Kolmogorov and V H Crespi Registry-dependent interlayer potential for graphitic systems Phys Rev B 71 235415 (2005) [19] M Reguzzoni A Fasolino E Molinari and M C Righi Potential energy surface for graphene on graphene Ab initio derivation analytical description and microscopic interpretation Phys Rev B 86 (2012) [20] M M van Wijk A Schuring M I Katsnelson and A Fasolino Moire Patterns as a Probe of Interplanar Interactions for Graphene on h-BN Phys Rev Lett 113 135504 (2014) [21] D Mandelli W Ouyang M Urbakh and O Hod The Princess and the Nanoscale Pea Long-Range Penetration of Surface Distortions into Layered Materials Stacks ACS Nano 13 7603 (2019) [22] E Koren E Loumlrtscher C Rawlings A W Knoll and U Duerig Adhesion and friction in mesoscopic graphite contacts Science 348 679 (2015) [23] E Koren I Leven E Loumlrtscher A Knoll O Hod and U Duerig Coherent commensurate electronic states at the interface between misoriented graphene layers Nat Nanotechnol 11 752 (2016)
11
[24] W Ouyang I Azuri D Mandelli A Tkatchenko L Kronik M Urbakh and O Hod Mechanical and Tribological Properties of Layered Materials under High Pressure Assessing the Importance of Many-Body Dispersion Effects J Chem Theory Comput 16 666 (2020) [25] W Ouyang D Mandelli M Urbakh and O Hod Nanoserpents Graphene Nanoribbon Motion on Two-Dimensional Hexagonal Materials Nano Lett 18 6009 (2018) [26] I Leven T Maaravi I Azuri L Kronik and O Hod Interlayer Potential for Grapheneh-BN Heterostructures J Chem Theory Comput 12 2896 (2016) [27] T Maaravi I Leven I Azuri L Kronik and O Hod Interlayer Potential for Homogeneous Graphene and Hexagonal Boron Nitride Systems Reparametrization for Many-Body Dispersion Effects J Phys Chem C 121 22826 (2017) [28] L H Liang and B Li Size-dependent thermal conductivity of nanoscale semiconducting systems Phys Rev B 73 153303 (2006) [29] P K Schelling S R Phillpot and P Keblinski Comparison of atomic-level simulation methods for computing thermal conductivity Phys Rev B 65 144306 (2002) [30] J Shiomi Nonequilirium molecular dynamics methods for lattice heat conduction calculations Annual Review of Heat Transfer 17 (2014) [31] G E Stedman Fermis Golden RulemdashAn Exercise in Quantum Field Theory Am J Phys 39 205 (1971) [32] L T Kong G Bartels C Campantildeaacute C Denniston and M H Muumlser Implementation of Greens function molecular dynamics An extension to LAMMPS Comput Phys Commun 180 1004 (2009) [33] S Plimpton Fast Parallel Algorithms for Short-Range Molecular Dynamics J Comput Phys 117 1 (1995) [34] M Harb et al The c-axis thermal conductivity of graphite film of nanometer thickness measured by time resolved X-ray diffraction Appl Phys Lett 101 233108 (2012) [35] H Zhang X Chen Y-D Jho and A J Minnich Temperature-Dependent Mean Free Path Spectra of Thermal Phonons Along the c-Axis of Graphite Nano Lett 16 1643 (2016) [36] D P Sellan E S Landry J E Turney A J H McGaughey and C H Amon Size effects in molecular dynamics thermal conductivity predictions Phys Rev B 81 214305 (2010) [37] S J Stuart A B Tutein and J A Harrison A reactive potential for hydrocarbons with intermolecular interactions J Chem Phys 112 6472 (2000)
S1
Supporting information for ldquoControllable Thermal Conductivity of
Twisted Graphene and Hexagonal Boron Nitride Interfacesrdquo
Wengen Ouyang1 Huasong Qin2 Michael Urbakh1 and Oded Hod1
1Department of Physical Chemistry School of Chemistry The Raymond and Beverly
Sackler Faculty of Exact Sciences and The Sackler Center for Computational
Molecular and Materials Science Tel Aviv University Tel Aviv 6997801 Israel
2State Key Laboratory for Strength and Vibration of Mechanical Structures School of
Aerospace Xirsquoan Jiaotong University Xirsquoan 710049 China
The Supporting information includes following sections
1 Methodology
2 Thermal conductivity of twisted bilayer graphene
3 Theory for calculating the phonon coupling of twisted bilayer graphene
4 Derivation of Fermirsquos golden rule
5 Temperature dependence of cross-plane thermal conductivity
S2
1 Methodology
11 Model system
The initial interlayer distance across the layered stack was set equal to 34 Aring and 33 Aring for graphite
and bulk h-BN respectively Periodic boundary conditions were applied in all directions It should be
noted that the lattice structure is rigorously periodic only at some specific twist angles the values of
which are listed in Table S1 in section 3 below While the cross-sectional area for each misfit angle
θ is different all systems considered have a contact area exceeding 12 nm2 which was shown to
provide converged results with respect to unit-cell dimensions [1] The intralayer interactions within
each graphene and h-BN layer were modeled via the second generation REBO potential [2] and
Tersoff potential [3] respectively The interlayer interactions between the layers of graphite and bulk
h-BN were described via our dedicated interlayer potential (ILP) [4] which is implemented in the
LAMMPS [5] suite of codes [6]
Fig S1 Schematic representation of the simulation setup (a) and steady-state temperature profile (b) respectively In panel (a) two identical AB-stacked graphite slabs (gray and orange respectively) are twisted with respect to each other to create a stacking fault of misfit angle θ A thermal bias is induced by applying Langevin thermostats to the two layers marked by dashed red (Thot) and green (Tcold) rectangles The arrows indicate the direction of the vertical heat flux Since periodic boundary conditions are applied also in the vertical direction two twisted interfaces are shown across which heat flows in opposite directions The steady-state temperature profiles are illustrated in panel (b) where N is the total number of layers in the model system and RAB dAB and Rθ and dθ mark the interfacial Kapitza resistance [89] and interlayer distance for contacting graphene layers with AB-stacking and misfit angle θ respectively The red lines in panel (b) mark the temperature variation across the twisted interface where the vertical axis corresponds to the position of the various layers along the stack and the horizontal axis marks the temperature of the various layers
S3
12 Simulation Protocol
All MD simulations were performed with the LAMMPS simulation package [5] The velocity-Verlet
algorithm with a time-step of 05 fs was used to propagate the equations of motion A Noseacute-Hoover
thermostat with a time constant of 025 ps was used for constant temperature simulations To maintain
a specified hydrostatic pressure the three translational vectors of the simulation cell were adjusted
independently by a Noseacute-Hoover barostat with a time constant of 10 ps [7] To relax the box we first
equilibrated the systems in the NPT ensemble at a temperature of T = 300 K and zero pressure for
250 ps (see Fig S2) After equilibration Langevin thermostats with damping coefficients 10 ps-1
were applied to the bottom and middle layer of the graphene stack (see Fig S1) with target
temperature Thot = 375 K (hot reservoir) and Tcold = 225 K (cold reservoir) respectively Then the
system was allowed to reach steady-state over a subsequent simulation period of 750 ps (see Fig S2)
during which the dynamics of all non-thermostated layers followed the NVE ensemble For the larger
model systems the length of the NPT and Langevin stages was doubled (for the 32 and 48 layers
systems) or tripled (for the 104 layers graphitic system) to ensure convergence of the obtained steady-
state Once steady-state was obtained the last 500 ps were used to calculate the thermal conductivity
of the twisted graphite and bulk h-BN The statistical errors were estimated using ten different data
sets each calculated over a time interval of 50 ps
Fig S2 Time evolution of the temperature of thermostated layers for 16 layers twisted graphite with misfit angle (a) θ = 0deg (b) θ = 509deg (c) θ = 1518deg and (d) θ = 3016deg Note that the thermal fluctuations increase with increasing the misfit angle due to the growing interfacial thermal resistance that enhances phonon back scattering at the twisted junction
S4
13 Calculation of the interfacial thermal resistance
According to Fourierrsquos law the cross-plane thermal conductivity (120581120581CP) of a twisted graphitic interface
of misfit angle θ can be calculated as
120581120581CP = 119876119860119860Δ119879119879Δ119911119911
(S1)
where 119876 is the heat flux A is the cross-section area and Δ119879119879Δ119911119911 is the temperature gradient along
the direction of heat flux (perpendicular to the basal plane in our case) Fig S1(b) shows a schematic
temperature profile along the z-direction where the vertical axis corresponds to the position of the
various layers along the stack and the horizontal axis marks the temperature of the various layers The
actual temperature profiles extracted from the NEMD simulations for twisted graphite with different
number of layers can be found in Fig S3 For Bernal-stacked graphite (ie θ = 0deg red circles in Fig
S3) only the linear region of the temperature profile was used to calculate 120581120581CP and the points
corresponding to the layers where the thermostats were applied were omitted (marked with green
triangle in Fig S3) The 120581120581CP of the system was calculated using Eq (S1) by averaging over the two
linear regions of the temperature profiles For the twisted case (θ ne 0deg) we found a sudden
temperature decrease Δ119879119879120579120579 at the position of the twisted interface (see black squares in Fig S3) 120581120581CP
in this case was calculated using the temperature gradient calculated for the same layer range as that
for θ = 0deg To characterize the thermal properties of the twisted interface the concept of interfacial
thermal resistance (ITR) ie Kapitza resistance [89] was introduced According the definition of
the Kapitza resistance [8] 119877119877 = 119860119860Δ119879119879119876 and noticing that Δ119879119879tot = (1198731198732 minus 3)Δ119879119879AB + Δ119879119879120579120579 and
Δ119911119911 = (1198731198732 minus 3)119889119889AB + 119889119889120579120579 Eq (1) can be rewritten considering two-resistors in series as [see Fig
S1(b)]
(1198731198732 minus 3)119877119877AB + 119877119877120579120579 = [(1198731198732 minus 3)119889119889AB + 119889119889120579120579] 120581120581CPfrasl (S2)
Here 119877119877AB 119889119889AB Δ119879119879AB and 119877119877120579120579 119889119889120579120579 Δ119879119879120579120579 are the ITR interlayer distance and temperature
difference for adjacent AB-stacked and twisted graphene layers respectively For the aligned contact
(120579120579 = 0deg) the ITR can be simply calculated as 119877119877AB = 119889119889AB120581120581AB Once 119877119877AB is known 119877119877120579120579 can be
calculated from Eq (S2) We note that the sharp temperature drop at the twisted interface indicates
that 119877119877120579120579 should be much larger than 119877119877AB
S5
Fig S3 Temperature profiles for graphitic stacks consisting of (a) 8 and (b) 16 layers The red circles and black squares represent the temperature profiles for the aligned (120579120579 = 0deg) and twisted (120579120579 =3016deg) junctions respectively Green triangles represent data points that were omitted in the 120581120581CP calculation
2 Thermal conductivity of twisted bilayer graphene
As comparison we also calculated the interfacial thermal conductivity (ITC) and ITR of twisted
bilayer graphene (tBLG) with the transient MD simulation approach [15-17] since the NEMD
simulation protocol used in the main text becomes invalid in this case In this protocol the system
was first equilibrated within the NPT ensemble at T = 200 K and zero pressure for 100 ps which was
followed by a 100 ps NVT ensemble equilibration stage and a 100 ps of NVE ensemble equilibration
stage After the system reached steady-state an ultrafast heat impulse was imposed on the top layer
of the t-BLG for 50 fs to increase the temperature of the top layer from 200 K to 400 K while that of
bottom layer of tBLG remained unchanged After the external heat source was removed thermal
energy flowed from the top layer to the bottom layer due to the temperature difference and the
temperature of both layers approached 300 K when quasi-steady-state was reached During the
thermal relaxation time interval (500 ps) the temperature and energy of the system sections were
recorded The ITR could then be extracted using the following equation [15-17]
part119864119864t120597120597120597120597
= 119860119860119877119877119879119879bot(119905119905) minus 119879119879top(119905119905) (S3)
where 119864119864t is the total energy of the top graphene layer R is the ITR of the tBLG A is the interfacial
cross-section area and 119879119879bot and 119879119879top are the instantaneous temperatures measured for the bottom
and top layers of the tBLG respectively Note that in Eq S3 we assume a linear dependence of the
heat flux on the temperature difference between the layers The ITC of the tBLG is simply defined as
ITC equiv 119889119889ITR where d is the average interlayer distance
S6
The ITC and ITR of tBLG as functions of misfit angle calculated with the transient MD simulation
protocol are illustrated in Fig S4 demonstrating similar misfit-angle dependence as that for the
NEMD protocol with Langevin thermostat exercised to obtain the results presented in the main text
This further validates the reliability of the simulation protocol adopted in the main text which is more
suitable to treat thick slabs and allows to obtain a true stead-state
Fig S4 Misfit-angle dependence of (a) ITC and (b) ITR for a twisted bilayer graphene obtained using the transient MD simulation approach
3 Theory for calculating the phonon-phonon coupling of twisted bilayer graphene
31 Brillouin Zone of supercell in tBLG
For tBLG the lattice structure is rigorously periodic only at some specific misfit angles θ where the
lattice vector 1199231199231 = 11989911989911199381199381 + 11989911989921199381199382 in the bottom layer equals the vector 1199231199232 = 11989811989811199381199381 + 11989811989821199381199382 in the
top layer with certain integers m1 m2 and n1 n2 Here 1199381199381 = 119886119886(10) and 1199381199382 = 11988611988612radic32 are
the primitive lattice vectors of the bottom layer and a is the lattice constant of monolayer graphene
Thus the exact superlattice period is then given by [18]
119871119871 = |11989911989911199381199381 + 11989911989921199381199382| = 11988611988611989911989912 + 11989911989922 + 11989911989911198991198992 = |1198991198991minus1198991198992|1198861198862 sin(1205791205792) (S4)
where θ is the angle between two lattice vectors 1199231199231 and 1199231199232 In the simulations below we always
rotated the supercell such that its lattice vector is 1199231199231 = 119871119871(10) and 1199231199232 = 11987111987112radic32 In this case
the corresponding reciprocal lattice vector of the moireacute superlattice satisfies the relation 119918119918119894119894 ∙ 119923119923119895119895 =
2120587120587120575120575119894119894119895119895 such that
1199181199181 = 4120587120587radic3119871119871
radic32
minus12 1199181199182 = 4120587120587
radic3119871119871(01) (S5)
S7
Both the lattice vectors and the corresponding reciprocal lattice vectors of the superlattice of the tBLG
are presented in Fig S5 Table S1 reports the parameters used to construct rhombus periodic
supercells of different misfit angles that can be duplicated to construct a rectangular periodic supercell
Table S1 The parameters used to construct periodic supercells of various misfit angles
120579120579 (deg) 119860119860 (nm2) 1198991198991 1198991198992 1198981198981 1198981198982
0 60383688 24 0 24 0
0696407 709613169 48 47 47 48
1121311 273718420 30 29 29 30
2000628 85648391 17 16 16 17
3006558 152322047 23 21 21 23
4048894 189013524 26 23 23 26
5085849 53255058 14 12 12 14
7926470 49376245 14 11 11 14
9998709 86277388 19 14 14 19
15178179 31554670 15 4 4 15
19932013 42876612 15 8 8 15
25039660 13942761 9 4 4 9
30158276 18974735 11 4 4 11
32204228 21805221 12 4 4 12
32 Special points for Brillouin Zone integration
The calculation of the sum over wave vector q in Eq (4) in the main text can be transformed to an
integral using the relation sum (⋯ )119954119954 = 1119881119881119887119887int (⋯ )BZ d119954119954 where 119881119881119887119887 = (2120587120587)3119881119881 is the volume of the
Brillouin Zone (BZ) and V is volume of the real-space unit-cell The calculation of integral is usually
inefficient since it requires calculating the value of the function over a large set of k points in the first
BZ To calculate such integrations more efficiently simple k-point meshes can be replaced by a
carefully selected set of special points in the BZ 119954119954119894119894 [19-22] over which the function is evaluated
The integral can then be estimated via
119868119868 = 1119881119881119887119887int 119891119891(119954119954)BZ d119954119954 asymp 1
119873119873sum 119908119908119894119894119891119891(119954119954119894119894)119894119894 (S6)
S8
where 119881119881119887119887 is the volume of the BZ 119908119908119894119894 is the weight of the ith data point and N normalizes the
weighting factors to unity 119873119873 = sum 119908119908119894119894119894119894 The set of selected 119954119954119894119894 forms a grid in the irreducible
Brillouin zone (IBZ) as is illustrated by the red points in Fig S5(b) The coordinates of these points
for a hexagonal lattice are presented in Eq (S7)
Fig S5 (a) Twisted bilayer graphene of misfit angle θ = 509deg L1 and L2 are the superlattice vectors (b) The corresponding first Brillouin Zone of (a) G1 and G2 are the reciprocal lattice vectors of the superlattice The triangle ΔΓMK represents the irreducible Brillouin zone Red circles mark the position of the special points used to evaluate the integral over the first Brillouin Zone
⎩⎪⎨
⎪⎧ 1199541199541 = 1
9 1205971205973 1199541199542 = 2
9 1205971205973 1199541199543 = 1
18 512059712059718
1199541199544 = 518
512059712059718 1199541199545 = 1
9 21205971205979 1199541199546 = 2
9 21205971205979
1199541199547 = 118
312059712059718 1199541199548 = 1
9 1205971205979 1199541199549 = 1
18 12059712059718
(S7)
Here 119905119905 = radic3 and the unit of the coordinates is 2120587120587119871119871 The weighting factors 119908119908119894119894 are [19]
119908119908124689 = 112
119908119908357 = 16 (S8)
Using Eqs (S6-S8) Eq (4) in the main text can be evaluated as follows
Γtot = 120587120587ℏ3
2sum 119908119908119896119896 sum
119890119890minus120573120573119864119864119954119954119948119948120582120582
119885119885
120588120588119864119864119954119954119948119948120582120582119881119881120582120582120582120582+31199031199032(119954119954119948119948)
2
1198641198641199541199541199481199481205821205822120582120582
9119896119896=1 (S9)
This equation was used to calculate the transition rate presented in Fig 3 in the main text
4 Derivation of Fermirsquos golden rule
The derivation of Fermirsquos golden rule is provided in a separate file (Fermirsquos Golden Rule for
phononspdf) due to its extent
S9
5 Temperature dependence of interfacial thermal conductivity
In the main text the target temperatures of the Langevin thermostats for the bottom and middle layers
of graphene and h-BN were set to 225 K and 375 K respectively After reaching the steady-state the
average temperature of the system was found to be ~300 K To check the effect of average temperature
on our results we calculated the cross-plane thermal conductivity (120581120581CP ) and the corresponding
interfacial thermal resistance (ITR) at a different temperature gradient (325 K ndash 475 K) resulting the
average steady-state temperature of ~400K The protocol described in Section 1 above was used to
perform these calculations as well Both ILP and Lennard-Jones (LJ) potential were tested for
graphite whereas for the bulk h-BN simulations only the ILP was used The results for graphite and
bulk h-BN are illustrated in Fig S6 and Fig S7 respectively For the ILP we find that the overall
values of 120581120581CP (ITR) decrease (increase) slightly with increasing average temperature which is
consistent with a recent experiment [10] The LJ potential calculations as well exhibit very week
dependence on average temperature within the range studied Altogether the layer dependences of
both quantities remain mostly insensitive to the average temperature The reason is that the thickness
of the graphite and h-BN model systems was chosen to be much smaller than their phonon mean free
path (~200 nm for graphite [110-13] and ~100 nm for bulk h-BN [14]) such that phonon transport is
dominated by phonon-boundary scattering and the Umklapp process only makes marginal
contribution [10]
S10
Fig S6 Layer dependence of 120581120581CP (a c) and ITR (b d) for Bernal-stacked graphite at average steady-state temperatures of 300 K (red circles) and 400 K (black squares) The left and right columns correspond to the 120581120581CP and ITR calculated with ILP and LJ potential respectively
Fig S7 Layer dependence of 120581120581CP (a) and ITR (b) for AArsquo-stacked h-BN at average temperatures of 300 K (red circles) and 400 K (blue squares)
S11
References [1] Z Wei J Yang W Chen K Bi D Li and Y Chen Phonon mean free path of graphite along the c-axis Appl Phys Lett 104 081903 (2014) [2] D W Brenner O A Shenderova J A Harrison S J Stuart B Ni and S B Sinnott A second-generation reactive empirical bond order (REBO) potential energy expression for hydrocarbons J Phys Condens Matter 14 783 (2002) [3] A Kınacı J B Haskins C Sevik and T Ccedilağın Thermal conductivity of BN-C nanostructures Phys Rev B 86 (2012) [4] W Ouyang I Azuri D Mandelli A Tkatchenko L Kronik M Urbakh and O Hod Mechanical and Tribological Properties of Layered Materials under High Pressure Assessing the Importance of Many-Body Dispersion Effects J Chem Theory Comput 16 666 (2020) [5] S Plimpton Fast Parallel Algorithms for Short-Range Molecular Dynamics J Comput Phys 117 1 (1995) [6] W Ouyang D Mandelli M Urbakh and O Hod Nanoserpents Graphene Nanoribbon Motion on Two-Dimensional Hexagonal Materials Nano Lett 18 6009 (2018) [7] W Shinoda M Shiga and M Mikami Rapid estimation of elastic constants by molecular dynamics simulation under constant stress Phys Rev B 69 134103 (2004) [8] G L Pollack Kapitza Resistance Rev Mod Phys 41 48 (1969) [9] H-S Yang G R Bai L J Thompson and J A Eastman Interfacial thermal resistance in nanocrystalline yttria-stabilized zirconia Acta Mater 50 2309 (2002) [10] Q Fu J Yang Y Chen D Li and D Xu Experimental evidence of very long intrinsic phonon mean free path along the c-axis of graphite Appl Phys Lett 106 031905 (2015) [11] J Chen J H Walther and P Koumoutsakos Strain Engineering of Kapitza Resistance in Few-Layer Graphene Nano Lett 14 819 (2014) [12] H Zhang X Chen Y-D Jho and A J Minnich Temperature-Dependent Mean Free Path Spectra of Thermal Phonons Along the c-Axis of Graphite Nano Lett 16 1643 (2016) [13] D P Sellan E S Landry J E Turney A J H McGaughey and C H Amon Size effects in molecular dynamics thermal conductivity predictions Phys Rev B 81 214305 (2010) [14] P Jiang X Qian R Yang and L Lindsay Anisotropic thermal transport in bulk hexagonal boron nitride Phys Rev Mater 2 064005 (2018) [15] J Zhang Y Hong and Y Yue Thermal transport across graphene and single layer hexagonal boron nitride J Appl Phys 117 134307 (2015) [16] B Liu F Meng C D Reddy J A Baimova N Srikanth S V Dmitriev and K Zhou Thermal transport in a graphenendashMoS2 bilayer heterostructure a molecular dynamics study RSC Advances 5 29193 (2015) [17] Z Ding Q-X Pei J-W Jiang W Huang and Y-W Zhang Interfacial thermal conductance in grapheneMoS2 heterostructures Carbon 96 888 (2016) [18] N N T Nam and M Koshino Lattice relaxation and energy band modulation in twisted bilayer graphene Phys Rev B 96 075311 (2017)
S12
[19] R Ramiacutearez and M C Boumlhm Simple geometric generation of special points in brillouin-zone integrations Two-dimensional bravais lattices Int J Quantum Chem 30 391 (1986) [20]S L Cunningham Special points in the two-dimensional Brillouin zone Phys Rev B 10 4988 (1974) [21] H J Monkhorst and J D Pack Special points for Brillouin-zone integrations Phys Rev B 13 5188 (1976) [22] D J Chadi Special points for Brillouin-zone integrations Phys Rev B 16 1746 (1977)
1
Fermirsquos golden rule for phonons
1 Basic theory for phonons 11 Basic notations
Let us consider a 3D crystal with a total of 119873119873 = 119873119873111987311987321198731198733 unit cells and periodic boundary conditions
To be specific let 119938119938119894119894 119894119894 = 123 be the lattice vectors that define the unit cell We index unit cells
with n = (n1n2n3) where each ni = 12 ∙∙∙ Ni and their locations are 119929119929119899119899 = sum 1198991198991198941198941199381199381198941198943119894119894=1 Assume that
there are r atoms in each unit cell which are indexed with 119904119904 = 1⋯ 119903119903 The mass and the equilibrium
distance of the sth atom are notated as 119872119872s and 119929119929s0 respectively Then the location of the sth atom in
the nth unit cell at time t can be expressed as
119955119955119899119899119899119899(119905119905) = 119929119929119899119899 + 119929119929s0 + 119958119958119899119899119899119899(119905119905) (11)
where 119958119958119899119899119899119899(119905119905) is its displacement from its equilibrium position
The Lagrangian for this classical problem can be written as
ℒ = sum sum 119872119872119904119904|119955119899119899119904119904|2(119905119905)2
119903119903119899119899=1
119873119873119899119899=1 minus 119881119881 (12)
where the second term is the sum of interactions between all pairs of atoms
Under the harmonic approximation ie expanding the total potential energy 119881119881 around the
equilibrium positions The Lagrangian can be simplified as
ℒ = sum sum 119872119872119904119904|119958119899119899119904119904|2(119905119905)2
119903119903119899119899=1
119873119873119899119899=1 minus 1
2sum sum sum 120601120601120572120572120572120572prime
119899119899119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime120572120572120572120572prime119899119899119899119899prime119899119899119899119899prime (13)
where 119906119906119899119899119899119899120572120572 120572120572 = 123 are the Cartesian coordinates of the displacement 119958119958119899119899119899119899(119905119905) and
120601120601120572120572120572120572prime 119899119899119899119899prime119904119904 119904119904prime = 1205971205972119881119881
120597120597119903119903119899119899119904119904119899119899119903119903119899119899prime119904119904prime119899119899primeeq
= 120601120601120572120572prime120572120572 119899119899prime119899119899119904119904prime 119904119904 = 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime (14)
Note that the first order term vanishes because we are expanding around the equilibrium positions
120601120601120572120572120572120572prime 119899119899119899119899prime119904119904 119904119904prime represents the component of the force acted on the sth atom in the nth unit cell along 120572120572
direction when the atom 119904119904prime in the unit cell 119899119899prime moves a unit displacement along 120572120572prime direction The
symmetries of 120601120601120572120572120572120572prime 119899119899 119899119899prime119904119904 119904119904prime appearing in Eq (14) arise from the intechangability of the second
derivative and the translational invariance of the interactions
12 Dynamical matrix
The equation of motion of the sth atom in the nth unit cell can be derived using the EulerndashLagrange
equation as follows
2
119872119872119899119899119906119899119899119899119899120572120572 = minussum 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119899119899prime119899119899prime120572120572prime 119906119906119899119899prime119899119899prime120572120572prime (15)
If we displace all atoms equally ie shifting 119906119906119899119899prime119899119899prime120572120572prime to 119906119906119899119899prime119899119899prime120572120572prime + 120575120575 the total force on the sth atom in
the nth unit cell does not change From the above equation we have
sum 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119899119899prime120572120572prime = 0 (16)
We are looking for normal modes (because any general solution can be written as a linear combination
of them) these are solutions where all atoms oscillate with the same frequency Moreover because
of the lattice structure we expect solutions to reflect this periodicity So we guess solutions of the
form
119906119906119899119899119899119899120572120572(119905119905) = 1119872119872119904119904
119890119890119899119899120572120572119890119890119894119894(119954119954∙119929119929119899119899minus120596120596119905119905) (17)
where 119890119890119899119899120572120572 are real-space solutions that will be determined later and 119954119954 is the wave-vector in
reciprocal space Substituting Eq (17) into Eq (15) we can get
1205961205962119890119890119899119899120572120572(119954119954) = sum 119863119863119899119899120572120572119899119899prime120572120572prime(119954119954)119890119890119899119899prime120572120572prime(119954119954)119899119899prime120572120572prime (18)
where
119863119863119899119899120572120572119899119899prime120572120572prime(119954119954) = sum 1
119872119872119904119904119872119872119904119904prime120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119890119890
minus119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime119899119899prime = sum 1
119872119872119904119904119872119872119904119904prime120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime 119890119890
minus119894119894119954119954∙119929119929119897119897119897119897 (19)
is called dynamical matrix (dimension 3119903119903 times 3119903119903) Note that we have defined the relative cell distance
vector 119929119929119897119897 equiv 119929119929119899119899 minus 119929119929119899119899prime and number index 119897119897 equiv 119899119899 minus 119899119899prime where the infinite sum over 119899119899prime can be
replaced by the sum over 119897119897 for any value of the index 119899119899 Note that dynamical matrix is Hermitian
symmetric (ie 119863119863119899119899120572120572119899119899prime120572120572prime(119954119954)
lowast= 119863119863119899119899prime120572120572prime
119899119899120572120572 (minus119954119954)) because ϕ is symmetric so all the eigenvalues of Eq (18)
(120596120596120582120582(119954119954) 120582120582 = 12⋯ 3119903119903) are real for each 119954119954 in the Brillouin zone (BZ) which is determined by
det1205961205961205821205822(119954119954)120575120575119899119899120572120572120575120575119899119899prime120572120572prime minus 119863119863119899119899120572120572119899119899prime120572120572prime(119954119954) = 0 (110)
Taking the conjugate of this equation we have
0 = det 1205961205961205821205822(119954119954)lowast120575120575119899119899120572120572120575120575119899119899prime120572120572prime minus 119863119863119899119899120572120572119899119899
prime120572120572prime(119954119954)lowast = det1205961205961205821205822(119954119954)120575120575119899119899120572120572120575120575119899119899prime120572120572prime minus 119863119863119899119899120572120572119899119899
prime120572120572prime(minus119954119954) (111)
while replacing 119954119954 by minus119954119954 in Eq (110) we have det1205961205961205821205822(minus119954119954)120575120575119899119899120572120572120575120575119899119899prime120572120572prime minus 119863119863119899119899120572120572119899119899prime120572120572prime(minus119954119954) = 0
Itrsquos clear that 1205961205961205821205822(119954119954) and 1205961205961205821205822(minus119954119954) obey the same equation thus we have
120596120596120582120582(minus119954119954) = 120596120596120582120582(119954119954) (112)
The corresponding eigenvectors are orthonormal
sum 119890119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582
prime119899119899120572120572 = 120575120575120582120582120582120582prime (113)
The complex conjugate of Eq (18) gives
1205961205962119890119890119899119899120572120572lowast (119954119954) = sum 119863119863119899119899120572120572119899119899prime120572120572prime(119954119954)
lowast119890119890119899119899prime120572120572primelowast (119954119954)119899119899prime120572120572prime = sum 119863119863119899119899prime120572120572prime
119899119899120572120572 (minus119954119954)119890119890119899119899prime120572120572primelowast (119954119954)119899119899prime120572120572prime (114)
3
While replacing 119954119954 by minus119954119954 in Eq (18) we get the following equation
1205961205962119890119890119899119899120572120572(minus119954119954) = sum 119863119863119899119899120572120572119899119899prime120572120572prime(minus119954119954)119890119890119899119899prime120572120572prime(minus119954119954)119899119899prime120572120572prime (115)
From Eq (114) and Eq (115) we see that eigenvectors 119890119890119899119899120572120572120582120582 (minus119954119954)lowast and 119890119890119899119899120572120572120582120582 (119954119954) obey the same
eigenvalue equation Since the eigenvectors are normalized we get the following property
119890119890119899119899120572120572120582120582 (minus119954119954)lowast
= 119890119890119899119899120572120572120582120582 (119954119954) (116)
2 Second quantization The general solution is a linear combination of all these normal modes thus we have
119906119906119899119899119899119899120572120572(119905119905) = sum 119862119862120582120582(119954119954)119873119873119872119872119904119904
119890119890119899119899120572120572120582120582 (119954119954)119890119890minus119894119894120596120596119954119954120582120582119905119905119890119890119894119894119954119954∙119929119929119899119899119954119954120582120582 = sum 119876119876120582120582(119954119954119905119905)119873119873119872119872119904119904
119890119890119899119899120572120572120582120582 (119954119954)119890119890119894119894119954119954∙119929119929119899119899119954119954120582120582 (21)
where the we define the normal coordinates as 119876119876120582120582(119954119954 119905119905) equiv 119862119862120582120582(119954119954)119890119890minus119894119894120596120596120582120582(119954119954)119905119905 in the eigenvectors
representation To ensure that the displacements 119906119906119899119899119899119899120572120572(119905119905) are real (namely 119906119906119899119899119899119899120572120572lowast (119905119905) = 119906119906119899119899119899119899120572120572(119905119905)) the
following relation on 119876119876120582120582(119954119954 119905119905) and 119890119890119899119899120572120572120582120582 is enforced
119876119876120582120582(119954119954 119905119905)119890119890119899119899120572120572120582120582 (119954119954)lowast
= 119876119876120582120582(minus119954119954 119905119905)119890119890119899119899120572120572120582120582 (minus119954119954) (22)
where we used the fact that the sum over 119954119954 runs symmetrically over both negative and positive
values Using Eq (116) we have
[119876119876120582120582(119954119954 119905119905)]lowast = 119876119876120582120582(minus119954119954 119905119905) (23)
21 Kinetic energy term
Using Eq (21) the kinetic energy of the system can be expressed as
119879119879 = 12sum 1198721198721198991198991199061198991198991198991198991205721205722 (119905119905)119899119899119899119899120572120572 = 1
2sum 119872119872119904119904
119873119873119872119872119904119904sum sum 119876120582120582(119954119954 119905119905)119876120582120582prime(119954119954prime 119905119905)119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582
prime(119954119954prime) sum 119890119890119894119894119954119954+119954119954prime∙119929119929119899119899119899119899119954119954prime120582120582prime119954119954120582120582119899119899120572120572 =
12sum sum 119876120582120582(119954119954 119905119905)119876120582120582prime(minus119954119954 119905119905)119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582
prime(minus119954119954)119954119954119954119954prime120582120582prime =119899119899120572120572
12sum 119876120582120582(119954119954 119905119905)119876120582120582prime
lowast (119954119954 119905119905) sum 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582prime(119954119954)
lowast119899119899120572120572 =119954119954120582120582120582120582prime
12sum 119876120582120582(119954119954 119905119905)119876120582120582lowast(119954119954 119905119905)119954119954120582120582 =
12sum 119876120582120582(119954119954 119905119905)119876120582120582(minus119954119954 119905119905)119954119954120582120582
ie
119879119879 = 12sum 119876120582120582(119954119954 119905119905)119876120582120582(minus119954119954 119905119905)119954119954120582120582 (24)
To derive Eq (24) we used Eqs (113) and (116) as well as the following equations
1119873119873sum 119890119890119894119894119954119954+119954119954prime∙119929119929119899119899119899119899 = 120575120575119954119954+119954119954prime120782120782 (25)
4
22 Potential energy term
Similarity the potential energy can be rewritten in terms of the normal coordinates as follows
119880119880 =12120601120601119899119899119899119899120572120572119899119899prime119899119899prime120572120572prime119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
120572120572120572120572prime119899119899119899119899prime119899119899119899119899prime=
12120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119899119899+119954119954prime∙119929119929119899119899prime
119873119873119872119872119899119899119872119872119899119899prime
[119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(119954119954prime 119905119905)] 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime120582120582prime (119954119954prime)
119954119954prime120582120582prime119954119954120582120582119899119899119899119899prime120572120572120572120572prime119899119899119899119899prime
=12120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894(119954119954+119954119954prime)∙119929119929119899119899119890119890minus119894119894119954119954prime∙119929119929119897119897
119873119873119872119872119899119899119872119872119899119899prime
[119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(119954119954prime 119905119905)] 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime120582120582prime (119954119954prime)
119954119954prime120582120582prime119954119954120582120582119899119899119897119897120572120572120572120572prime119899119899119899119899prime
=12120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890minus119894119894119954119954prime∙119929119929119897119897
119872119872119899119899119872119872119899119899prime[119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(119954119954prime 119905119905)] 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (119954119954prime)1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899
119899119899=120575120575119902119902+119902119902prime=0
119954119954prime120582120582prime119954119954120582120582119897119897120572120572120572120572prime119899119899119899119899prime
=12120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894119954119954119929119929119897119897
119872119872119899119899119872119872119899119899prime [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)] 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (minus119954119954)119954119954120582120582120582120582prime119897119897120572120572120572120572prime119899119899119899119899prime
=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]119890119890119899119899120572120572120582120582 (119954119954)
119899119899120572120572
119863119863119899119899120572120572119899119899prime120572120572prime(minus119954119954)119890119890119899119899prime120572120572prime
120582120582prime (minus119954119954)119899119899prime120572120572prime119954119954120582120582120582120582prime
=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]119890119890119899119899120572120572120582120582 (119954119954)
119899119899120572120572
1205961205961205821205822(minus119954119954)119890119890119899119899120572120572120582120582prime (minus119954119954)
119954119954120582120582120582120582prime
=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]1205961205961205821205822(119954119954)120575120575120582120582120582120582prime =119954119954120582120582120582120582prime
121205961205961205821205822(119954119954)119876119876120582120582(119954119954 119905119905)119876119876120582120582(minus119954119954 119905119905)
119954119954120582120582
ie
119880119880 = 12sum 1205961205961205821205822(119954119954)119876119876120582120582(119954119954 119905119905)119876119876120582120582(minus119954119954 119905119905)119954119954120582120582 (26)
where in above deriviation the orthogonality of its eigenvectores and the symmetry of its eigenvalues
are used Thus the Lagrangian reads as
ℒ = 119879119879 minus 119881119881 = 12sum 119876120582120582(minus119954119954 119905119905)119876120582120582(119954119954 119905119905) minus 120596120596120582120582
2(119954119954)119876119876120582120582(minus119954119954 119905119905)119876119876120582120582(119954119954 119905119905)119954119954120582120582 (27)
Using the relation 119875119875120582120582(119954119954 119905119905) = 120597120597ℒ part119876120582120582(119954119954 119905119905) the Hamiltonian of the system then can be written as
119867119867 = 119879119879 + 119881119881 = 12sum [119875119875120582120582(minus119954119954 119905119905)119875119875120582120582(119954119954 119905119905) + 120596120596120582120582
2(119954119954)119876119876120582120582(minus119954119954 119905119905)119876119876120582120582(119954119954 119905119905)]119954119954120582120582 (28)
Now we quantize H by asking the momenta and coordinates to be operators
119867119867 = 12sum 119875119875minus119954119954120582120582119875119875119954119954120582120582 + 120596120596119954119954120582120582
2 119876119876minus119954119954120582120582119876119876119954119954120582120582119954119954120582120582 (29)
here 119875119875119954119954120582120582 and 119876119876119954119954120582120582 obey the following commutation relations
119876119876119954119954120582120582119875119875119954119954prime120582120582prime = 119894119894ℏ120575120575119954119954119954119954prime120575120575120582120582120582120582prime
119876119876119954119954120582120582119876119876119954119954prime120582120582prime = 0 119875119875119954119954120582120582119875119875119954119954prime120582120582prime = 0 (210)
Similar to the case of ordinary quantum harmonic oscillators it is convenient to define ladder
operators for each mode as follows
119876119876119954119954120582120582 = ℏ
2120596120596119954119954120582120582119887119887119954119954120582120582 + 119887119887minus119954119954120582120582
dagger 119875119875119954119954120582120582 = 119894119894ℏ120596120596119954119954120582120582
2119887119887119954119954120582120582
dagger minus 119887119887minus119954119954120582120582 (211)
where 119887119887119954119954120582120582dagger and 119887119887119954119954120582120582 are Bosonic creation and annihilation operators for phonons with momentum
119954119954 branch index 120582120582 and frequency 120596120596119954119954120582120582 which obeys the Bosonic commutation relation
5
119887119887119954119954120582120582 119887119887119954119954prime120582120582prime
dagger = 120575120575119954119954119954119954prime120575120575120582120582120582120582prime
119887119887119954119954120582120582 119887119887119954119954prime120582120582prime = 0 119887119887119954119954120582120582dagger 119887119887119954119954prime120582120582prime
dagger = 0 (212)
Substituting Eqs (211) into (29) and using the properties Eq (210) and (212) we have
119867119867 =12119875119875minus119954119954120582120582119875119875119954119954120582120582 +120596120596119954119954120582120582
2 119876119876minus119954119954120582120582119876119876119954119954120582120582119954119954120582120582
=12minus
ℏ120596120596119954119954120582120582
2119887119887minus119954119954120582120582
dagger minus 119887119887119954119954120582120582119887119887119954119954120582120582dagger minus 119887119887minus119954119954120582120582+ 120596120596119954119954120582120582
2 ℏ2120596120596119954119954120582120582
119887119887minus119954119954120582120582 + 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 + 119887119887minus119954119954120582120582
dagger 119954119954120582120582
=ℏ4120596120596119954119954120582120582minus119887119887minus119954119954120582120582
dagger 119887119887119954119954120582120582dagger minus 119887119887minus119954119954120582120582
dagger 119887119887minus119954119954120582120582 minus 119887119887119954119954120582120582119887119887119954119954120582120582dagger + 119887119887119954119954120582120582119887119887minus119954119954120582120582
119954119954120582120582
+ 119887119887minus119954119954120582120582119887119887119954119954120582120582 + 119887119887minus119954119954120582120582119887119887minus119954119954120582120582dagger + 119887119887119954119954120582120582
dagger 119887119887119954119954120582120582 + 119887119887119954119954120582120582dagger 119887119887minus119954119954120582120582
dagger
=ℏ4120596120596119954119954120582120582119887119887minus119954119954120582120582
dagger 119887119887minus119954119954120582120582 + 119887119887minus119954119954120582120582119887119887minus119954119954120582120582dagger + 119887119887119954119954120582120582119887119887119954119954120582120582
dagger + 119887119887119954119954120582120582dagger 119887119887119954119954120582120582
119954119954120582120582
=ℏ41205961205961199541199541205821205822119887119887minus119954119954120582120582
dagger 119887119887minus119954119954120582120582 + 1+ 2119887119887119954119954120582120582dagger 119887119887119954119954120582120582 + 1
119954119954120582120582
= ℏ120596120596119954119954120582120582 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 +
12
119954119954120582120582
Therefore we finally get the quantized representation of non-interacting phonons
119867119867 = sum ℏ120596120596119954119954120582120582 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 + 1
2119954119954120582120582 (213)
The operator of atom displacements (Eq 114) is expressed in terms of the phonon operators by
119906119906119899119899119899119899120572120572 = sum ℏ
2119873119873119872119872119904119904120596120596119954119954120582120582119890119890119899119899120572120572120582120582 119890119890119894119894119954119954∙119929119929119899119899119887119887119954119954120582120582 + 119887119887minus119954119954120582120582
dagger 119954119954120582120582 (214)
These equations will be used below
3 Inter-phonon coupling within harmonic approximation 31 Hamiltonian with inter-phonon coupling
In this section we consider systems that consist of two (or more) covalently bonded units that are
weakly coupled between them Unlike previous studies that considered phonon-phonon couplings
resulting from anharmonicity effects [1] here all phonon mode considered are Harmonic and the
couplings arise from the division of the entire system into subunits The Hamiltonian of the whole
system can be written as
119867119867 = 1198671198671 + 1198671198672 + 11986711986712 (31)
To derive the expressions of 1198671198671 1198671198672 and 11986711986712 in Eq (31) we consider the Hamiltonian of the
whole system written as the function of the atomic displacements
119867119867 = 119879119879 + 119881119881 = sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)2119899119899119899119899120572120572 + 1
2sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime120572120572120572120572prime119899119899119899119899prime119899119899119899119899prime (32)
6
Letrsquos assume that subsystem I and II contain atoms with indeices ranging from 119904119904 = 12⋯ 1199031199032 and
119904119904 = 1199031199032
+ 1 1199031199032
+ 2⋯ 119903119903 respectively Then the kinetic energy term in Eq (32) can be rewritten as
119879119879 = sum sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)2
1199031199032119899119899=1 + sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)
2119903119903119899119899=1199031199032+1
119899119899120572120572 (33)
While the potential energy term is
119881119881 =12 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
1199031199032
119899119899119899119899prime=1
+ 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
119903119903
119899119899119899119899prime=1199031199032+1
+ 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime119899119899119899119899prime
+ 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
1199031199032
119899119899prime=1
119903119903
119899119899=1199031199032+1
= 11988111988111 + 11988111988122 + 11988111988112 + 11988111988121
Or equivalently
⎩⎪⎪⎨
⎪⎪⎧ 11988111988111 = 1
2sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
1199031199032119899119899119899119899prime=1120572120572120572120572prime119899119899119899119899prime
11988111988122 = 12sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
119903119903119899119899119899119899prime=1199031199032+1120572120572120572120572prime119899119899119899119899prime
11988111988112 = 12sum sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
119903119903119899119899prime=1199031199032+1
1199031199032119899119899=1120572120572120572120572prime119899119899119899119899prime
11988111988121 = 12sum sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
1199031199032119899119899prime=1
119903119903119899119899=1199031199032+1120572120572120572120572prime119899119899119899119899prime
(34)
32 Second quantization
We may now quantize this Hamiltonian and write it in the basis of the eigenstates of the coupled
subunits In second quantization the atomic displacement operators of the two subunits are given in
the following form
119906119906119899119899119899119899120572120572 =
⎩⎨
⎧ sum ℏ
2119873119873119872119872119904119904ω119954119954120582120582119890119890119899119899120572120572120582120582 119890119890119894119894119954119954∙119929119929119899119899119886119886119954119954120582120582 + 119886119886minus119954119954120582120582
dagger 119954119954120582120582 119904119904 isin 1 1199031199032
sum ℏ
2119873119873119872119872119904119904120596120596119954119954prime120582120582prime119890119899119899120572120572120582120582
prime119890119890119894119894119954119954prime∙119929119929119899119899 119886119886 119954119954prime120582120582prime + 119886119886minus119954119954prime120582120582primedagger 119954119954prime120582120582prime 119904119904 isin 119903119903
2+ 1 119903119903
(35)
Here we use different notations for the creation and annihilation operators (119886119886119954119954120582120582119886119886minus119954119954120582120582dagger and 119886119886119954119954120582120582119886119886minus119954119954120582120582
dagger )
eigenvalues (ω119954119954120582120582120596120596119954119954120582120582) and eigenvectors (119890119890119899119899120572120572120582120582 119890119899119899120572120572120582120582 ) for the two subunits Note that 119890119890119899119899120572120572120582120582 and 119890119899119899120572120572120582120582 are
of the dimensions of the whole system however their non-zero elements appear only on the relevant
subunits such that they obey the following relations sum 119890119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582
prime119899119899120572120572 = 120575120575120582120582120582120582prime sum 119890119899119899120572120572120582120582
lowast119890119899119899120572120572120582120582
prime119899119899120572120572 =
120575120575120582120582120582120582prime sum 119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582
prime119899119899120572120572 = sum 119890119890119899119899120572120572120582120582
lowast119890119899119899120572120572120582120582
prime119899119899120572120572 = 0 Defining the normal coordinates of the two subunits as
119876119876119954119954120582120582 equiv ℏ
2ω119954119954120582120582119886119886119954119954120582120582 + 119886119886minus119954119954120582120582
dagger 119876119876119954119954120582120582 equiv ℏ
2ω119954119954120582120582119886119886119954119954120582120582 + 119886119886minus119954119954120582120582
dagger (36)
and the corresponding momenta operators as
7
119875119875119954119954120582120582 = 119894119894ℏω119954119954120582120582
2119886119886119954119954120582120582
dagger minus 119886119886minus119954119954120582120582 119875119875119954119954120582120582 = 119894119894ℏω1199541199541205821205822
119886119886119954119954120582120582dagger minus 119886119886minus119954119954120582120582 (37)
Eq (35) reads
119906119906119899119899119899119899120572120572 = sum 119890119890119904119904119899119899120582120582 (119954119954)
119873119873119872119872119904119904119890119890119894119894119954119954∙119929119929119899119899119876119876119954119954120582120582119954119954120582120582 119904119904 isin 1 119903119903
2
sum 119890119904119904119899119899120582120582 (119954119954)119873119873119872119872119904119904
119890119890119894119894119954119954∙119929119929119899119899119876119876119954119954120582120582119954119954120582120582 119904119904 isin 1199031199032
+ 1 119903119903 (38)
The second quantized kinetic energy operator is then written as
119879119879 = 1198791198791 + 1198791198792 = 12sum 119875119875119954119954120582120582119875119875minus119954119954120582120582119954119954120582120582 + 1
2sum 119875119875119954119954prime120582120582prime119875119875minus119954119954prime120582120582prime119954119954prime120582120582prime (39)
Correspondingly the various potential energy terms are obtained by substituting Eq (38) in Eq (34)
11988111988111 =12 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119899119899119899119899prime
119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime120582120582prime (119954119954prime)
119872119872119899119899119872119872119899119899prime
1119873119873119890119890119894119894119954119954∙119929119929119899119899+119894119894119954119954prime∙119929119929119899119899prime119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954120582120582119954119954prime120582120582prime
1199031199032
119899119899119899119899prime=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954119954119954prime120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (119954119954prime)1119873119873 119890119890
119894119894(119954119954+119954119954prime)119929119929119899119899prime
119899119899prime120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime
119872119872119899119899119872119872119899119899prime119899119899
1199031199032
119899119899119899119899prime=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954119954119954prime120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (119954119954prime) 120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime
1199031199032
119899119899119899119899prime=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime
119954119954120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (minus119954119954)120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
1199031199032
119899119899119899119899prime=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime
119954119954120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954) 119863119863119899119899120572120572119899119899
prime120572120572prime(minus119954119954)119890119890119899119899prime120572120572prime120582120582prime (minus119954119954)
120572120572prime
1199031199032
119899119899prime=1
1199031199032
119899119899=1120572120572
=12 ω119954119954120582120582prime
2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582120582120582prime
119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582prime (119954119954)
lowast
1199031199032
119899119899=1
=120572120572
12ω119954119954120582120582
2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582
ie
11988111988111 = 12sum ω119954119954120582120582
2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582 (310)
Going from the second line to the third line we used the fact that the sum over 119899119899prime runs between plusmninfin
and the summand depends only on the difference between 119899119899 and 119899119899prime hence the sum is independent
of the value of the index 119899119899 Therefore we can replace the sum over 119899119899prime by a sum over 119897119897 equiv 119899119899 minus 119899119899prime
amd define 119929119929119897119897 equiv 119929119929119899119899 minus 119929119929119899119899prime Following the same procedure we can get the corresponding
expressions for the second diagonal term
11988111988122 = 12sum 1205961205961199541199541205821205822 119876119876119954119954120582120582119876119876minus119954119954120582120582119954119954120582120582 (311)
Similarly for the off-diagonal terms we get
8
11988111988112 =12 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119899119899119899119899prime
119903119903
119899119899prime=1199031199032+1
119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime
120582120582prime (119954119954prime)
119872119872119899119899119872119872119899119899prime
1119873119873119890119890119894119894119954119954∙119929119929119899119899+119894119894119954119954prime∙119929119929119899119899prime119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954120582120582119954119954prime120582120582prime
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954119954119954prime120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime
120582120582prime (119954119954prime)1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime
119872119872119899119899119872119872119899119899prime119899119899
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954119954119954prime120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime
120582120582prime (119954119954prime) 120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime
119954119954120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime
120582120582prime (minus119954119954)120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954
119890119899119899prime120572120572prime120582120582prime (119954119954)
lowast120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119890119890119899119899120572120572120582120582 (119954119954)119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954
Where we have defined
119881119881120582120582120582120582prime(119954119954) equiv sum sum sum 119890119899119899prime120572120572prime120582120582prime (119954119954)
lowast119881119881119899119899120572120572119899119899
prime120572120572prime(119954119954)119890119890119899119899120572120572120582120582 (119954119954)119903119903119899119899prime=1199031199032+1
1199031199032119899119899=1120572120572120572120572prime (312)
where
119881119881119899119899120572120572119899119899prime120572120572prime(119954119954) equiv sum 119890119890119894119894119954119954∙119929119929119897119897
119872119872119904119904119872119872119904119904prime119897119897 120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime (313)
Itrsquos easy to show that 119881119881120582120582120582120582prime(119954119954) has the following property
119881119881120582120582120582120582primelowast (119954119954) = 119881119881120582120582120582120582prime(minus119954119954) (314)
Following the same procedure we have
9
11988111988121 =12 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119899119899119899119899prime
119903119903
119899119899=1199031199032+1
119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (119954119954prime)
119872119872119899119899119872119872119899119899prime
1119873119873119890119890119894119894119954119954∙119929119929119899119899+119894119894119954119954prime∙119929119929119899119899prime119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954120582120582119954119954prime120582120582prime
1199031199032
119899119899prime=1120572120572120572120572prime
=12 120601120601120572120572prime120572120572
119899119899prime minus 119899119899119904119904prime 119904119904
119899119899prime119899119899
119903119903
119899119899prime=1199031199032+1
119890119899119899prime120572120572prime120582120582prime (119954119954prime)119890119890119899119899120572120572120582120582 (119954119954)
119872119872119899119899prime119872119872119899119899
1119873119873 119890119890
119894119894119954119954prime∙119929119929119899119899prime+119894119894119954119954∙119929119929119899119899119876119876119954119954prime120582120582prime119876119876119954119954120582120582119954119954prime120582120582prime119954119954120582120582
1199031199032
119899119899=1120572120572prime120572120572
=12 119876119876119954119954prime120582120582prime119876119876119954119954120582120582
119954119954119954119954prime120582120582120582120582prime119890119899119899prime120572120572prime120582120582prime (119954119954prime)119890119890119899119899120572120572120582120582 (119954119954)
1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime
119872119872119899119899119872119872119899119899prime119899119899
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954prime120582120582prime119876119876119954119954120582120582
119954119954119954119954prime120582120582120582120582prime119890119899119899prime120572120572prime120582120582prime (119954119954prime)119890119890119899119899120572120572120582120582 (119954119954) 120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954prime120582120582prime119876119876minus119954119954prime120582120582119890119899119899prime120572120572prime
120582120582prime (119954119954prime)119890119890119899119899120572120572120582120582 (minus119954119954prime)120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890minus119894119894119954119954prime∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897119954119954prime120582120582120582120582prime
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954prime120582120582prime119876119876119954119954prime120582120582
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954prime 119890119890119899119899120572120572120582120582 (119954119954prime)
lowast120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890minus119894119894119954119954prime∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119890119899119899prime120572120572prime120582120582prime (119954119954prime)
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582prime119876119876119954119954120582120582
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954
119890119890119899119899120572120572120582120582 (119954119954)lowast120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890minus119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119890119899119899prime120572120572prime120582120582prime (119954119954)
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger dagger
3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954
119890119899119899prime120572120572prime120582120582prime (119954119954)
lowast120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119890119890119899119899120572120572120582120582 (119954119954)119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
lowast
=
⎩⎨
⎧12 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954⎭⎬
⎫dagger
= 11988111988112dagger
Define
1198671198671 = 11987911987911 + 119881119881111198671198672 = 11987911987922 + 1198811198812211986711986712 = 11988111988112 + 11988111988121
(315)
we finally get the expressions of 1198671198671 1198671198672 and 11986711986712 in Eq (31) as follows
⎩⎪⎨
⎪⎧ 1198671198671 = sum sum ℏ120596120596119954119954120582120582 119886119886119954119954120582120582
dagger 119886119886119954119954120582120582 + 1231199031199032
120582120582=1119954119954
1198671198672 = sum sum ℏ120596120596119954119954120582120582prime 119886119886119954119954120582120582primedagger 119886119886119954119954120582120582prime + 1
23119903119903
120582120582prime=1+31199031199032119954119954
11986711986712 = 12sum sum sum 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903120582120582prime=31199031199032 +1
31199031199032120582120582=1 + h c 119954119954
(316)
Here hc means the Hermitian conjugate Since the indices of 119886119886119954119954120582120582 119886119886119954119954120582120582prime and 119876119876119954119954120582120582 119876119876119954119954120582120582prime belong to
the two system sections we can define an abbreviated notation 119886119886119954119954120582120582 and 119876119876119954119954120582120582 using the index 120582120582 to
identify which subsystem they belong to In this case the Hamiltonian of the coupled system can be
10
simplified as follows
119867119867 = 1198671198670 + 119867119867119862119862 (317)
where
1198671198670 = sum sum ℏ120596120596119954119954120582120582 119886119886119954119954120582120582
dagger 119886119886119954119954120582120582 + 123119903119903
120582120582=1119954119954
119867119867119862119862 = 12sum sum sum 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903120582120582prime=31199031199032 +1
31199031199032120582120582=1 + ℎ 119888119888 119954119954
(318)
4 Greenrsquos function 41 Dynamics of the ladder operators
In order to describe the dynamics of the ladder operators appearing in the Hamiltonian of Eq (317)
we express them in the Heisenberg picture as follows
119886119886119901119901(120591120591) = 119890119890119894119894ℏ119867119867119905119905119886119886119901119901119890119890
minus119894119894ℏ119867119867119905119905 = 119890119890
119867119867120591120591ℏ 119886119886119901119901119890119890
minus119867119867120591120591ℏ (41)
where we define the imaginary time 120591120591 equiv 119894119894119905119905 and introduce the notation 119953119953 equiv (119954119954 120582120582) and 119953119953 equiv (minus119954119954 120582120582)
The corresponding equation of motion for the ladder operators is given by
ℏ part119886119886119953119953(120591120591)
120597120597120591120591= 119867119867119886119886119953119953(120591120591) (42)
For the uncoupled system (119867119867119862119862 = 0) this gives
ℏpart119886119886119953119953(120591120591)120597120597120591120591
= 1198671198670119886119886119953119953(120591120591) = 1198671198670 1198901198901198671198670120591120591ℏ 119886119886119953119953119890119890
minus1198671198670120591120591ℏ = 1198671198670119890119890
1198671198670120591120591ℏ 119886119886119953119953119890119890
minus1198671198670120591120591ℏ minus 119890119890
1198671198670120591120591ℏ 119886119886119953119953119890119890
minus1198671198670120591120591ℏ 1198671198670
= 1198901198901198671198670120591120591ℏ 1198671198670119886119886119953119953 minus 1198861198861199531199531198671198670119890119890
minus1198671198670120591120591ℏ = 119890119890
1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890
minus1198671198670120591120591ℏ
ie
ℏ part119886119886119953119953(120591120591)
120597120597120591120591= 119890119890
1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890
minus1198671198670120591120591ℏ (43)
The commutator on the right-hand-side of Eq (43) can be evaluated as follows
1198671198670119886119886119953119953prime = sum ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953 + 1
2119953119953 119886119886119953119953prime = sum ℏ120596120596119953119953119886119886119953119953
dagger119886119886119953119953119886119886119953119953prime119953119953 = sum ℏ120596120596119953119953119886119886119953119953dagger119886119886119953119953119886119886119953119953prime minus 119886119886119953119953prime119886119886119953119953
dagger119886119886119953119953119953119953 =
sum ℏ120596120596119953119953119886119886119953119953dagger119886119886119953119953119886119886119953119953prime minus 120575120575119953119953prime119953119953 + 119886119886119953119953
dagger119886119886119953119953prime119886119886119953119953119953119953 = minusℏ120596120596119953119953prime119886119886119953119953prime + sum ℏ120596120596119953119953119886119886119953119953dagger119886119886119953119953119886119886119953119953prime minus 119886119886119953119953
dagger119886119886119953119953119886119886119953119953prime119953119953 = minusℏ120596120596119953119953prime119886119886119953119953prime
ie
1198671198670119886119886119953119953prime = minusℏ120596120596119953119953prime119886119886119953119953prime (44)
Therefore we have
ℏ part119886119886119953119953(120591120591)
120597120597120591120591= 119890119890
1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890
minus1198671198670120591120591ℏ = minusℏ120596120596119953119953119890119890
1198671198670120591120591ℏ 119886119886119953119953119890119890
minus1198671198670120591120591ℏ = minusℏ120596120596119953119953119886119886119953119953(120591120591) (45)
11
The solution of Eq (45) is 119886119886119953119953(120591120591) = 119886119886119953119953(0)119890119890minus120596120596119953119953120591120591 = 119886119886119953119953119890119890minus120596120596119953119953120591120591 For 119886119886119953119953dagger(120591120591) we have
1198671198670119886119886119953119953primedagger (0) = 1198671198670119886119886119953119953prime
dagger = ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953 +
12
119953119953
119886119886119953119953primedagger = ℏ120596120596119953119953119886119886119953119953
dagger119886119886119953119953119886119886119953119953primedagger
119953119953
= ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953119886119886119953119953prime
dagger minus 119886119886119953119953primedagger 119886119886119953119953
dagger119886119886119953119953119953119953
= ℏ120596120596119953119953 119886119886119953119953dagger 120575120575119953119953119953119953prime + 119886119886119953119953prime
dagger 119886119886119953119953 minus 119886119886119953119953primedagger 119886119886119953119953
dagger119886119886119953119953119953119953
= ℏ120596120596119953119953prime119886119886119953119953primedagger + ℏ120596120596119953119953 119886119886119953119953
dagger119886119886119953119953primedagger 119886119886119953119953 minus 119886119886119953119953prime
dagger 119886119886119953119953dagger119886119886119953119953
119953119953
= ℏ120596120596119953119953prime119886119886119953119953primedagger
In summary we have
119886119886119953119953(120591120591) = 119886119886119953119953(0)119890119890minus120596120596119953119953120591120591 = 119886119886119953119953119890119890minus120596120596119953119953120591120591
119886119886119953119953dagger(120591120591) = 119886119886119953119953
dagger(0)119890119890120596120596119953119953120591120591 = 119886119886119953119953dagger119890119890120596120596119953119953120591120591 (46)
42 Greenrsquos function
To describe thermal transport properties between different system sections we use the formalism of
thermal (or imaginary time) phonon Greenrsquos function [2] To this end we define the thermal Greenrsquos
function for phonons as
119866119866119953119953119953119953prime(120591120591 120591120591prime) = minuslang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime)rang = minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime) (47)
where 119879119879120591120591 is the time ordering operator and 120588120588119867119867 is the statistical operator for the grand canonical
ensemble (note that the chemical potential for phonons is zero)
120588120588119867119867 = 119890119890minus120573120573119867119867119885119885119867119867 (48)
where the partition function is given by 119885119885119867119867 equiv Tr119890119890minus120573120573119867119867 and 120573120573 = 1119896119896119861119861119879119879 with 119896119896119861119861 being
Boltzmannrsquos constant and 119879119879 the temperature For time independent Hamiltonians 119866119866119953119953119953119953prime(120591120591 120591120591prime)
depends only on 120591120591 minus 120591120591prime ie
119866119866119953119953119953119953prime(120591120591 120591120591prime) = 119866119866119953119953119953119953prime(120591120591 minus 120591120591prime 0) (49)
To show this we shall first assume that 120591120591 gt 120591120591prime such that
119866119866119953119953119953119953prime(120591120591 120591120591prime) = minuslang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime)rang = minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime) = minusTr 120588120588119867119867119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime)
= minusTr 120588120588119867119867119890119890119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591ℏ 119890119890
119867119867120591120591primeℏ 119886119886119953119953prime
dagger 119890119890minus119867119867120591120591primeℏ = minusTr 120588120588119867119867119890119890
minus119867119867120591120591primeℏ 119890119890
119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591minus120591120591prime
ℏ 119886119886119953119953primedagger
= minusTr 120588120588119867119867119890119890119867119867120591120591minus120591120591prime
ℏ 119886119886119953119953119890119890minus119867119867120591120591minus120591120591prime
ℏ 119886119886119953119953primedagger = minusTr 120588120588119867119867119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953prime
dagger (0)
= minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953primedagger (0) = minus lang119879119879120591120591119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953prime
dagger (0)rang = 119866119866119953119953119953119953prime(120591120591 minus 120591120591prime 0)
Where we have used the commutativity of 120588120588119867119867 and 119890119890plusmn119867119867120591120591prime
ℏ and the invariance of the trace operation
12
towards cyclic permutations Similarly for 120591120591 lt 120591120591prime we have
119866119866119953119953119953119953prime(120591120591 120591120591prime) = minuslang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime)rang = minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime) = minusTr 120588120588119867119867119886119886119953119953primedagger (120591120591prime)119886119886119953119953(120591120591)
= minusTr 120588120588119867119867119890119890119867119867120591120591primeℏ 119886119886119953119953prime
dagger 119890119890minus119867119867120591120591primeℏ 119890119890
119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591ℏ = minusTr 120588120588119867119867119886119886119953119953prime
dagger 119890119890119867119867120591120591minus120591120591prime
ℏ 119886119886119953119953119890119890minus119867119867120591120591ℏ 119890119890
119867119867120591120591primeℏ
= minusTr 120588120588119867119867119886119886119953119953primedagger 119890119890
119867119867120591120591minus120591120591primeℏ 119886119886119953119953119890119890
minus119867119867120591120591minus120591120591prime
ℏ = minusTr 120588120588119867119867119886119886119953119953primedagger (0)119886119886119953119953(120591120591 minus 120591120591prime)
= minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953primedagger (0)119886119886119953119953(120591120591 minus 120591120591prime) = minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953prime
dagger (0)
= minuslang119879119879120591120591119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953primedagger (0)rang = 119866119866119953119953119953119953prime(120591120591 minus 120591120591prime 0)
For simplicity we will introduce the notation 119866119866119953119953119953119953prime(120591120591 0) equiv 119866119866119953119953119953119953prime(120591120591) We further note that when using
imaginary time 119866119866119953119953119953119953prime(120591120591) is a periodic functions in the domain [minus120573120573ℏ120573120573ℏ] with a period of 120573120573ℏ (see
Page 236 of Ref [2])
119866119866119953119953119953119953prime(120591120591 + 120573120573ℏ 0) = 119866119866119953119953119953119953prime(120591120591 0) 120591120591 lt 0119866119866119953119953119953119953prime(120591120591 minus 120573120573ℏ 0) = 119866119866119953119953119953119953prime(120591120591 0) 120591120591 gt 0 (410)
To show this we shall again assume first that minus120573120573ℏ lt 120591120591 lt 0 to write that is
119866119866119953119953119953119953prime(120591120591 + 120573120573ℏ 0) = minus lang119879119879120591120591119886119886119953119953(120591120591 + 120573120573ℏ)119886119886119953119953primedagger (0)rang = minuslang119886119886119953119953(120591120591 + 120573120573ℏ)119886119886119953119953prime
dagger (0)rang
= minusTr 120588120588119867119867119890119890119867119867(120591120591+120573120573ℏ)
ℏ 119886119886119953119953119890119890minus119867119867(120591120591+120573120573ℏ)
ℏ 119886119886119953119953prime = minusTr 120588120588119867119867119890119890120573120573119867119867119890119890
119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591ℏ 119890119890minus120573120573119867119867119886119886119953119953prime
= minusTr120588120588119867119867119890119890120573120573119867119867119886119886119953119953(120591120591)119890119890minus120573120573119867119867119886119886119953119953prime(0) = minus
1119885119885119867119867
Tr119890119890minus120573120573119867119867119890119890120573120573119867119867119886119886119953119953(120591120591)119890119890minus120573120573119867119867119886119886119953119953prime(0)
= minus1119885119885119867119867
Tr119886119886119953119953(120591120591)119890119890minus120573120573119867119867119886119886119953119953prime(0) = minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119886119886119953119953prime(0)119886119886119953119953(120591120591)
= minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime(0) = minusTr120588120588119867119867119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime = 119866119866119953119953119953119953prime(120591120591 0)
Similarly for 120573120573ℏ gt 120591120591 gt 0 we have
119866119866119953119953119953119953prime(120591120591 minus 120573120573ℏ 0) = minus lang119879119879120591120591119886119886119953119953(120591120591 minus 120573120573ℏ)119886119886119953119953primedagger (0)rang = minuslang119886119886119953119953prime
dagger (0)119886119886119953119953(120591120591 minus 120573120573ℏ)rang
= minusTr 120588120588119867119867119886119886119953119953prime119890119890119867119867(120591120591minus120573120573ℏ)
ℏ 119886119886119953119953119890119890minus119867119867(120591120591minus120573120573ℏ)
ℏ = minusTr 120588120588119867119867119886119886119953119953prime119890119890minus120573120573119867119867119890119890
119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591ℏ 119890119890120573120573119867119867
= minusTr120588120588119867119867119886119886119953119953prime(0)119890119890minus120573120573119867119867119886119886119953119953(120591120591)119890119890120573120573119867119867 = minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119886119886119953119953prime(0)119890119890minus120573120573119867119867119886119886119953119953(120591120591)119890119890120573120573119867119867
= minus1119885119885119867119867
Tr119886119886119953119953prime(0)119890119890minus120573120573119867119867119886119886119953119953(120591120591) = minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119886119886119953119953(120591120591)119886119886119953119953prime(0)
= minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime(0) = minusTr120588120588119867119867119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime = 119866119866119953119953119953119953prime(120591120591 0)
13
Therefore 119866119866119953119953119953119953prime(120591120591) can be expanded as a Fourier series in the domain [0120573120573ℏ] as follows
119866119866119953119953119953119953prime(120591120591) = 1120573120573ℏsum 119890119890minus119894119894120596120596119899119899120591120591119866119866119953119953119953119953prime(119894119894120596120596119899119899)infinminusinfin (411)
where 120596120596119899119899 = 2120587120587119899119899120573120573ℏ
and the associated Fourier coefficient is given by
119866119866119953119953119953119953prime(119894119894120596120596119899119899) = int d120591120591119890119890119894119894120596120596119899119899120591120591119866119866119953119953119953119953prime(120591120591)120573120573ℏ0 120596120596119899119899 = 2119899119899120587120587
120573120573ℏ (412)
Having proven the translational time invariance and the periodicity of the Greenrsquos functios we can
now calculate it for the uncoupled system
119866119866119953119953119953119953prime0 (120591120591 120591120591prime) = minuslang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime)rang0 == minus lang119886119886119953119953(0)119890119890minus120596120596119953119953120591120591119886119886119953119953prime
dagger (0)119890119890120596120596119953119953prime120591120591primerang 120591120591 minus 120591120591prime gt 0
minus lang119886119886119953119953primedagger (0)119890119890120596120596119953119953prime120591120591
prime119886119886119953119953(0)119890119890minus120596120596119953119953120591120591rang 120591120591 minus 120591120591prime lt 0
= minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591
primelang119886119886119953119953(0)119886119886119953119953prime
dagger (0)rang 120591120591 minus 120591120591prime gt 0
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang 120591120591 minus 120591120591prime lt 0
= minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591
primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0) + 119886119886119953119953(0)119886119886119953119953primedagger (0)rang 120591120591 minus 120591120591prime gt 0
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang 120591120591 minus 120591120591prime lt 0
= minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591
primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang + 120575120575119953119953119953119953prime 120591120591 minus 120591120591prime gt 0
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang 120591120591 minus 120591120591prime lt 0
ie
119866119866119953119953119953119953prime0 (120591120591 120591120591prime) =
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang + 120575120575119953119953119953119953prime 120591120591 minus 120591120591prime gt 0
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang 120591120591 minus 120591120591prime lt 0 (413)
where we have used Eq (46) for 119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime) and Commutation relation 119886119886119953119953(0)119886119886119953119953prime
dagger (0) = 120575120575119953119953119953119953prime
To calculate lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang we first prove following equation
119890119890119860119860119861119861119890119890minus119860119860 = sum 1119899119899119860(119899119899)119861119861infin
119899119899=0 (414)
where 119860(119896119896)119861119861 equiv 119860 119860(119896119896minus1)119861119861 To prove this identity we define the operator 119891119891(119905119905) = 119890119890119905119905A119861119861119890119890minus119905119905119860119860
and taylor expand it around 119905119905 = 0
119891119891(119905119905) = 119891119891(0) + 119905119905119891119891prime(0) + 1199051199052
2119891119891primeprime(0) + ⋯ = sum 119905119905119899119899
119899119899119889119889119899119899119891119889119889119905119905119899119899
119905119905=0
infin119899119899=0 (415)
The corresponding derivatives are given by
⎩⎪⎨
⎪⎧ 119891119891prime(119905119905) = 119890119890119905119905119860119860119860119861119861119890119890minus119905119905119860119860 minus 119890119890119905119905119860119860119861119861119860119890119890minus119905119905119860119860 = 119890119890119905119905119860119860119860119861119861119890119890minus119905119905119860119860
119891119891primeprime(119905119905) = 119890119890119905119905119860119860119860119860119861119861 minus 119860119861119861119860119890119890minus119905119905119860119860 = 119890119890119905119905119860119860119860(2)119861119861119890119890minus119905119905119860119860
⋮119891119891(119899119899)(119905119905) = 119890119890119905119905119860119860119860(119899119899)119861119861119890119890minus119905119905119860119860
(416)
14
Substituting Eq (416) into Eq (415) we have
119890119890119905119905A119861119861119890119890minus119905119905119860119860 = sum 119905119905119899119899
119899119899119860(119899119899)119861119861infin
119899119899=0 (417)
Itrsquos clear that Eq (413) is a spectial case of Eq (416) with 119905119905 = 1 With this we can proceed as
follows
lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang =
11198851198851198671198670
Tr119890119890minus1205731205731198671198670119886119886119953119953primedagger (0)119886119886119953119953(0) =
11198851198851198671198670
Tr119890119890minus1205731205731198671198670119886119886119953119953primedagger (0)1198901198901205731205731198671198670119890119890minus1205731205731198671198670119886119886119953119953(0)
=11198851198851198671198670
Tr (minus120573120573)119899119899
1198991198991198671198670
(119899119899)119886119886119953119953primedagger (0)
infin
119899119899=0
119890119890minus1205731205731198671198670119886119886119953119953(0)
Note that 1198671198670119886119886119953119953primedagger (0) = ℏ120596120596119953119953prime119886119886119953119953prime
dagger (0) we have
1198671198670(119899119899)119886119886119953119953prime
dagger (0) = ℏ120596120596119953119953prime119899119899119886119886119953119953primedagger (0) (418)
such that
lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang =
11198851198851198671198670
Tr (minus120573120573)119899119899
1198991198991198671198670
(119899119899)119886119886119953119953primedagger (0)
infin
119899119899=0
119890119890minus1205731205731198671198670119886119886119953119953(0)
=11198851198851198671198670
Tr minus120573120573ℏ120596120596119953119953prime
119899119899
119899119899119886119886119953119953primedagger (0)
infin
119899119899=0
119890119890minus1205731205731198671198670119886119886119953119953(0)
= 119890119890minus120573120573ℏ120596120596119953119953prime11198851198851198671198670
Tr119890119890minus1205731205731198671198670119886119886119953119953(0)119886119886119953119953primedagger (0) = 119890119890minus120573120573ℏ120596120596119953119953primelang119886119886119953119953(0)119886119886119953119953prime
dagger (0)rang
= 119890119890minus120573120573ℏ120596120596119953119953primelang119886119886119953119953primedagger (0)119886119886119953119953(0)rang + 120575120575119953119953119953119953prime
therefore we obtain lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang 119890119890120573120573ℏ120596120596119953119953prime minus 1 = 120575120575119953119953119953119953prime For 119953119953prime ne 119953119953 and general 119890119890120573120573ℏ120596120596119953119953prime we have
lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang = 0 For 119953119953prime = 119953119953 we obtain lang119886119886119953119953
dagger(0)119886119886119953119953(0)rang = 119890119890120573120573ℏ120596120596119953119953 minus 1minus1
Hence we have
lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang = 120575120575119953119953prime119953119953
1119890119890120573120573ℏ120596120596119953119953minus1
equiv 120575120575119953119953prime119953119953119899119899119861119861120596120596119953119953 (419)
where 119899119899119861119861120596120596119953119953 is the Bose-Einstein distribution for phonons Substituting Eq (418) in Eq (413)
we obtain
119866119866119953119953119953119953prime0 (120591120591 120591120591prime) =
minus120575120575119953119953119953119953prime1 + 119899119899119861119861120596120596119953119953119890119890minus120596120596119953119953120591120591minus120591120591prime 120591120591 minus 120591120591prime gt 0
minus120575120575119953119953119953119953prime119899119899119861119861120596120596119953119953119890119890minus120596120596119953119953120591120591minus120591120591prime 120591120591 minus 120591120591prime lt 0 (420)
Note that only those Greenrsquos functions of the form 119866119866119953119953119953119953prime0 (120591120591 120591120591prime) = 1198661198661199531199530(120591120591 minus 120591120591prime)120575120575119953119953119953119953prime are non-zero due
to the orthogonality of normal modes Looking at the non-vanishing terms and setting 120591120591prime = 0
without loss of generality we can calculate the Fourier transform of 1198661198661199531199530(120591120591) from Eq (412)
1198661198661199531199530(119894119894120596120596119899119899) = int d1205911205911198901198901198941198941205961205961198991198991205911205911198661198661199531199530(120591120591)120573120573ℏ0 = minusint d120591120591119890119890119894119894120596120596119899119899120591120591120573120573ℏ
0 1 + 119899119899119861119861120596120596119953119953119890119890minus120596120596119953119953120591120591 = minus1+119899119899119861119861120596120596119953119953119894119894120596120596119899119899minus120596120596119953119953
119890119890119894119894120596120596119899119899minus120596120596119953119953120573120573ℏ minus
15
1 = minus1+ 1
119890119890120573120573ℏ120596120596119953119953minus1
119894119894120596120596119899119899minus120596120596119953119953119890119890119894119894
2120587120587119899119899120573120573ℏ 120573120573ℏ119890119890minus120573120573ℏ120596120596119953119953 minus 1 = minus 119890119890120573120573ℏ120596120596119953119953
119894119894120596120596119899119899minus120596120596119953119953
119890119890minus120573120573ℏ120596120596119953119953minus1
119890119890120573120573ℏ120596120596119953119953minus1= minus 1
119894119894120596120596119899119899minus120596120596119953119953
1minus119890119890120573120573ℏ120596120596119953119953
119890119890120573120573ℏ120596120596119953119953minus1= 1
119894119894120596120596119899119899minus120596120596119953119953
ie
1198661198661199531199530(119894119894120596120596119899119899) == 1119894119894120596120596119899119899minus120596120596119953119953
(421)
5 Fermirsquos golden rule 51 Interaction picture
Next we can proceed with calculating the Greenrsquos function of the coupled system To this end we
define the coupling Hamiltonian operator term in the interaction picture as
119867119867119862119862119868119868 (120591120591) = 1198901198901198671198670120591120591ℏ 119867119867119862119862119890119890
minus1198671198670120591120591ℏ (51)
The time evolution of 119867119867119862119862(120591120591) is then given by
ℏ part119867119867119862119862119868119868 (120591120591)120597120597120591120591
= 1198671198670119867119867119862119862119868119868 (120591120591) (52)
Note that the Greenrsquos function defined above was given in the Heisenberg picture To proceed we
need to transform it to the interaction picture
119866119866119953119953119953119953prime(120591120591 0) = minus lang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953primedagger (0)rang = minusTr 120588120588119867119867119879119879120591120591 119890119890
119867119867120591120591ℏ 119886119886119953119953(0)119890119890minus
119867119867120591120591ℏ 119886119886119953119953prime
dagger (0)
= minusTr 120588120588119867119867119879119879120591120591 119890119890119867119867120591120591ℏ 119890119890minus
1198671198670120591120591ℏ 119890119890
1198671198670120591120591ℏ 119886119886119953119953(0)119890119890minus
1198671198670120591120591ℏ 119890119890
1198671198670120591120591ℏ 119890119890minus
119867119867120591120591ℏ 119886119886119953119953prime
dagger (0)
= minusTr 120588120588119867119867119879119879120591120591 119880119880(0 120591120591)119886119886119953119953119868119868 (120591120591)119880119880(120591120591 0)119886119886119953119953primedagger (0)
where
119886119886119953119953119868119868 (120591120591) equiv 1198901198901198671198670120591120591ℏ 119886119886119953119953(0)119890119890minus
1198671198670120591120591ℏ (53)
is the operator in interaction picture and the operator 119880119880 is defined by
119880119880(1205911205911 1205911205912) equiv 11989011989011986711986701205911205911ℏ 119890119890minus
119867119867 (1205911205911minus1205911205912)ℏ 119890119890minus
11986711986701205911205912ℏ (54)
Note that while 119880119880 is not unitary it satisfies the following group property
119880119880(1205911205911 1205911205912)119880119880(1205911205912 1205911205913) = 119880119880(1205911205911 1205911205913) (55)
and the boundary condition 119880119880(1205911205911 1205911205911) = 1 In addition the 120591120591 derivative of 119880119880 is simply
ℏpart119880119880(120591120591 120591120591prime)
120597120597120591120591= 1198671198670119890119890
1198671198670120591120591ℏ 119890119890minus
119867119867120591120591minus120591120591primeℏ minus 119890119890
1198671198670120591120591ℏ 119867119867119890119890minus
119867119867120591120591minus120591120591primeℏ 119890119890minus
1198671198670120591120591primeℏ
= 1198901198901198671198670120591120591ℏ 1198671198670 minus 119867119867119890119890minus
1198671198670120591120591ℏ 119890119890
1198671198670120591120591ℏ 119890119890minus
119867119867120591120591minus120591120591primeℏ 119890119890minus
1198671198670120591120591primeℏ = 119890119890
1198671198670120591120591ℏ minus119867119867119862119862119890119890
minus1198671198670120591120591ℏ 119880119880(120591120591 120591120591prime)
= minus119867119867119862119862119868119868 (120591120591)119880119880(120591120591 120591120591prime)
16
ie
ℏ part119880119880120591120591120591120591prime120597120597120591120591
= minus119867119867119862119862119868119868 (120591120591)119880119880(120591120591 120591120591prime) (56)
The solution of Eq (56) is (see page 235 of Ref [2])
119880119880(120591120591 120591120591prime) = summinus1ℏ
119899119899
119899119899 int 1198891198891205911205911120591120591120591120591prime ⋯int 119889119889120591120591119899119899
120591120591120591120591prime 119879119879120591120591119867119867119862119862119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119899119899)infin
119899119899=0 (57)
The exact thermal Greens function now may be rewritten in the interaction picture as
119866119866119953119953119953119953prime(120591120591 0) = minus1119885119885119867119867
Tr 119890119890minus120573120573119867119867119879119879120591120591119880119880(0 120591120591)119886119886119953119953119868119868 (120591120591)119880119880(120591120591 0)119886119886119953119953primedagger (0)
= minusTr 119890119890minus120573120573119867119867119879119879120591120591 119880119880(0 120591120591)119886119886119953119953119868119868 (120591120591)119880119880(120591120591 0)119886119886119953119953prime
dagger (0)
Tr119890119890minus120573120573119867119867
= minusTr 119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)119879119879120591120591 119886119886119953119953119868119868 (120591120591)119880119880(0 120591120591)119880119880(120591120591 0)119886119886119953119953prime
dagger (0)
Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)
= minusTr 119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)119879119879120591120591 119886119886119953119953119868119868 (120591120591)119880119880(00)119886119886119953119953prime
dagger (0)
Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)=
= minusTr 119890119890minus1205731205731198671198670119879119879120591120591 119880119880(120573120573ℏ 0)119886119886119953119953119868119868 (120591120591)119886119886119953119953prime
dagger (0)
Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)
Where we used the fact that we are free to change the order of the operators within the time ordering
operation (see pages 241-242 of Ref [2])
52 Wickrsquos theorem
Thus the Greenrsquos function can be expanded as [2]
119866119866119953119953119953119953prime(120591120591 0) = minussum 1
119898119898minus1ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)119886119886119953119953119868119868 (120591120591)119886119886
119953119953primedagger (0)rang0infin
119898119898=0
sum 1119898119898minus
1ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)rang0infin
119898119898=0 (58)
where lang⋯ rang0 represents the ensemble average with respect to the non-interacting basis
Tr119890119890minus1205731205731198671198670(⋯ ) Or explicitly
119866119866119953119953119953119953prime(120591120591 0) = minus119866119866119953119953119953119953prime0 (120591120591)minus1ℏint 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953primedagger (0)rang0+
12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0+⋯
1minus1ℏint 1198891198891205911205911120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)rang0+12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0+⋯
(59)
For consiceness we introduce the following notation
119863119863119898119898 equiv 1119898119898minus 1
ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)rang0 (510)
To simplify the calculation in Eq (58) we adopt Wickrsquos theorem (see pages 237-241 of Ref
[2])which can be expressed as follows
lang119879119879120591120591119860119861119861119862119863119863 ⋯119884119884119885rang0 = lang119879119879120591120591119860119861119861rang0lang119879119879120591120591119862119863119863rang0⋯ lang119879119879120591120591119884119884119885rang0 + lang119879119879120591120591119860119862rang0lang119879119879120591120591119861119861119863119863rang0⋯ lang119879119879120591120591119883119883119885rang0 + ⋯ (511)
17
Here 119860119861119861 hellip 119884119884 119885 represent 119886119886119953119953(120591120591) or 119886119886119953119953dagger(120591120591) In Eq (511) each term corresponds to a particular
pairing of the operators 119860119861119861119862119863119863 ⋯119884119884119885 and all possible pairings are taken into account Here the only
non-vanishing propagators will have the form [see Eq (420)]
lang119879119879120591120591 119886119886119953119953primedagger (120591120591prime)119886119886119953119953(120591120591)rang0 = minuslang119879119879120591120591 119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime)rang0 = 119866119866119953119953119953119953prime0 (120591120591 120591120591prime) = 1198661198661199531199530(120591120591 minus 120591120591prime)120575120575119953119953119953119953prime (512)
Therefore the second term in the numerator of Eq (59) can be calculated as
1198681198682 = minus1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0
= minus1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 lang119879119879120591120591119886119886119953119953prime
dagger (0)119886119886119953119953(120591120591)rang0 minus1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
= 119866119866119953119953119953119953prime0 (120591120591 0) minus
1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 minus
1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
The first term in above equation is called disconnected part since the pairing is performed separately
on 119867119867119862119862(1205911205911) and 119886119886119953119953(120591120591)119886119886119953119953primedagger (0) All other terms have pairs that mix creation and annihilation operators
of the Hamiltonian with 119886119886119953119953(120591120591) or 119886119886119953119953primedagger (0) and are said to have connected party For simplicity we
include all these terms in the notation lang119879119879120591120591119867119867119862119862(1205911205911)119886119886119953119953(120591120591)119886119886119953119953primedagger (0)rang0119888119888 Furthermore we define 119866119866119953119953119953119953prime
(1) (120591120591) equiv
minus1ℏ int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862(1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 Then we have
1198681198683 = minus1ℏint 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 = 119866119866119953119953119953119953prime0 (120591120591 0)1198631198631 + 119866119866119953119953119953119953prime
(1) (120591120591) (513)
Using this method the third term in the numerator of Eq (59) can be calculated as
1198681198683 = 12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 = 12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0 lang119879119879120591120591119886119886119953119953prime
dagger (0)119886119886119953119953(120591120591)rang0 +
12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 + 12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 +
12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 = 119866119866119953119953119953119953prime0 (120591120591 0) 1
2ℏ2 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0 +
12ℏ2 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 + 12ℏ2 int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 +
12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
Here the last term is denoted as 119866119866119953119953119953119953prime(2) (120591120591)
Using Eq (513) the second and third terms on the right hand above equation can be simplified as
follows
18
12ℏ2
1198891198891205911205912120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 +1
2ℏ2 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
=12 minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0
minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
+12 minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0
minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
=12 minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0
minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
+12 minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0
minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
= minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0
minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 = 1198631198631119866119866119953119953119953119953prime(1) (120591120591)
Where we performed the following integration variables interchange 1205911205911 ⟷ 1205911205912 Thus we have
1198681198683 = 119866119866119953119953119953119953prime(2) (120591120591) + 119866119866119953119953119953119953prime
(1) (120591120591)1198631198631 + 119866119866119953119953119953119953prime0 (120591120591 0)1198631198632 (514)
Similarity we can calculate the fourth term of Eq (59) as
1198681198684 = minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 =
minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0 lang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 + 3 times
minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 + 3 times
minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 +
minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 =
119866119866119953119953119953119953prime0 (120591120591 0) minus1
6ℏ3 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0+
12ℏ2 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0
minus1ℏ int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888+
12ℏ2 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 minus1ℏ int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0+ 119866119866119953119953119953119953prime
(3) (120591120591) = 119866119866119953119953119953119953prime(3) (120591120591) +
119866119866119953119953119953119953prime(2) (120591120591)1198631198631 + 119866119866119953119953119953119953prime
(1) (120591120591)1198631198632 + 119866119866119953119953119953119953prime0 (120591120591 0)1198631198633
Higher order terms can be treated in the same manner (see pages 95-96 of Ref [2]) Then when
substituting 1198681198682 1198681198683 1198681198684 and all higher order terms into Eq (59) we can simplify the numerator as
follows
119866119866119953119953119953119953prime0 (120591120591) minus
1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0
+1
2ℏ2 1198891198891205911205911120573120573ℏ
0 1198891198891205911205912120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 + ⋯
= 119866119866119953119953119953119953prime0 (120591120591) + 119866119866119953119953119953119953prime
(1) (120591120591) + 119866119866119953119953119953119953prime(2) (120591120591) + ⋯ (1 + 1198631198631 + 1198631198632 + ⋯ )
Noting that the expression (1 + 1198631198631 + 1198631198632 + ⋯ ) is exactly canceled with the denominator the
Greenrsquos function can be simplified as
19
119866119866119953119953119953119953prime(120591120591) = 119866119866119953119953119953119953prime0 (120591120591) + 119866119866119953119953119953119953prime
(1) (120591120591) + 119866119866119953119953119953119953prime(2) (120591120591) + ⋯ (515)
where
119866119866119953119953119953119953prime
(1) (120591120591) = minus1ℏ int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888
119866119866119953119953119953119953prime(2) (120591120591) = 1
2ℏ2 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888 (516)
53 Calculation of Greenrsquos function
531 First order appriximation In Eq (516) we go back to the full notation 119886119886119954119954120582120582 to distinguish phonons of different branches
then 119866119866119953119953119953119953prime(1) (120591120591) is calculated as
119866119866119953119953119953119953prime(1) (120591120591) = minus
1ℏ
12 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591 119881119881119895119895119895119895prime(119948119948)119876119876119948119948119895119895(1205911205911)119876119876119948119948119895119895prime
dagger (1205911205911)3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ 119881119881119895119895119895119895primelowast (119948119948)119876119876119948119948119895119895prime(1205911205911)119876119876119948119948119895119895
dagger (1205911205911)3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= minus12ℏ
1198891198891205911205911120573120573ℏ
0lang119879119879120591120591
ℏ119881119881119895119895119895119895prime(119948119948)2120596120596119948119948119895119895120596120596119948119948119895119895prime
119886119886119948119948119895119895(1205911205911) + 119886119886minus119948119948119895119895dagger (1205911205911) 119886119886minus119948119948119895119895prime(1205911205911) + 119886119886119948119948119895119895prime
dagger (1205911205911)3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ ℏ119881119881119895119895119895119895prime
lowast (119948119948)
2120596120596119948119948119895119895120596120596119948119948119895119895prime119886119886119948119948119895119895prime(1205911205911) + 119886119886minus119948119948119895119895prime
dagger (1205911205911) 119886119886minus119948119948119895119895(1205911205911) + 119886119886119948119948119895119895dagger (1205911205911)
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
According to Wickrsquos theorem [2] the terms that contain 119886119886119948119948119895119895119886119886minus119948119948119895119895prime and 119886119886minus119948119948119895119895dagger 119886119886119948119948119895119895prime
dagger are equal to zero
thus the first term of above equation are calculated as
11989211989211 = minus14
119881119881119895119895119895119895prime(119948119948)
120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119886119886minus119948119948119895119895
dagger (1205911205911)119886119886minus119948119948119895119895prime(1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ lang119879119879120591120591119886119886119948119948119895119895(1205911205911)119886119886119948119948119895119895primedagger (1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888
= minus14
119881119881119895119895119895119895prime(119948119948)
120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119886119886minus119948119948119895119895
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus119948119948119895119895prime(1205911205911)rang0
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ lang119879119879120591120591119886119886119948119948119895119895primedagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)119886119886119948119948119895119895(1205911205911)rang0
= minus14
119881119881119895119895119895119895prime(119948119948)
120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
01198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954minus119948119948120575120575120582120582119895119895119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119954119954primeminus119948119948120575120575120582120582prime119895119895prime3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ 1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119948119948120575120575120582120582119895119895prime119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119954119954prime119948119948120575120575120582120582prime119895119895
20
Simplification of above equation gives
11989211989211 =
⎩⎪⎨
⎪⎧minus119881119881120582120582120582120582prime(minus119954119954)
4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954120582120582
0 (120591120591 1205911205911)119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903
2lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582(119954119954)
4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954120582120582
0 (120591120591 1205911205911)119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582prime le 3119903119903
2lt 120582120582 le 3119903119903
0 else
(517)
Similarity the second term of 119866119866119953119953119953119953prime(1) (120591120591) is calculated as
11989211989212 = minus14
minus119881119881119895119895119895119895primelowast (119948119948)
4120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119886119886119948119948119895119895prime(1205911205911)119886119886119948119948119895119895
dagger (1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ lang119879119879120591120591119886119886minus119948119948119895119895primedagger (1205911205911)119886119886minus119948119948119895119895(1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888
= minus14
minus119881119881119895119895119895119895primelowast (119948119948)
4120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)119886119886119948119948119895119895prime(1205911205911)rang0 lang119879119879120591120591119886119886119948119948119895119895dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus119948119948119895119895(1205911205911)rang0 lang119879119879120591120591119886119886minus119948119948119895119895prime
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
ie
11989211989212 =
⎩⎪⎨
⎪⎧
minus119881119881120582120582120582120582primelowast (119954119954)
4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954prime120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903
2lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582lowast (minus119954119954)
4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954prime120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119954119954prime 1 le 120582120582prime le 3119903119903
2lt 120582120582 le 3119903119903
0 else
(518)
Note that since 119881119881120582120582120582120582primelowast (119954119954) = 119881119881120582120582120582120582prime(minus119954119954) [Eq (314)] the first order approximation of Greenrsquos function
can be written as
119866119866119954119954120582120582119954119954prime120582120582prime(1) (120591120591) =
⎩⎪⎨
⎪⎧minus119881119881
120582120582120582120582prime(minus119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ
0 1198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582 le3119903119903
2lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582
(119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ
0 1198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582prime le3119903119903
2lt 120582120582 le 3119903119903
0 else
(519)
To derive the Greenrsquos function in frequency domain we use the expression for Fourier series then
we have
1198891198891205911205911120573120573ℏ
01198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0) = 1198891198891205911205911120573120573ℏ
0
1120573120573ℏ
119890119890minus119894119894120596120596119899119899(120591120591minus1205911205911)1198661198661199541199541205821205820 (119894119894120596120596119899119899)
119899119899
1120573120573ℏ
119890119890minus119894119894120596120596119899119899prime1205911205911119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899prime)
119899119899prime
=1
(120573120573ℏ)2119890119890minus119894119894120596120596119899119899120591120591 1198891198891205911205911120573120573ℏ
0119890119890119894119894120596120596119899119899minus120596120596119899119899prime1205911205911 119866119866119954119954120582120582
0 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899prime)
119899119899119899119899prime=
=1
(120573120573ℏ)2119890119890minus119894119894120596120596119899119899120591120591[120573120573ℏ120575120575119899119899119899119899prime]1198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899prime)119899119899119899119899prime
=1120573120573ℏ
119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899)119899119899
According to the definition of Fourier series [see Eq (412)] we get the Fourier coefficient of
21
119866119866119954119954120582120582119954119954prime120582120582prime(1) (120591120591)
119866119866119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) =
⎩⎪⎨
⎪⎧minus119881119881120582120582120582120582prime(minus119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582prime1198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime 1 le 120582120582 le 31199031199032
lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582(119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582prime1198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime 1 le 120582120582prime le 31199031199032
lt 120582120582 le 3119903119903
0 else
(520)
where 119866119866119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) = 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) times Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) times 119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899) and Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) is defined as
Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) =
⎩⎪⎨
⎪⎧minus119881119881120582120582120582120582prime(minus119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582prime120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903
2lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582(119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582prime120575120575119954119954119954119954prime 1 le 120582120582prime le 3119903119903
2lt 120582120582 le 3119903119903
0 else
(521)
532 second order approximation The second order approximation of the Greenrsquos function in Eq (516) 119866119866119954119954120582120582119954119954prime120582120582prime
(2) (120591120591) is calculated as
119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) =
18ℏ2 1198891198891205911205911
120573120573ℏ
0 1198891198891205911205912120573120573ℏ
0lang119879119879120591120591
⎣⎢⎢⎡ 11988111988111989511989511198951198951prime(119948119948120783120783)1198761198761199481199481207831207831198951198951(1205911205911)1198761198761199481199481207831207831198951198951prime
dagger (1205911205911)3119903119903
1198951198951prime=31199031199032 +1
31199031199032
1198951198951=1119948119948120783120783
+ 11988111988111989411989411198941198941primelowast (119948119948120783120783)1198761198761199481199481207831207831198941198941prime(1205911205911)1198761198761199481199481207831207831198941198941
dagger (1205911205911)3119903119903
1198941198941prime=31199031199032 +1
31199031199032
1198941198941=1119948119948120783120783 ⎦⎥⎥⎤
⎣⎢⎢⎡ 11988111988111989511989521198951198952prime(119948119948120784120784)1198761198761199481199481207841207841198951198952(1205911205912)1198761198761199481199481207841207841198951198952prime
dagger (1205911205912)3119903119903
1198951198952prime=31199031199032 +1
31199031199032
1198951198952=1119948119948120784120784
+ 11988111988111989411989421198941198942primelowast (119948119948120784120784)1198761198761199481199481207841207841198941198942prime(1205911205912)1198761198761199481199481207841207841198941198942
dagger (1205911205912)3119903119903
1198941198942prime=31199031199032 +1
31199031199032
1198941198942=1119948119948120784120784 ⎦⎥⎥⎤119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888 = 1198921198921 + 1198921198922 + 1198921198923 + 1198921198924
where
1198921198921 = 18ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0
sum sum 11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989511989521198951198952prime(119948119948120784120784)1le1198951198952le
31199031199032 lt1198951198952
primele311990311990311994811994812078412078411989511989521198951198952prime
1le1198951198951le31199031199032 lt1198951198951
primele311990311990311994811994812078312078311989511989511198951198951prime
times lang1198791198791205911205911198761198761199481199481207831207831198951198951(1205911205911)1198761198761199481199481207831207831198951198951primedagger (1205911205911)1198761198761199481199481207841207841198951198952(1205911205912)1198761198761199481199481207841207841198951198952prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(522)
1198921198922 = 18ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0
sum sum 11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942primelowast (119948119948120784120784)
1le1198941198942le31199031199032 lt1198941198942
primele311990311990311994811994812078412078411989411989421198941198942prime
1le1198951198951le31199031199032 lt1198951198951
primele311990311990311994811994812078312078311989511989511198951198951prime
times lang1198791198791205911205911198761198761199481199481207831207831198951198951(1205911205911)1198761198761199481199481207831207831198951198951primedagger (1205911205911)1198761198761199481199481207841207841198941198942prime (1205911205912)1198761198761199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(523)
1198921198923 = 18ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0
sum sum 11988111988111989411989411198941198941primelowast (119948119948120783120783)11988111988111989511989521198951198952prime(119948119948120784120784)
1le1198951198952le31199031199032 lt1198951198952
primele311990311990311994811994812078412078411989511989521198951198952prime
1le1198941198941le31199031199032 lt1198941198941
primele311990311990311994811994812078312078311989411989411198941198941prime
times lang1198791198791205911205911198761198761199481199481207831207831198941198941prime (1205911205911)1198761198761199481199481207831207831198941198941dagger (1205911205911)1198761198761199481199481207841207841198951198952(1205911205912)1198761198761199481199481207841207841198951198952prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(524)
22
1198921198924 = 18ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0
sum sum 11988111988111989411989411198941198941primelowast (119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)1le1198941198942le
31199031199032 lt1198941198942
primele311990311990311994811994812078412078411989411989421198941198942prime
1le1198941198941le31199031199032 lt1198941198941
primele311990311990311994811994812078312078311989411989411198941198941prime
times lang1198791198791205911205911198761198761199481199481207831207831198941198941prime (1205911205911)1198761198761199481199481207831207831198941198941dagger (1205911205911)1198761198761199481199481207841207841198941198942prime (1205911205912)1198761198761199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(525)
In what follows we will expand the term 1198921198922 The expansion of all other terms follows the same lines
and eventually results in the same expression Expanding the product 11987611987611994811994812078312078311989511989511198761198761199481199481207831207831198951198951primedagger 1198761198761199481199481207841207841198941198942prime1198761198761199481199481207841207841198941198942
dagger we can
rewrite Eq (523) as follows
1198921198922 = 132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)119881119881
11989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032 lt1198941198942
primele311990311990311994811994812078412078411989411989421198941198942prime
1le1198951198951le31199031199032 lt1198951198951
primele311990311990311994811994812078312078311989511989511198951198951prime
times 119868119868 (526)
Where
119868119868 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911) + 119886119886minus1199481199481207831207831198951198951dagger (1205911205911) 119886119886minus1199481199481207831207831198951198951prime(1205911205911) + 1198861198861199481199481207831207831198951198951prime
dagger (1205911205911) 1198861198861199481199481207841207841198941198942prime (1205911205912) + 119886119886minus1199481199481207841207841198941198942primedagger (1205911205912) 119886119886minus1199481199481207841207841198941198942(1205911205912) + 1198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0c
= lang119879119879120591120591 1198861198861199481199481207831207831198951198951119886119886minus1199481199481207831207831198951198951prime + 11988611988611994811994812078312078311989511989511198861198861199481199481207831207831198951198951primedagger + 119886119886minus1199481199481207831207831198951198951
dagger 119886119886minus1199481199481207831207831198951198951prime + 119886119886minus1199481199481207831207831198951198951dagger 1198861198861199481199481207831207831198951198951prime
dagger 12059112059111198861198861199481199481207841207841198941198942prime119886119886minus1199481199481207841207841198941198942 + 1198861198861199481199481207841207841198941198942prime1198861198861199481199481207841207841198941198942
dagger + 119886119886minus1199481199481207841207841198941198942primedagger 119886119886minus1199481199481207841207841198941198942
+ 119886119886minus1199481199481207841207841198941198942primedagger 1198861198861199481199481207841207841198941198942
dagger 1205911205912119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0c
= lang119879119879120591120591 11988611988611994811994812078312078311989511989511198861198861199481199481207831207831198951198951primedagger + 119886119886minus1199481199481207831207831198951198951
dagger 119886119886minus1199481199481207831207831198951198951prime12059112059111198861198861199481199481207841207841198941198942prime1198861198861199481199481207841207841198941198942
dagger + 119886119886minus1199481199481207841207841198941198942primedagger 119886119886minus11994811994812078412078411989411989421205911205912
119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
Where the symbol [hellip ]1205911205911 signifies that the operators in the brackets are given in the interaction
picture The last equality in above equation comes from the fact that the contractions of the product
of the operators are equal to zero when the number of creation and annihilation operators are not the
same (see Wickrsquos theorem in Ref [2]) Using Wickrsquos theorem [2] we are able to calculate the above
equation term by term as follows
⎩⎪⎨
⎪⎧ 1198681198681 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951prime
dagger (1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)1198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888
1198681198682 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
1198681198683 = lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)1198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
1198681198684 = lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(527)
We shall now calculate them term by term
23
1198681198681 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951primedagger (1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)1198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942
dagger (1205911205912)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
+ lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951prime
dagger (1205911205911)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0
+ lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)1198861198861199481199481207841207841198941198942prime(1205911205912)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0119888119888
= 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199481199481207841207841198951198952
0 (1205911205912 1205911205912)12057512057511989411989421198941198942prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582
+ 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 1198661198661199481199481207831207831198951198951
0 (1205911205911 1205911205911)12057512057511989511989511198951198951prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582
+ 1198661198661199481199481207831207831198951198951prime0 (1205911205912 1205911205911)1205751205751199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951prime ∙ 119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582
= 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582
+ 1198661198661199481199481207831207831198951198951prime0 (1205911205912 1205911205911)1205751205751199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951prime ∙ 119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582
The last equality results from the fact that 1198951198951 and 1198951198951prime are indices of different ranges (belonging to
different subsystems) and the same holds for 1198941198942 and 1198941198942prime Similarity
1198681198684 = lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0
= 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 119866119866minus11994811994812078412078411989411989420 (1205911205912 1205911205912)12057512057511989411989421198941198942prime
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 119866119866minus1199481199481207831207831198951198951prime 0 (1205911205911 1205911205912)1205751205751199481199481207831207831199481199481207841207841205751205751198951198951prime1198941198942prime
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205911 1205911205911)12057512057511989511989511198951198951prime
+ 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205912 1205911205911)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942
= 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 119866119866minus1199481199481207831207831198951198951prime 0 (1205911205911 1205911205912)1205751205751199481199481207831207831199481199481207841207841205751205751198951198951prime1198941198942prime
+ 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205912 1205911205911)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942
Where again the last equality results from the fact that 1198951198951 and 1198951198951prime are indices of different ranges
(belonging to different subsystems) and the same holds for 1198941198942 and 1198941198942prime Similarly we obtain 1198681198682 =
1198681198683 = 0 for the same reason as shown below
24
1198681198682 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951prime
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0
+ lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)1198861198861199481199481207831207831198951198951(1205911205911)rang0
+ lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0
= 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582 ∙ 119866119866minus11994811994812078412078411989411989420 (1205911205912 1205911205912)12057512057511989411989421198941198942prime
+ 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)120575120575minus11994811994812078412078411994811994812078312078312057512057511989511989511198941198942prime ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205911)12057512057511989511989511198951198951prime
+ 1198661198661199481199481207831207831198951198951prime0 (1205911205912 1205911205911)120575120575minus11994811994812078412078411994811994812078312078312057512057511989411989421198951198951prime ∙ 119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 = 0
1198681198683 = lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)1198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0
+ lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0
+ lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0
= 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 11986611986611994811994812078412078411989411989420 (1205911205912 1205911205912)12057512057511989411989421198941198942prime
+ 11986611986611994811994812078412078411989411989420 (1205911205911 1205911205912)120575120575minus1199481199481207831207831199481199481207841207841205751205751198951198951prime1198941198942 ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205911 1205911205911)12057512057511989511989511198951198951prime
+ 1198661198661199481199481207841207841198941198942prime0 (1205911205912 1205911205911)120575120575minus1199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951 ∙ 119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 = 0
The two non-vanishing terms (1198681198681 and 1198681198684) produce the following contributions to 1198921198922 of Eq (526)
11989211989221 = 132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime1198681198681 =
132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime1198661198661199481199481207831207831198951198951
0 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙
1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582 + 1
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime1198661198661199481199481207831207831198951198951prime
0 (1205911205912 1205911205911)1205751205751199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951prime ∙
119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 =
⎩⎪⎨
⎪⎧
120575120575119954119954119954119954prime
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast (119954119954prime)
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (1205911205911 1205911205912) ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
120575120575119954119954119954119954prime
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032+1 1198661198661199541199541198951198951prime
0 (1205911205912 1205911205911) ∙ 119866119866119954119954prime120582120582prime0 (1205911205911 0)119866119866119954119954120582120582
0 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032
0 else
and
25
11989211989224 = 132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime1198681198684 =
132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙
119866119866minus1199481199481207831207831198951198951prime 0 (1205911205911 1205911205912)1205751205751199481199481207831207831199481199481207841207841205751205751198951198951prime1198941198942prime + 1
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙
1198661198661199541199541205821205820 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 119866119866minus1199481199481207831207831198951198951
0 (1205911205912 1205911205911)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 =
⎩⎪⎨
⎪⎧120575120575119954119954119954119954prime
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032+1 1198661198661199541199541198951198951prime
0 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime0 (1205911205912 0)119866119866119954119954120582120582
0 (120591120591 1205911205911) 120582120582 120582120582prime isin 1 31199031199032
120575120575119954119954119954119954prime
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast (119954119954prime)
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
0 else
Here we used the following properties 120596120596minus119954119954120582120582 = 120596120596119954119954120582120582 and 1198811198811205821205821198951198951prime(minus119954119954) = 1198811198811205821205821198951198951primelowast (119954119954) [see Eqs (112) and
(314)]
The equivalence of 11989211989221 and 11989211989224 can be seen by changing the integration variables 1205911205912 harr 1205911205911
Substituting 11989211989221 and 11989211989224 into Eq (526) 1198921198922 is given by
1198921198922 =
⎩⎪⎨
⎪⎧
120575120575119954119954119954119954prime
16 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast (119954119954prime)
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime
0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
120575120575119954119954119954119954prime
16 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032+1 1198661198661199541199541198951198951prime
0 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime0 (1205911205911 0)119866119866119954119954120582120582
0 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032
0 else
(528)
By changing the indices and integration variables itrsquos straightforward to show that the other three
terms (119892119892111989211989231198921198924 ) are identical to 1198921198922 thus the final expression of 119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) [Eq (516)) can be
calculated as
119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) = 41198921198922 =
⎩⎪⎨
⎪⎧120575120575119954119954119954119954prime
4 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032
120575120575119954119954119954119954prime
4 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast 119954119954prime
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime
0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 31199031199032
+ 13119903119903
0 else
(529)
To calculate 119866119866119954119954120582120582119954119954prime120582120582prime(2) in frequency space we expand 119866119866119954119954120582120582119954119954prime120582120582prime
(2) (120591120591) in a Fourier series
119866119866119954119954120582120582119954119954prime120582120582prime
(2) (120591120591) = 1120573120573ℏsum 119890119890minus119894119894120596120596119899119899120591120591119866119866119954119954120582120582119954119954prime120582120582prime
(2) (119894119894120596120596119899119899)119899119899
119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = int d120591120591119890119890119894119894120596120596119899119899120591120591119866119866119954119954120582120582119954119954prime120582120582prime
(2) (120591120591)120573120573ℏ0
(530)
Using Eq (421) we get for 120582120582 120582120582prime isin 1 31199031199032
26
119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) =
120575120575119954119954119954119954prime4
1198891198891205911205911120573120573ℏ
0 1198891198891205911205912120573120573ℏ
0
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1198661198661199541199541198951198951prime0 (1205911205912 1205911205911)119866119866119954119954120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912)
=120575120575119954119954119954119954prime
4
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1198891198891205911205911120573120573ℏ
0 1198891198891205911205912120573120573ℏ
0
1120573120573ℏ
119890119890minus1198941198941205961205961198991198992(1205911205912minus1205911205911)1198661198661199541199541198951198951prime0 1198941198941205961205961198991198992
1198991198992
∙1120573120573ℏ
119890119890minus11989411989412059612059611989911989911205911205911119866119866119954119954120582120582prime0 1198941198941205961205961198991198991
1198991198991
∙1120573120573ℏ
119890119890minus119894119894120596120596119899119899(120591120591minus1205911205912)1198661198661199541199541205821205820 (119894119894120596120596119899119899)
119899119899
=120575120575119954119954119954119954prime
4
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1120573120573ℏ
119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954120582120582prime
0 11989411989412059612059611989911989911198661198661199541199541198951198951prime0 1198941198941205961205961198991198992
11989911989911989911989911198991198992⎩⎪⎨
⎪⎧ 1120573120573ℏ
1198891198891205911205911120573120573ℏ
0119890119890minus1198941198941205961205961198991198991minus12059612059611989911989921205911205911
times1120573120573ℏ
1198891198891205911205912120573120573ℏ
0119890119890minus1198941198941205961205961198991198992minus1205961205961198991198991205911205912
⎭⎪⎬
⎪⎫
=120575120575119954119954119954119954prime
4
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1120573120573ℏ
119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)
11989911989911989911989911198991198992
119866119866119954119954120582120582prime0 11989411989412059612059611989911989911198661198661199541199541198951198951prime
0 1198941198941205961205961198991198992120575120575119899119899111989911989921205751205751198991198991198991198992
=120575120575119954119954119954119954prime120573120573ℏ
119890119890minus119894119894120596120596119899119899120591120591
119899119899
1198661198661199541199541205821205820 (119894119894120596120596119899119899)
14
119881119881120582120582prime1198951198951prime(119954119954
prime)1198811198811205821205821198951198951primelowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899)119866119866119954119954120582120582prime
0 (119894119894120596120596119899119899)
Comparing above expression with Eq (530) the Fourier transform of 119866119866119954119954120582120582119954119954120582120582prime(2) (120591120591) reads as
119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) 120575120575119954119954119954119954prime
4sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 119866119866119954119954120582120582prime
0 (119894119894120596120596119899119899) (531)
where we have introduced the notation
Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) equiv
120575120575119954119954119954119954prime
4sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) (532)
and 119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954120582120582prime(2) (119894119894120596120596119899119899) ∙ 119866119866119954119954120582120582prime
0 (119894119894120596120596119899119899) Similarly for 120582120582 120582120582prime isin 31199031199032
+ 13119903119903 we
have
Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) =
120575120575119954119954119954119954prime
4sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast 119954119954prime
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (119894119894120596120596119899119899) (533)
and all together we have for Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899)
Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) =
⎩⎪⎪⎨
⎪⎪⎧120575120575119954119954119954119954prime
4sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 1 3119903119903
2
120575120575119954119954119954119954prime
4sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast 119954119954prime
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
0 else
(534)
27
533 Dysonrsquos equation Substituting Eqs (521) and (534) into Eq (516) and performing a Fourier transform we have
119866119866119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) = 119866119866119954119954120582120582119954119954prime120582120582prime0 (119894119894120596120596119899119899) + 119866119866119954119954120582120582119954119954prime120582120582prime
(1) (119894119894120596120596119899119899) + 119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) + ⋯
= 1198661198661199541199541205821205820 (119894119894120596120596119899119899)120575120575119954119954119954119954prime120575120575120582120582120582120582prime + 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) ∙ 119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899) + 1198661198661199541199541205821205820 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime
(2) (119894119894120596120596119899119899)
∙ 119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899) + ⋯ = 119866119866119954119954120582120582
0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime120575120575120582120582120582120582prime + 1198661198661199541199541205821205820 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) ∙ 119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899)
or
119866119866119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) = 1198661198661199541199541205821205820 (119894119894120596120596119899119899)120575120575119954119954119954119954prime120575120575120582120582120582120582prime + 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) ∙ 119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899) (535)
Eq (535) is the so-called Dysonrsquos equation (see Ref [2]) where
Σ119954119954120582120582119954119954120582120582prime(119894119894120596120596119899119899) = Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) + Σ119954119954120582120582119954119954prime120582120582prime
(2) (119894119894120596120596119899119899) + ⋯
is called self-energy and Σ119954119954120582120582119954119954120582120582prime(119899119899) (119894119894120596120596119899119899)119899119899 = 12⋯ is the self-energy of nth order approximation Up
to the second order approximation the self-energy is written as
Σ119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) =
⎩⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎧ minus
120575120575119954119954119954119954prime
2
119881119881120582120582120582120582primelowast (119954119954)
120596120596119954119954120582120582120596120596119954119954120582120582prime 1 le 120582120582 le 3119903119903
2lt 120582120582prime le 3119903119903
minus120575120575119954119954119954119954prime
2
119881119881120582120582prime120582120582(119954119954)
120596120596119954119954120582120582120596120596119954119954120582120582prime 1 le 120582120582prime le 3119903119903
2lt 120582120582 le 3119903119903
120575120575119954119954119954119954prime
4sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 1 3119903119903
2
120575120575119954119954119954119954prime
4sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast 119954119954prime
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
(536)
54 Fermirsquos golden Rule
To get the Fermirsquos golden Rule we need to calculate the retarded Greenrsquos function which can be
calculated from 119866119866119954119954120582120582119954119954120582120582prime(119894119894120596120596119899119899) by analytic continuation to the real axis via 119894119894120596120596119899119899 rarr 120596120596 + 119894119894119894119894 with an
infinitesimal positive 119894119894 [12]
119866119866119954119954120582120582119954119954120582120582prime119903119903 (120596120596) equiv 119866119866119954119954120582120582119954119954120582120582prime(119894119894120596120596119899119899 rarr 120596120596 + 119894119894119894119894) 119894119894 rarr 0 (537)
Similarity the retarded self-energy is defined as
Σ119954119954120582120582119954119954120582120582prime119903119903 (120596120596) equiv Σ119954119954120582120582119954119954120582120582prime(119894119894120596120596119899119899 rarr 120596120596 + 119894119894119894119894) 119894119894 rarr 0 (538)
The transition rate of phonons of branch 120582120582 at 119954119954 Γ119954119954120582120582(120596120596) is related to the retarded self energy
Σ119954119954120582120582119903119903 (120596120596) via [13]
Γ119954119954120582120582(120596120596) = minus2ImΣ119954119954120582120582119954119954120582120582119903119903 (120596120596) (539)
Using the expression for 1198661198661199541199541205821205820 (119894119894120596120596119899119899) [Eq (421)] and focused on the second order approximation [13]
we have
28
Σ119954119954120582120582119954119954120582120582119903119903 (119894119894120596120596119899119899) =
⎩⎪⎨
⎪⎧14sum
1198811198811205821205821198951198951prime(119954119954)
2
1205961205961199541199541198951198951prime120596120596119954119954120582120582
31199031199031198951198951prime=
31199031199032 +1
1
120596120596minus1205961205961199541199541198951198951prime +119894119894119894119894
120582120582 isin 1 31199031199032
14sum 1198811198811198951198951120582120582(119954119954)2
1205961205961199541199541198951198951120596120596119954119954120582120582
311990311990321198951198951=1
1120596120596minus1205961205961199541199541198951198951+119894119894119894119894
120582120582 isin 31199031199032
+ 13119903119903
(540)
Using the relation
Im 1
120596120596minus1205961205961199541199541198951198951prime +119894119894119894119894
= minus120587120587120575120575120596120596 minus 1205961205961199541199541198951198951prime (541)
The transition rate reads as
Γ119954119954120582120582(120596120596) =
⎩⎪⎨
⎪⎧1205871205872sum
1198811198811205821205821198951198951prime(119954119954)
2
1205961205961199541199541198951198951prime120596120596119954119954120582120582
31199031199031198951198951prime=
31199031199032 +1
120575120575120596120596 minus 1205961205961199541199541198951198951prime 120582120582 isin 1 31199031199032
1205871205872sum 1198811198811198951198951120582120582(119954119954)2
1205961205961199541199541198951198951120596120596119954119954120582120582
311990311990321198951198951=1
120575120575120596120596 minus 1205961205961199541199541198951198951 120582120582 isin 31199031199032
+ 13119903119903
(542)
Or equivalently
Γ119954119954120582120582(119864119864 = ℏ120596120596) =
⎩⎪⎨
⎪⎧120587120587ℏ3
2sum
1198811198811205821205821198951198951prime(119954119954)
2
1198641198641199541199541198951198951prime119864119864119954119954120582120582
31199031199031198951198951prime=
31199031199032 +1
120575120575119864119864 minus 1198641198641199541199541198951198951prime 120582120582 isin 1 31199031199032
120587120587ℏ3
2sum 1198811198811198951198951120582120582(119954119954)2
1198641198641199541199541198951198951119864119864119954119954120582120582
311990311990321198951198951=1
120575120575119864119864 minus 1198641198641199541199541198951198951 120582120582 isin 31199031199032
+ 13119903119903
(543)
Γ119954119954120582120582(119864119864) represents the probability per unit time of a transition with energy 119864119864 from phonon branch 120582120582
at wave number 119954119954 in subsystem I (II) to a set of phonon branches in subsystem II (I) with the same
wave number Here state 119954119954120582120582 in one subsystem is coupled to state 1199541199541198951198951prime in the other subsystem via
1198811198811205821205821198951198951prime(119954119954) Since the two expressions in Eq (543) are completely equivalent just representing
transitions of opposite directions we consider only the first one in what follows Assuming that
subsystem II sufficiently large such that its density of states is nearly continuous the sum over its
states in the above expression can be replaced by an integral over the energy sum (⋯ )119895119895 rarr
int(⋯ )120588120588119864119864119954119954d119864119864119954119954 giving
Γ119954119954120582120582(119864119864) = 120587120587ℏ3
2 intd119864119864119954119954prime120588120588119864119864119954119954prime 119881119881120582120582119895119895prime(119954119954)
2119864119864119954119954119895119895prime=119864119864119954119954prime
119864119864119954119954prime119864119864119954119954120582120582120575120575119864119864 minus 119864119864119954119954prime = 120587120587ℏ3
2120588120588(119864119864)
119881119881120582120582119895119895prime(119954119954)2119864119864119954119954119895119895prime=119864119864
119864119864119864119864119954119954120582120582 (544)
Eq (544) represents the transition rate of the phonons with energy 119864119864 from branch 120582120582 and wave
number 119954119954 in subsystem I to the continuous manifold of states in subsystem II The total transition
rate between the two subsystems is then given by the sum of transition rates from all states in
subsystem I weighted by their phonon populations to the manifold of states in subsystem II
Assuming that the two subsystems are weakly coupled such that the width of state 119954119954120582120582 in subsystem
I is small and the probability to leave it at an energy other than 119864119864119954119954120582120582 is negligible we can now write
29
total transition rate as
Γtot = 1119885119885119890119890minus120573120573119864119864119954119954120582120582Γ119954119954120582120582119864119864119954119954120582120582
119954119954120582120582
=120587120587ℏ3
2
119890119890minus120573120573119864119864119954119954120582120582
119885119885
120588120588119864119864119954119954120582120582 119881119881120582120582119895119895prime(119954119954)2119864119864119954119954119895119895prime=119864119864119954119954120582120582
1198641198641199541199541205821205822119954119954120582120582
=120587120587ℏ3
2
119890119890minus120573120573119864119864119954119954120582120582
119885119885
120588120588119864119864119954119954120582120582 119881119881120582120582120582120582+31199031199032(119954119954)
2
1198641198641199541199541205821205822119954119954120582120582
Finally we get
Γtot = Γ119954119954120582120582(119864119864) =120587120587ℏ3
2sum 119890119890minus120573120573119864119864119954119954120582120582
119885119885
120588120588119864119864119954119954120582120582119881119881120582120582120582120582+31199031199032(119954119954)
2
1198641198641199541199541205821205822119954119954120582120582 (545)
where 119885119885 = sum 119890119890minus120573120573119864119864119954119954120582120582119954119954120582120582 is the partition function Here we choose the index of phonon branches 120582120582
such that the phonon branch with lower energy has a smaller index ie 1198641198641199541199541205821205821 lt 1198641198641199541199541205821205822 for index 1205821205821 lt
1205821205822 Note that phonon branches 120582120582 and 1198951198951prime belong to the subsystem I (with index from 1 to 31199031199032) and
subsystem II (with index from 1 to 1 + 31199031199032 to 3119903119903) respectively we choose 119895119895prime = 120582120582 + 31199031199032
to make
sure that 119864119864119954119954119895119895prime = 119864119864119954119954120582120582 since phonon energy of two uncoupled system is identical Eq (545) is the
Fermirsquos golden rule for inter-phonon coupling
References
[1] Y Xu J-S Wang W Duan B-L Gu and B Li Nonequilibrium Greens function method for phonon-phonon interactions and ballistic-diffusive thermal transport Phys Rev B 78 224303 (2008) [2] A L Fetter and J D Walecka Quantum theory of many-particle systems (Courier Corporation 2012) [3] G E Stedman Fermis Golden RulemdashAn Exercise in Quantum Field Theory Am J Phys 39 205 (1971)
2
Graphene is considered to be one of the most promising heat dissipating materials in nanoelectronics
[1] due to its ultrahigh in-plane room-temperature thermal conductivity of ~3000-5000 Wm-1K-1 [23]
This however can be hindered by graphene-substrate interactions that may lead to a substantial
reduction of the heat-transport due to phonon leakage across the graphene-substrate interface and
strong interfacial scattering of flexural phonon modes [4] Such undesirable substrate effects can be
reduced by considering multilayer graphene stacks These are expected to effectively isolate the top
graphene layers from the substrate due to the considerably lower cross-plane thermal conductivity
(~68 Wm-1K-1) [5] while exhibiting high in-plane conductivity that can be tuned via the stack
thickness [6-13] Anisotropic thermal conductivity is also observed for bulk hexagonal boron nitride
(h-BN) with the in-plane and cross-plane thermal conductivity in the range of 390-420 Wm-1K-1 and
25-48 Wm-1K-1 respectively [1415]
Efficient is-situ tuning of the thermal conductivity of such graphitic structures can be achieved by
controlling the twist angle between adjacent layers within the stack This has been recently
computationally demonstrated for finite-sized nanoscale few-layer graphene junctions [1617] Two
factors however limit the applicability of these results (i) the simulations were performed using
simplistic isotropic interlayer potentials that are known to be inaccurate for simulating the interlayer
interactions in layered materials [18-21] and (ii) the relevance of the results for large-scale interfaces
is questionable due to significant edge scattering effects inherent to the small finite-sized model
systems studied
To address these issues we investigate the interlayer thermal conductivity of graphene and h-BN
stacks of varying thicknesses and twist angles This allows us to gain fundamental understanding of
the heat transport mechanisms in layered materials stacks and identify feasible means to control it
Our model system consists of two contacting identical AB (AArsquo)-stacked graphite (h-BN) slabs
whose interfacing graphene (h-BN) layers are twisted with respect to each other to create a stacking
fault of misfit angle θ (see Figure 1) Recent experiments demonstrated fine control over the misfit
angle in such setups [2223] The thickness of the entire construction is varied between 27 nm-35 nm
(8-104 layers) and periodic boundary conditions are applied in all directions Heat transport
simulations are performed using state-of-the-art anisotropic interlayer potentials [18-21] applied to
the twisted stacks These potentials were shown to capture well the structural dynamic heat
dissipation and phonon spectrum of graphitic and h-BN layered systems [24-27] A thermal bias is
induced by applying Langevin thermostats with different temperatures to two layers residing on
opposite sides away from the twisted interface (see Sec 1 of the Supporting Information (SI) for
further details)
3
Figure 1 Schematic representation of the simulation setup Two identical AB-stacked graphite slabs
(gray and orange respectively) are twisted with respect to each other to create a stacking fault of misfit
angle θ A thermal bias is induced by applying Langevin thermostats to the two layers marked by
dashed red (Thot) and green (Tcold) rectangles The arrows indicate the direction of the vertical heat
flux Since periodic boundary conditions are applied also in the vertical direction two twisted
interfaces are shown across which heat flows in opposite directions
We start by studying the effect of the misfit angle on the cross-plane thermal conductivity of the
twisted graphite and h-BN stacks Figure 2 presents the dependence of the cross-plane thermal
conductivity of the entire stack on the misfit angle for model systems consisting of 8 (red circles) and
16 (black triangles) layers for (a) graphite and (b) h-BN A pronounced dependence of the cross-plane
thermal conductivity (120581120581CP) of the entire graphitic stack is clearly evident which above a misfit angle
of sim 5deg 120581120581CP drops by a factor of 3-4 with respect to the value obtained for the aligned contact
Similar misfit-angle dependence of 120581120581CP is obtained for twisted bilayer graphene (tBLG) using the
transient MD simulation approach (see Sec 2 of the SI) We note that this sharp drop for graphite is
steeper and that the overall reduction is higher than those previously obtained using Lennard-Jones
interlayer potentials in finite model systems [1617] The corresponding cross-plane thermal
conductivity of the commensurate h-BN stack is found to be approximately double that of graphite
for the same number of layers Notably it reduces more gradually with the twist angle and saturates
at sim 15deg with an overall two-three fold reduction
The thermal conductivity of both graphite and h-BN stacks is found to increase when doubling their
thickness To identify the source of this thickness dependence we plot in Figure 2(c-d) the interfacial
thermal resistance (ITR) (see Sec 12 of the SI for the definition) associated with the twisted junction
formed between the contacting graphene or h-BN layers of the two optimally-stacked slabs Note that
unlike 120581120581CP which measures the conductivity of the entire stack the ITR corresponds to the heat
transport resistance of the two adjacent layers forming the twisted interface Two important
4
observations can be made (i) the ITR is weakly dependent on the stack thickness indicating that the
thickness dependence arises from the conductivity of the optimally-stacked interfacing slabs
Specifically in the thickness range considered the heat conductivity grows with slab thickness due to
reduction of phonon-phonon interactions and increased contribution of long wave-length phonons
below the mean-free path [28-30] (ii) the ITR strongly depends on the twist angle demonstrating a
~10-fold (4-fold) increase when the twist angle at the graphene (h-BN) interface is varied from 0deg
to 15deg This clearly indicates that the twist angle can be utilized to control the cross-plane thermal
conductivity of hexagonal two-dimensional (2D) materials and to effectively thermally isolate the top
layers from the underlying substrate
Figure 2 Twist-angle dependence of the cross-plane thermal conductivity of the entire stack (a b)
and the interfacial thermal resistance (c d) of the twisted contact formed between the optimally-
stacked slabs of graphite (a c) and bulk h-BN (b d) respectively Red circles and black triangles
correspond to the results obtained using 8 and 16 layer models respectively
The strong dependence of the cross-plane thermal conductivity of graphene and h-BN on the stacking
fault twist angle is related to the degree of coupling between the phonon modes of the two contacting
layers at the twisted interface Note that the term ldquocouplingrdquo used herein is not related to the standard
notion of phonon-phonon couplings due to anharmonic effects Instead we regard to the off-diagonal
terms of the Hessian when represented in the basis of the harmonic phonon modes of the isolated
5
layers To demonstrate this we write the dynamical matrix (the mass-reduced Fourier transform of
the force constant matrix) in block form as follows
120625120625(119954119954) = 12062512062511(119954119954) 12062512062512(119954119954)12062512062521(119954119954) 12062512062522(119954119954) (1)
where 12062512062511(119954119954) and 12062512062522(119954119954) are the block matrices relating to the first and second layer and 12062512062512(119954119954)
and 12062512062521(119954119954) = 12062512062512dagger (119954119954) all evaluated at wave-vector 119954119954 The interlayer phonon-phonon couplings are
obtained by diagonalizing separately 12062512062511(119954119954) and 12062512062522(119954119954) such that 12062512062511(119954119954) =
1199321199321dagger(119954119954)12062512062511(119954119954)1199321199321(119954119954) and 12062512062522(119954119954) = 1199321199322
dagger(119954119954)12062512062522(119954119954)1199321199322(119954119954) are diagonal matrices containing the
frequencies (120596120596119894119894) of the phonon modes of the two layers and 1199321199321(119954119954) and 1199321199322(119954119954) are unitary
matrices of the corresponding eigenvectors We now construct a global block diagonal transformation
matrix of the form
119932119932(119954119954) = 1199321199321(119954119954) 120782120782120782120782 1199321199322(119954119954) (2)
and transform the full dynamical matrix as follows
119932119932dagger(119954119954)120625120625(119954119954)119932119932(119954119954) = 12062512062511(119954119954) 12062512062512(119954119954)12062512062521(119954119954) 12062512062522(119954119954)
(3)
where 12062512062512(119954119954) = 1199321199321dagger(119954119954)12062512062512(119954119954)1199321199322(119954119954) and 12062512062521(119954119954) = 12062512062512
dagger (119954119954) are the interlayer phonon-phonon
coupling blocks Naturally when the two layers are infinitely separated these coupling blocks vanish
and the diagonal blocks converge to those of the isolated layers
The overall coupling between the two layers can be obtained from the individual phonon-phonon
coupling matrix elements via Fermirsquos golden rule [31] which reads as (see Sec 4 of SI for a detailed
derivation)
Γtot = 120587120587ℏ3
2sum 119890119890minus120573120573119864119864119954119954120582120582
119885119885
120588120588119864119864119954119954120582120582119881119881120582120582120582120582+31199031199032(119954119954)
2
1198641198641199541199541205821205822119954119954120582120582 (4)
where 119885119885 = sum 119890119890minus120573120573119864119864119954119954120582120582119954119954120582120582 is the partition function 119864119864119954119954120582120582 is the energy of phonons at branch 120582120582 with
wave number 119954119954 120588120588119864119864119954119954120582120582 is the density-of-states (DOS) at 119864119864119954119954120582120582 and 119881119881120582120582120582120582+31199031199032(119954119954)
2 is the coupling
matrix element between branches of phonons of similar energy in the two layers whose number of
atoms in one unit cell is 119903119903
Using Eq (4) we can rationalize the misfit angle dependence of the heat flux across the twisted
interface from the calculated inter-phonon coupling To that end we performed room temperature
(300K) simulations (technical details can be found in Sec 3 of SI) for tBLG with different misfit
angles using the Greenrsquos function molecular dynamics (GFMD) developed by Kong et al [32] as
6
implemented in LAMMPS [33] The simulations allow us to evaluate the dynamical matrix from
which the phonon-phonon couplings can be extracted (see details in Sec 3 of SI) and the overall heat
transfer rate calculated Figure 3 shows the resulting heat transfer rate (normalized to is value for the
aligned contact (120579120579 = 0) ) as a function of the misfit angle compared to the interfacial thermal
conductivity defined as the inverse of the ITR presented in Figure 2(c) ITC equiv 1ITR The
remarkable agreement between the calculated interfacial thermal conductivity and Fermirsquos golden
rule results indicate that the dependence of the interlayer phonon-phonon couplings on the misfit
angle is responsible for the strong angle dependence of the interfacial conductivity Notably the sharp
heat conductivity drop at misfit angles in the range of 0deg-5deg as well as the small conductivity for larger
misfit angles are well captured by Fermirsquos golden rule
Figure 3 Comparison between Fermirsquos golden rule results (open blue squares) for the interfacial heat-
transfer rate of a tBLG and the calculated interfacial thermal conductivity at various misfit angles
ITC simulation results are presented for both 8 layers (open red circles) and 16 layers (open black
triangles) showing similar behavior For comparison purposes all data sets are normalized to their
value obtained for the aligned contact
To correlate our results with experimentally measured thermal conductivities that are often obtained
for thick samples we repeated our calculations for increasing stack thicknesses at fixed misfit angles
Figure 4 presents results for the calculated heat conductivity of (a) graphite and (b) h-BN stacks either
aligned (open red circles) or twisted by 120579120579 = 3016deg (open black diamond symbols) as a function of
number of layers in the stack As discussed above for both systems the misoriented stack exhibits
lower heat conductivity compared to the aligned system however its thickness dependence is
considerably stronger This can be attributed to the significantly higher interface resistance of the
twisted interface that when plugged in Eq (S2) of the SI for the overall conductivity induces stronger
7
thickness dependence
Comparing our calculated heat conductivities for the aligned contact (open red circles) to available
experimental data for ~35 nm thick graphite slabs [34] (dashed green line) we find that at the thickest
model system considered of 104 graphene layers (~34 nm thick) the calculated value of 085plusmn005
W m sdot Kfrasl is in remarkable agreement with the measured value of ~07 W m sdot Kfrasl Furthermore
experimental values for bulk graphite [5] indicate that the thermal conductivity continues to grow up
to ~68 W m sdot Kfrasl (black dash-dotted line) which is consistent with the general trend of the calculated
heat conductivity that does not saturate for the thickest model system considered These results
strongly enforce the validity of our force-field and model systems to model the heat conductivity of
twisted layered materials interfaces Available experimental results for the heat conductivity of bulk
h-BN are marked by the dashed-dotted black and dashed-green lines in Figure 4(b) In line with our
findings for the graphitic interface our calculated finite slab heat conductivities for the aligned
interface (open red circles) continue to grow with the number of layers and are consistently below the
bulk value
Figure 4 Thickness dependence of the thermal conductivity 120581120581CP of aligned (open red circles) and
twisted by 3016deg (open black diamond symbols) graphite (a) and h-BN (b) stacks Blue squares
represent results obtained using the isotropic Lennard-Jones potential for the aligned contacts The
green dashed and black dash-dotted lines represent experimental results measured for graphite (a)
(Refs [534]) and bulk h-BN (b) (Refs [1415]) Note that both axis scales are logarithmic Error bar
estimation procedure is discussed in Sec 1 of the SI
Another important factor that may affect the interlayer thermal transport properties of 2D material
stacks is the average temperature of the system which was taken to be ~300K in all abovementioned
simulations To evaluate the sensitivity of our results towards this parameter we repeated the heat
conductivity and interfacial resistance calculations of optimally stacked graphite and h-BN stacks for
an average temperature of 400 K The results presented in see Sec 4 of the SI indicate that 120581120581CP and
8
ITR weakly depend on the average temperature over the entire thickness range considered This
conclusion is consistent with recent experimental findings [11] We may attribute this to the fact that
the thickness of the graphite and h-BN stack model considered is much smaller than their phonon
mean-free path (~200 nm for graphite [1011133536] and ~100 nm for bulk h-BN [15]) such that
the phonon transport is dominated by phonon-boundary scattering which weakly depends on
temperature in the range considered Therefore temperature dependent Umklapp processes have only
marginal contribution to our results [11]
Finally we note that previous calculations of the heat conductivity of twisted graphitic interfaces
relied on Lennard-Jones (LJ) potentials describing the interlayer interactions [8-1013] To
demonstrate the importance of using registry-dependent interlayer potentials we have repeated our
calculations of the heat conductivity of graphitic slabs with the REBO intralayer potential augmented
by LJ interlayer interactions [37] (120576120576 = 284 meV120590120590 = 34 Å ) We find that the calculated heat
conductivities obtained using the LJ interlayer potential are consistently higher than those obtained
by our ILP and that the difference between them grows with the model system thickness Notably the
heat conductivity obtained using the LJ potential for a graphitic slab of thickness ~34 nm is 154
W m sdot Kfrasl overestimating the experimental value by more than a factor of 2
The excellent agreement of our ILP calculations with experimental data of nanoscale graphitic stacks
therefore demonstrates the reliability of our predictions for the strong interfacial misfit angle
dependence of cross-layer thermal conductivity in graphite and h-BN The observed sharp
conductivity decrease of twisted graphitic interfaces at misfit angles lt 5deg opens the way to control
the thermal evacuation rate and thermal isolation of active layers in graphene-based electronic and
mechanical devices The revealed underlying mechanism suggests that design rules can be obtained
by carefully tailoring the phonon-phonon couplings across the twisted interface While the misfit
angle dependence of h-BN is found to be weaker than that of graphite the overall thermal
conductivity of the former is found to be higher This may be utilized to achieve higher conductivity
and controllability in twisted heterogeneous junctions of layered materials
ASSOCIATED CONTENT
Supporting Information
The supporting Information section includes a description of the methodology thermal conductivity
of twisted bilayer graphene theory for calculating the phonon coupling of twisted bilayer graphene
9
derivation of Fermirsquos golden rule and temperature dependence of the cross-plane thermal
conductivity
AUTHOR INFORMATION
Corresponding Author
E-mail urbakhtauextauacil
Notes
The authors declare no competing financial interest
ACKNOWLEDGMENTS
The authors would like to thank Prof Abraham Nitzan Dr Guy Cohen and Dr Yiming Pan for
helpful discussions WO acknowledges the financial support from a fellowship program for
outstanding postdoctoral researchers from China and India in Israeli Universities and the support from
the National Natural Science Foundation of China (Nos 11890673 and 11890674) HQ
acknowledges the financial support from the National Natural Science Foundation of China (No
11890674) MU acknowledges the financial support of the Israel Science Foundation Grant
No 114118 and the ISF-NSFC joint grant 319119 OH is grateful for the generous financial
support of the Israel Science Foundation under grant no 158617 and the Naomi Foundation for
generous financial support via the 2017 Kadar Award This work is supported in part by COST Action
MP1303
10
References
[1] A A Balandin Thermal properties of graphene and nanostructured carbon materials Nat Mater 10 569 (2011) [2] S Ghosh I Calizo D Teweldebrhan E P Pokatilov D L Nika A A Balandin W Bao F Miao and C N Lau Extremely high thermal conductivity of graphene Prospects for thermal management applications in nanoelectronic circuits Appl Phys Lett 92 151911 (2008) [3] A A Balandin S Ghosh W Bao I Calizo D Teweldebrhan F Miao and C N Lau Superior thermal conductivity of single-layer graphene Nano Lett 8 902 (2008) [4] J H Seol et al Two-Dimensional Phonon Transport in Supported Graphene Science 328 213 (2010) [5] R Taylor The thermal conductivity of pyrolytic graphite J Philosophical Magazine 13 157 (1966) [6] S Ghosh W Bao D L Nika S Subrina E P Pokatilov C N Lau and A A Balandin Dimensional crossover of thermal transport in few-layer graphene Nat Mater 9 555 (2010) [7] Z Wang R Xie C T Bui D Liu X Ni B Li and J T L Thong Thermal Transport in Suspended and Supported Few-Layer Graphene Nano Lett 11 113 (2011) [8] Z Wei Z Ni K Bi M Chen and Y Chen Interfacial thermal resistance in multilayer graphene structures Phys Lett A 375 1195 (2011) [9] Y Ni Y Chalopin and S Volz Significant thickness dependence of the thermal resistance between few-layer graphenes Appl Phys Lett 103 061906 (2013) [10] J Chen J H Walther and P Koumoutsakos Strain Engineering of Kapitza Resistance in Few-Layer Graphene Nano Lett 14 819 (2014) [11] Q Fu J Yang Y Chen D Li and D Xu Experimental evidence of very long intrinsic phonon mean free path along the c-axis of graphite Appl Phys Lett 106 031905 (2015) [12] G Fugallo A Cepellotti L Paulatto M Lazzeri N Marzari and F Mauri Thermal Conductivity of Graphene and Graphite Collective Excitations and Mean Free Paths Nano Lett 14 6109 (2014) [13] Z Wei J Yang W Chen K Bi D Li and Y Chen Phonon mean free path of graphite along the c-axis Appl Phys Lett 104 081903 (2014) [14] E K Sichel R E Miller M S Abrahams and C J Buiocchi Heat capacity and thermal conductivity of hexagonal pyrolytic boron nitride Phys Rev B 13 4607 (1976) [15] P Jiang X Qian R Yang and L Lindsay Anisotropic thermal transport in bulk hexagonal boron nitride Phys Rev Mater 2 064005 (2018) [16] M-H Wang Y-E Xie and Y-P Chen Thermal transport in twisted few-layer graphene Chinese Physics B 26 116503 (2017) [17] X Nie L Zhao S Deng Y Zhang and Z Du How interlayer twist angles affect in-plane and cross-plane thermal conduction of multilayer graphene A non-equilibrium molecular dynamics study Int J Heat Mass Tran 137 161 (2019) [18] A N Kolmogorov and V H Crespi Registry-dependent interlayer potential for graphitic systems Phys Rev B 71 235415 (2005) [19] M Reguzzoni A Fasolino E Molinari and M C Righi Potential energy surface for graphene on graphene Ab initio derivation analytical description and microscopic interpretation Phys Rev B 86 (2012) [20] M M van Wijk A Schuring M I Katsnelson and A Fasolino Moire Patterns as a Probe of Interplanar Interactions for Graphene on h-BN Phys Rev Lett 113 135504 (2014) [21] D Mandelli W Ouyang M Urbakh and O Hod The Princess and the Nanoscale Pea Long-Range Penetration of Surface Distortions into Layered Materials Stacks ACS Nano 13 7603 (2019) [22] E Koren E Loumlrtscher C Rawlings A W Knoll and U Duerig Adhesion and friction in mesoscopic graphite contacts Science 348 679 (2015) [23] E Koren I Leven E Loumlrtscher A Knoll O Hod and U Duerig Coherent commensurate electronic states at the interface between misoriented graphene layers Nat Nanotechnol 11 752 (2016)
11
[24] W Ouyang I Azuri D Mandelli A Tkatchenko L Kronik M Urbakh and O Hod Mechanical and Tribological Properties of Layered Materials under High Pressure Assessing the Importance of Many-Body Dispersion Effects J Chem Theory Comput 16 666 (2020) [25] W Ouyang D Mandelli M Urbakh and O Hod Nanoserpents Graphene Nanoribbon Motion on Two-Dimensional Hexagonal Materials Nano Lett 18 6009 (2018) [26] I Leven T Maaravi I Azuri L Kronik and O Hod Interlayer Potential for Grapheneh-BN Heterostructures J Chem Theory Comput 12 2896 (2016) [27] T Maaravi I Leven I Azuri L Kronik and O Hod Interlayer Potential for Homogeneous Graphene and Hexagonal Boron Nitride Systems Reparametrization for Many-Body Dispersion Effects J Phys Chem C 121 22826 (2017) [28] L H Liang and B Li Size-dependent thermal conductivity of nanoscale semiconducting systems Phys Rev B 73 153303 (2006) [29] P K Schelling S R Phillpot and P Keblinski Comparison of atomic-level simulation methods for computing thermal conductivity Phys Rev B 65 144306 (2002) [30] J Shiomi Nonequilirium molecular dynamics methods for lattice heat conduction calculations Annual Review of Heat Transfer 17 (2014) [31] G E Stedman Fermis Golden RulemdashAn Exercise in Quantum Field Theory Am J Phys 39 205 (1971) [32] L T Kong G Bartels C Campantildeaacute C Denniston and M H Muumlser Implementation of Greens function molecular dynamics An extension to LAMMPS Comput Phys Commun 180 1004 (2009) [33] S Plimpton Fast Parallel Algorithms for Short-Range Molecular Dynamics J Comput Phys 117 1 (1995) [34] M Harb et al The c-axis thermal conductivity of graphite film of nanometer thickness measured by time resolved X-ray diffraction Appl Phys Lett 101 233108 (2012) [35] H Zhang X Chen Y-D Jho and A J Minnich Temperature-Dependent Mean Free Path Spectra of Thermal Phonons Along the c-Axis of Graphite Nano Lett 16 1643 (2016) [36] D P Sellan E S Landry J E Turney A J H McGaughey and C H Amon Size effects in molecular dynamics thermal conductivity predictions Phys Rev B 81 214305 (2010) [37] S J Stuart A B Tutein and J A Harrison A reactive potential for hydrocarbons with intermolecular interactions J Chem Phys 112 6472 (2000)
S1
Supporting information for ldquoControllable Thermal Conductivity of
Twisted Graphene and Hexagonal Boron Nitride Interfacesrdquo
Wengen Ouyang1 Huasong Qin2 Michael Urbakh1 and Oded Hod1
1Department of Physical Chemistry School of Chemistry The Raymond and Beverly
Sackler Faculty of Exact Sciences and The Sackler Center for Computational
Molecular and Materials Science Tel Aviv University Tel Aviv 6997801 Israel
2State Key Laboratory for Strength and Vibration of Mechanical Structures School of
Aerospace Xirsquoan Jiaotong University Xirsquoan 710049 China
The Supporting information includes following sections
1 Methodology
2 Thermal conductivity of twisted bilayer graphene
3 Theory for calculating the phonon coupling of twisted bilayer graphene
4 Derivation of Fermirsquos golden rule
5 Temperature dependence of cross-plane thermal conductivity
S2
1 Methodology
11 Model system
The initial interlayer distance across the layered stack was set equal to 34 Aring and 33 Aring for graphite
and bulk h-BN respectively Periodic boundary conditions were applied in all directions It should be
noted that the lattice structure is rigorously periodic only at some specific twist angles the values of
which are listed in Table S1 in section 3 below While the cross-sectional area for each misfit angle
θ is different all systems considered have a contact area exceeding 12 nm2 which was shown to
provide converged results with respect to unit-cell dimensions [1] The intralayer interactions within
each graphene and h-BN layer were modeled via the second generation REBO potential [2] and
Tersoff potential [3] respectively The interlayer interactions between the layers of graphite and bulk
h-BN were described via our dedicated interlayer potential (ILP) [4] which is implemented in the
LAMMPS [5] suite of codes [6]
Fig S1 Schematic representation of the simulation setup (a) and steady-state temperature profile (b) respectively In panel (a) two identical AB-stacked graphite slabs (gray and orange respectively) are twisted with respect to each other to create a stacking fault of misfit angle θ A thermal bias is induced by applying Langevin thermostats to the two layers marked by dashed red (Thot) and green (Tcold) rectangles The arrows indicate the direction of the vertical heat flux Since periodic boundary conditions are applied also in the vertical direction two twisted interfaces are shown across which heat flows in opposite directions The steady-state temperature profiles are illustrated in panel (b) where N is the total number of layers in the model system and RAB dAB and Rθ and dθ mark the interfacial Kapitza resistance [89] and interlayer distance for contacting graphene layers with AB-stacking and misfit angle θ respectively The red lines in panel (b) mark the temperature variation across the twisted interface where the vertical axis corresponds to the position of the various layers along the stack and the horizontal axis marks the temperature of the various layers
S3
12 Simulation Protocol
All MD simulations were performed with the LAMMPS simulation package [5] The velocity-Verlet
algorithm with a time-step of 05 fs was used to propagate the equations of motion A Noseacute-Hoover
thermostat with a time constant of 025 ps was used for constant temperature simulations To maintain
a specified hydrostatic pressure the three translational vectors of the simulation cell were adjusted
independently by a Noseacute-Hoover barostat with a time constant of 10 ps [7] To relax the box we first
equilibrated the systems in the NPT ensemble at a temperature of T = 300 K and zero pressure for
250 ps (see Fig S2) After equilibration Langevin thermostats with damping coefficients 10 ps-1
were applied to the bottom and middle layer of the graphene stack (see Fig S1) with target
temperature Thot = 375 K (hot reservoir) and Tcold = 225 K (cold reservoir) respectively Then the
system was allowed to reach steady-state over a subsequent simulation period of 750 ps (see Fig S2)
during which the dynamics of all non-thermostated layers followed the NVE ensemble For the larger
model systems the length of the NPT and Langevin stages was doubled (for the 32 and 48 layers
systems) or tripled (for the 104 layers graphitic system) to ensure convergence of the obtained steady-
state Once steady-state was obtained the last 500 ps were used to calculate the thermal conductivity
of the twisted graphite and bulk h-BN The statistical errors were estimated using ten different data
sets each calculated over a time interval of 50 ps
Fig S2 Time evolution of the temperature of thermostated layers for 16 layers twisted graphite with misfit angle (a) θ = 0deg (b) θ = 509deg (c) θ = 1518deg and (d) θ = 3016deg Note that the thermal fluctuations increase with increasing the misfit angle due to the growing interfacial thermal resistance that enhances phonon back scattering at the twisted junction
S4
13 Calculation of the interfacial thermal resistance
According to Fourierrsquos law the cross-plane thermal conductivity (120581120581CP) of a twisted graphitic interface
of misfit angle θ can be calculated as
120581120581CP = 119876119860119860Δ119879119879Δ119911119911
(S1)
where 119876 is the heat flux A is the cross-section area and Δ119879119879Δ119911119911 is the temperature gradient along
the direction of heat flux (perpendicular to the basal plane in our case) Fig S1(b) shows a schematic
temperature profile along the z-direction where the vertical axis corresponds to the position of the
various layers along the stack and the horizontal axis marks the temperature of the various layers The
actual temperature profiles extracted from the NEMD simulations for twisted graphite with different
number of layers can be found in Fig S3 For Bernal-stacked graphite (ie θ = 0deg red circles in Fig
S3) only the linear region of the temperature profile was used to calculate 120581120581CP and the points
corresponding to the layers where the thermostats were applied were omitted (marked with green
triangle in Fig S3) The 120581120581CP of the system was calculated using Eq (S1) by averaging over the two
linear regions of the temperature profiles For the twisted case (θ ne 0deg) we found a sudden
temperature decrease Δ119879119879120579120579 at the position of the twisted interface (see black squares in Fig S3) 120581120581CP
in this case was calculated using the temperature gradient calculated for the same layer range as that
for θ = 0deg To characterize the thermal properties of the twisted interface the concept of interfacial
thermal resistance (ITR) ie Kapitza resistance [89] was introduced According the definition of
the Kapitza resistance [8] 119877119877 = 119860119860Δ119879119879119876 and noticing that Δ119879119879tot = (1198731198732 minus 3)Δ119879119879AB + Δ119879119879120579120579 and
Δ119911119911 = (1198731198732 minus 3)119889119889AB + 119889119889120579120579 Eq (1) can be rewritten considering two-resistors in series as [see Fig
S1(b)]
(1198731198732 minus 3)119877119877AB + 119877119877120579120579 = [(1198731198732 minus 3)119889119889AB + 119889119889120579120579] 120581120581CPfrasl (S2)
Here 119877119877AB 119889119889AB Δ119879119879AB and 119877119877120579120579 119889119889120579120579 Δ119879119879120579120579 are the ITR interlayer distance and temperature
difference for adjacent AB-stacked and twisted graphene layers respectively For the aligned contact
(120579120579 = 0deg) the ITR can be simply calculated as 119877119877AB = 119889119889AB120581120581AB Once 119877119877AB is known 119877119877120579120579 can be
calculated from Eq (S2) We note that the sharp temperature drop at the twisted interface indicates
that 119877119877120579120579 should be much larger than 119877119877AB
S5
Fig S3 Temperature profiles for graphitic stacks consisting of (a) 8 and (b) 16 layers The red circles and black squares represent the temperature profiles for the aligned (120579120579 = 0deg) and twisted (120579120579 =3016deg) junctions respectively Green triangles represent data points that were omitted in the 120581120581CP calculation
2 Thermal conductivity of twisted bilayer graphene
As comparison we also calculated the interfacial thermal conductivity (ITC) and ITR of twisted
bilayer graphene (tBLG) with the transient MD simulation approach [15-17] since the NEMD
simulation protocol used in the main text becomes invalid in this case In this protocol the system
was first equilibrated within the NPT ensemble at T = 200 K and zero pressure for 100 ps which was
followed by a 100 ps NVT ensemble equilibration stage and a 100 ps of NVE ensemble equilibration
stage After the system reached steady-state an ultrafast heat impulse was imposed on the top layer
of the t-BLG for 50 fs to increase the temperature of the top layer from 200 K to 400 K while that of
bottom layer of tBLG remained unchanged After the external heat source was removed thermal
energy flowed from the top layer to the bottom layer due to the temperature difference and the
temperature of both layers approached 300 K when quasi-steady-state was reached During the
thermal relaxation time interval (500 ps) the temperature and energy of the system sections were
recorded The ITR could then be extracted using the following equation [15-17]
part119864119864t120597120597120597120597
= 119860119860119877119877119879119879bot(119905119905) minus 119879119879top(119905119905) (S3)
where 119864119864t is the total energy of the top graphene layer R is the ITR of the tBLG A is the interfacial
cross-section area and 119879119879bot and 119879119879top are the instantaneous temperatures measured for the bottom
and top layers of the tBLG respectively Note that in Eq S3 we assume a linear dependence of the
heat flux on the temperature difference between the layers The ITC of the tBLG is simply defined as
ITC equiv 119889119889ITR where d is the average interlayer distance
S6
The ITC and ITR of tBLG as functions of misfit angle calculated with the transient MD simulation
protocol are illustrated in Fig S4 demonstrating similar misfit-angle dependence as that for the
NEMD protocol with Langevin thermostat exercised to obtain the results presented in the main text
This further validates the reliability of the simulation protocol adopted in the main text which is more
suitable to treat thick slabs and allows to obtain a true stead-state
Fig S4 Misfit-angle dependence of (a) ITC and (b) ITR for a twisted bilayer graphene obtained using the transient MD simulation approach
3 Theory for calculating the phonon-phonon coupling of twisted bilayer graphene
31 Brillouin Zone of supercell in tBLG
For tBLG the lattice structure is rigorously periodic only at some specific misfit angles θ where the
lattice vector 1199231199231 = 11989911989911199381199381 + 11989911989921199381199382 in the bottom layer equals the vector 1199231199232 = 11989811989811199381199381 + 11989811989821199381199382 in the
top layer with certain integers m1 m2 and n1 n2 Here 1199381199381 = 119886119886(10) and 1199381199382 = 11988611988612radic32 are
the primitive lattice vectors of the bottom layer and a is the lattice constant of monolayer graphene
Thus the exact superlattice period is then given by [18]
119871119871 = |11989911989911199381199381 + 11989911989921199381199382| = 11988611988611989911989912 + 11989911989922 + 11989911989911198991198992 = |1198991198991minus1198991198992|1198861198862 sin(1205791205792) (S4)
where θ is the angle between two lattice vectors 1199231199231 and 1199231199232 In the simulations below we always
rotated the supercell such that its lattice vector is 1199231199231 = 119871119871(10) and 1199231199232 = 11987111987112radic32 In this case
the corresponding reciprocal lattice vector of the moireacute superlattice satisfies the relation 119918119918119894119894 ∙ 119923119923119895119895 =
2120587120587120575120575119894119894119895119895 such that
1199181199181 = 4120587120587radic3119871119871
radic32
minus12 1199181199182 = 4120587120587
radic3119871119871(01) (S5)
S7
Both the lattice vectors and the corresponding reciprocal lattice vectors of the superlattice of the tBLG
are presented in Fig S5 Table S1 reports the parameters used to construct rhombus periodic
supercells of different misfit angles that can be duplicated to construct a rectangular periodic supercell
Table S1 The parameters used to construct periodic supercells of various misfit angles
120579120579 (deg) 119860119860 (nm2) 1198991198991 1198991198992 1198981198981 1198981198982
0 60383688 24 0 24 0
0696407 709613169 48 47 47 48
1121311 273718420 30 29 29 30
2000628 85648391 17 16 16 17
3006558 152322047 23 21 21 23
4048894 189013524 26 23 23 26
5085849 53255058 14 12 12 14
7926470 49376245 14 11 11 14
9998709 86277388 19 14 14 19
15178179 31554670 15 4 4 15
19932013 42876612 15 8 8 15
25039660 13942761 9 4 4 9
30158276 18974735 11 4 4 11
32204228 21805221 12 4 4 12
32 Special points for Brillouin Zone integration
The calculation of the sum over wave vector q in Eq (4) in the main text can be transformed to an
integral using the relation sum (⋯ )119954119954 = 1119881119881119887119887int (⋯ )BZ d119954119954 where 119881119881119887119887 = (2120587120587)3119881119881 is the volume of the
Brillouin Zone (BZ) and V is volume of the real-space unit-cell The calculation of integral is usually
inefficient since it requires calculating the value of the function over a large set of k points in the first
BZ To calculate such integrations more efficiently simple k-point meshes can be replaced by a
carefully selected set of special points in the BZ 119954119954119894119894 [19-22] over which the function is evaluated
The integral can then be estimated via
119868119868 = 1119881119881119887119887int 119891119891(119954119954)BZ d119954119954 asymp 1
119873119873sum 119908119908119894119894119891119891(119954119954119894119894)119894119894 (S6)
S8
where 119881119881119887119887 is the volume of the BZ 119908119908119894119894 is the weight of the ith data point and N normalizes the
weighting factors to unity 119873119873 = sum 119908119908119894119894119894119894 The set of selected 119954119954119894119894 forms a grid in the irreducible
Brillouin zone (IBZ) as is illustrated by the red points in Fig S5(b) The coordinates of these points
for a hexagonal lattice are presented in Eq (S7)
Fig S5 (a) Twisted bilayer graphene of misfit angle θ = 509deg L1 and L2 are the superlattice vectors (b) The corresponding first Brillouin Zone of (a) G1 and G2 are the reciprocal lattice vectors of the superlattice The triangle ΔΓMK represents the irreducible Brillouin zone Red circles mark the position of the special points used to evaluate the integral over the first Brillouin Zone
⎩⎪⎨
⎪⎧ 1199541199541 = 1
9 1205971205973 1199541199542 = 2
9 1205971205973 1199541199543 = 1
18 512059712059718
1199541199544 = 518
512059712059718 1199541199545 = 1
9 21205971205979 1199541199546 = 2
9 21205971205979
1199541199547 = 118
312059712059718 1199541199548 = 1
9 1205971205979 1199541199549 = 1
18 12059712059718
(S7)
Here 119905119905 = radic3 and the unit of the coordinates is 2120587120587119871119871 The weighting factors 119908119908119894119894 are [19]
119908119908124689 = 112
119908119908357 = 16 (S8)
Using Eqs (S6-S8) Eq (4) in the main text can be evaluated as follows
Γtot = 120587120587ℏ3
2sum 119908119908119896119896 sum
119890119890minus120573120573119864119864119954119954119948119948120582120582
119885119885
120588120588119864119864119954119954119948119948120582120582119881119881120582120582120582120582+31199031199032(119954119954119948119948)
2
1198641198641199541199541199481199481205821205822120582120582
9119896119896=1 (S9)
This equation was used to calculate the transition rate presented in Fig 3 in the main text
4 Derivation of Fermirsquos golden rule
The derivation of Fermirsquos golden rule is provided in a separate file (Fermirsquos Golden Rule for
phononspdf) due to its extent
S9
5 Temperature dependence of interfacial thermal conductivity
In the main text the target temperatures of the Langevin thermostats for the bottom and middle layers
of graphene and h-BN were set to 225 K and 375 K respectively After reaching the steady-state the
average temperature of the system was found to be ~300 K To check the effect of average temperature
on our results we calculated the cross-plane thermal conductivity (120581120581CP ) and the corresponding
interfacial thermal resistance (ITR) at a different temperature gradient (325 K ndash 475 K) resulting the
average steady-state temperature of ~400K The protocol described in Section 1 above was used to
perform these calculations as well Both ILP and Lennard-Jones (LJ) potential were tested for
graphite whereas for the bulk h-BN simulations only the ILP was used The results for graphite and
bulk h-BN are illustrated in Fig S6 and Fig S7 respectively For the ILP we find that the overall
values of 120581120581CP (ITR) decrease (increase) slightly with increasing average temperature which is
consistent with a recent experiment [10] The LJ potential calculations as well exhibit very week
dependence on average temperature within the range studied Altogether the layer dependences of
both quantities remain mostly insensitive to the average temperature The reason is that the thickness
of the graphite and h-BN model systems was chosen to be much smaller than their phonon mean free
path (~200 nm for graphite [110-13] and ~100 nm for bulk h-BN [14]) such that phonon transport is
dominated by phonon-boundary scattering and the Umklapp process only makes marginal
contribution [10]
S10
Fig S6 Layer dependence of 120581120581CP (a c) and ITR (b d) for Bernal-stacked graphite at average steady-state temperatures of 300 K (red circles) and 400 K (black squares) The left and right columns correspond to the 120581120581CP and ITR calculated with ILP and LJ potential respectively
Fig S7 Layer dependence of 120581120581CP (a) and ITR (b) for AArsquo-stacked h-BN at average temperatures of 300 K (red circles) and 400 K (blue squares)
S11
References [1] Z Wei J Yang W Chen K Bi D Li and Y Chen Phonon mean free path of graphite along the c-axis Appl Phys Lett 104 081903 (2014) [2] D W Brenner O A Shenderova J A Harrison S J Stuart B Ni and S B Sinnott A second-generation reactive empirical bond order (REBO) potential energy expression for hydrocarbons J Phys Condens Matter 14 783 (2002) [3] A Kınacı J B Haskins C Sevik and T Ccedilağın Thermal conductivity of BN-C nanostructures Phys Rev B 86 (2012) [4] W Ouyang I Azuri D Mandelli A Tkatchenko L Kronik M Urbakh and O Hod Mechanical and Tribological Properties of Layered Materials under High Pressure Assessing the Importance of Many-Body Dispersion Effects J Chem Theory Comput 16 666 (2020) [5] S Plimpton Fast Parallel Algorithms for Short-Range Molecular Dynamics J Comput Phys 117 1 (1995) [6] W Ouyang D Mandelli M Urbakh and O Hod Nanoserpents Graphene Nanoribbon Motion on Two-Dimensional Hexagonal Materials Nano Lett 18 6009 (2018) [7] W Shinoda M Shiga and M Mikami Rapid estimation of elastic constants by molecular dynamics simulation under constant stress Phys Rev B 69 134103 (2004) [8] G L Pollack Kapitza Resistance Rev Mod Phys 41 48 (1969) [9] H-S Yang G R Bai L J Thompson and J A Eastman Interfacial thermal resistance in nanocrystalline yttria-stabilized zirconia Acta Mater 50 2309 (2002) [10] Q Fu J Yang Y Chen D Li and D Xu Experimental evidence of very long intrinsic phonon mean free path along the c-axis of graphite Appl Phys Lett 106 031905 (2015) [11] J Chen J H Walther and P Koumoutsakos Strain Engineering of Kapitza Resistance in Few-Layer Graphene Nano Lett 14 819 (2014) [12] H Zhang X Chen Y-D Jho and A J Minnich Temperature-Dependent Mean Free Path Spectra of Thermal Phonons Along the c-Axis of Graphite Nano Lett 16 1643 (2016) [13] D P Sellan E S Landry J E Turney A J H McGaughey and C H Amon Size effects in molecular dynamics thermal conductivity predictions Phys Rev B 81 214305 (2010) [14] P Jiang X Qian R Yang and L Lindsay Anisotropic thermal transport in bulk hexagonal boron nitride Phys Rev Mater 2 064005 (2018) [15] J Zhang Y Hong and Y Yue Thermal transport across graphene and single layer hexagonal boron nitride J Appl Phys 117 134307 (2015) [16] B Liu F Meng C D Reddy J A Baimova N Srikanth S V Dmitriev and K Zhou Thermal transport in a graphenendashMoS2 bilayer heterostructure a molecular dynamics study RSC Advances 5 29193 (2015) [17] Z Ding Q-X Pei J-W Jiang W Huang and Y-W Zhang Interfacial thermal conductance in grapheneMoS2 heterostructures Carbon 96 888 (2016) [18] N N T Nam and M Koshino Lattice relaxation and energy band modulation in twisted bilayer graphene Phys Rev B 96 075311 (2017)
S12
[19] R Ramiacutearez and M C Boumlhm Simple geometric generation of special points in brillouin-zone integrations Two-dimensional bravais lattices Int J Quantum Chem 30 391 (1986) [20]S L Cunningham Special points in the two-dimensional Brillouin zone Phys Rev B 10 4988 (1974) [21] H J Monkhorst and J D Pack Special points for Brillouin-zone integrations Phys Rev B 13 5188 (1976) [22] D J Chadi Special points for Brillouin-zone integrations Phys Rev B 16 1746 (1977)
1
Fermirsquos golden rule for phonons
1 Basic theory for phonons 11 Basic notations
Let us consider a 3D crystal with a total of 119873119873 = 119873119873111987311987321198731198733 unit cells and periodic boundary conditions
To be specific let 119938119938119894119894 119894119894 = 123 be the lattice vectors that define the unit cell We index unit cells
with n = (n1n2n3) where each ni = 12 ∙∙∙ Ni and their locations are 119929119929119899119899 = sum 1198991198991198941198941199381199381198941198943119894119894=1 Assume that
there are r atoms in each unit cell which are indexed with 119904119904 = 1⋯ 119903119903 The mass and the equilibrium
distance of the sth atom are notated as 119872119872s and 119929119929s0 respectively Then the location of the sth atom in
the nth unit cell at time t can be expressed as
119955119955119899119899119899119899(119905119905) = 119929119929119899119899 + 119929119929s0 + 119958119958119899119899119899119899(119905119905) (11)
where 119958119958119899119899119899119899(119905119905) is its displacement from its equilibrium position
The Lagrangian for this classical problem can be written as
ℒ = sum sum 119872119872119904119904|119955119899119899119904119904|2(119905119905)2
119903119903119899119899=1
119873119873119899119899=1 minus 119881119881 (12)
where the second term is the sum of interactions between all pairs of atoms
Under the harmonic approximation ie expanding the total potential energy 119881119881 around the
equilibrium positions The Lagrangian can be simplified as
ℒ = sum sum 119872119872119904119904|119958119899119899119904119904|2(119905119905)2
119903119903119899119899=1
119873119873119899119899=1 minus 1
2sum sum sum 120601120601120572120572120572120572prime
119899119899119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime120572120572120572120572prime119899119899119899119899prime119899119899119899119899prime (13)
where 119906119906119899119899119899119899120572120572 120572120572 = 123 are the Cartesian coordinates of the displacement 119958119958119899119899119899119899(119905119905) and
120601120601120572120572120572120572prime 119899119899119899119899prime119904119904 119904119904prime = 1205971205972119881119881
120597120597119903119903119899119899119904119904119899119899119903119903119899119899prime119904119904prime119899119899primeeq
= 120601120601120572120572prime120572120572 119899119899prime119899119899119904119904prime 119904119904 = 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime (14)
Note that the first order term vanishes because we are expanding around the equilibrium positions
120601120601120572120572120572120572prime 119899119899119899119899prime119904119904 119904119904prime represents the component of the force acted on the sth atom in the nth unit cell along 120572120572
direction when the atom 119904119904prime in the unit cell 119899119899prime moves a unit displacement along 120572120572prime direction The
symmetries of 120601120601120572120572120572120572prime 119899119899 119899119899prime119904119904 119904119904prime appearing in Eq (14) arise from the intechangability of the second
derivative and the translational invariance of the interactions
12 Dynamical matrix
The equation of motion of the sth atom in the nth unit cell can be derived using the EulerndashLagrange
equation as follows
2
119872119872119899119899119906119899119899119899119899120572120572 = minussum 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119899119899prime119899119899prime120572120572prime 119906119906119899119899prime119899119899prime120572120572prime (15)
If we displace all atoms equally ie shifting 119906119906119899119899prime119899119899prime120572120572prime to 119906119906119899119899prime119899119899prime120572120572prime + 120575120575 the total force on the sth atom in
the nth unit cell does not change From the above equation we have
sum 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119899119899prime120572120572prime = 0 (16)
We are looking for normal modes (because any general solution can be written as a linear combination
of them) these are solutions where all atoms oscillate with the same frequency Moreover because
of the lattice structure we expect solutions to reflect this periodicity So we guess solutions of the
form
119906119906119899119899119899119899120572120572(119905119905) = 1119872119872119904119904
119890119890119899119899120572120572119890119890119894119894(119954119954∙119929119929119899119899minus120596120596119905119905) (17)
where 119890119890119899119899120572120572 are real-space solutions that will be determined later and 119954119954 is the wave-vector in
reciprocal space Substituting Eq (17) into Eq (15) we can get
1205961205962119890119890119899119899120572120572(119954119954) = sum 119863119863119899119899120572120572119899119899prime120572120572prime(119954119954)119890119890119899119899prime120572120572prime(119954119954)119899119899prime120572120572prime (18)
where
119863119863119899119899120572120572119899119899prime120572120572prime(119954119954) = sum 1
119872119872119904119904119872119872119904119904prime120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119890119890
minus119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime119899119899prime = sum 1
119872119872119904119904119872119872119904119904prime120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime 119890119890
minus119894119894119954119954∙119929119929119897119897119897119897 (19)
is called dynamical matrix (dimension 3119903119903 times 3119903119903) Note that we have defined the relative cell distance
vector 119929119929119897119897 equiv 119929119929119899119899 minus 119929119929119899119899prime and number index 119897119897 equiv 119899119899 minus 119899119899prime where the infinite sum over 119899119899prime can be
replaced by the sum over 119897119897 for any value of the index 119899119899 Note that dynamical matrix is Hermitian
symmetric (ie 119863119863119899119899120572120572119899119899prime120572120572prime(119954119954)
lowast= 119863119863119899119899prime120572120572prime
119899119899120572120572 (minus119954119954)) because ϕ is symmetric so all the eigenvalues of Eq (18)
(120596120596120582120582(119954119954) 120582120582 = 12⋯ 3119903119903) are real for each 119954119954 in the Brillouin zone (BZ) which is determined by
det1205961205961205821205822(119954119954)120575120575119899119899120572120572120575120575119899119899prime120572120572prime minus 119863119863119899119899120572120572119899119899prime120572120572prime(119954119954) = 0 (110)
Taking the conjugate of this equation we have
0 = det 1205961205961205821205822(119954119954)lowast120575120575119899119899120572120572120575120575119899119899prime120572120572prime minus 119863119863119899119899120572120572119899119899
prime120572120572prime(119954119954)lowast = det1205961205961205821205822(119954119954)120575120575119899119899120572120572120575120575119899119899prime120572120572prime minus 119863119863119899119899120572120572119899119899
prime120572120572prime(minus119954119954) (111)
while replacing 119954119954 by minus119954119954 in Eq (110) we have det1205961205961205821205822(minus119954119954)120575120575119899119899120572120572120575120575119899119899prime120572120572prime minus 119863119863119899119899120572120572119899119899prime120572120572prime(minus119954119954) = 0
Itrsquos clear that 1205961205961205821205822(119954119954) and 1205961205961205821205822(minus119954119954) obey the same equation thus we have
120596120596120582120582(minus119954119954) = 120596120596120582120582(119954119954) (112)
The corresponding eigenvectors are orthonormal
sum 119890119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582
prime119899119899120572120572 = 120575120575120582120582120582120582prime (113)
The complex conjugate of Eq (18) gives
1205961205962119890119890119899119899120572120572lowast (119954119954) = sum 119863119863119899119899120572120572119899119899prime120572120572prime(119954119954)
lowast119890119890119899119899prime120572120572primelowast (119954119954)119899119899prime120572120572prime = sum 119863119863119899119899prime120572120572prime
119899119899120572120572 (minus119954119954)119890119890119899119899prime120572120572primelowast (119954119954)119899119899prime120572120572prime (114)
3
While replacing 119954119954 by minus119954119954 in Eq (18) we get the following equation
1205961205962119890119890119899119899120572120572(minus119954119954) = sum 119863119863119899119899120572120572119899119899prime120572120572prime(minus119954119954)119890119890119899119899prime120572120572prime(minus119954119954)119899119899prime120572120572prime (115)
From Eq (114) and Eq (115) we see that eigenvectors 119890119890119899119899120572120572120582120582 (minus119954119954)lowast and 119890119890119899119899120572120572120582120582 (119954119954) obey the same
eigenvalue equation Since the eigenvectors are normalized we get the following property
119890119890119899119899120572120572120582120582 (minus119954119954)lowast
= 119890119890119899119899120572120572120582120582 (119954119954) (116)
2 Second quantization The general solution is a linear combination of all these normal modes thus we have
119906119906119899119899119899119899120572120572(119905119905) = sum 119862119862120582120582(119954119954)119873119873119872119872119904119904
119890119890119899119899120572120572120582120582 (119954119954)119890119890minus119894119894120596120596119954119954120582120582119905119905119890119890119894119894119954119954∙119929119929119899119899119954119954120582120582 = sum 119876119876120582120582(119954119954119905119905)119873119873119872119872119904119904
119890119890119899119899120572120572120582120582 (119954119954)119890119890119894119894119954119954∙119929119929119899119899119954119954120582120582 (21)
where the we define the normal coordinates as 119876119876120582120582(119954119954 119905119905) equiv 119862119862120582120582(119954119954)119890119890minus119894119894120596120596120582120582(119954119954)119905119905 in the eigenvectors
representation To ensure that the displacements 119906119906119899119899119899119899120572120572(119905119905) are real (namely 119906119906119899119899119899119899120572120572lowast (119905119905) = 119906119906119899119899119899119899120572120572(119905119905)) the
following relation on 119876119876120582120582(119954119954 119905119905) and 119890119890119899119899120572120572120582120582 is enforced
119876119876120582120582(119954119954 119905119905)119890119890119899119899120572120572120582120582 (119954119954)lowast
= 119876119876120582120582(minus119954119954 119905119905)119890119890119899119899120572120572120582120582 (minus119954119954) (22)
where we used the fact that the sum over 119954119954 runs symmetrically over both negative and positive
values Using Eq (116) we have
[119876119876120582120582(119954119954 119905119905)]lowast = 119876119876120582120582(minus119954119954 119905119905) (23)
21 Kinetic energy term
Using Eq (21) the kinetic energy of the system can be expressed as
119879119879 = 12sum 1198721198721198991198991199061198991198991198991198991205721205722 (119905119905)119899119899119899119899120572120572 = 1
2sum 119872119872119904119904
119873119873119872119872119904119904sum sum 119876120582120582(119954119954 119905119905)119876120582120582prime(119954119954prime 119905119905)119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582
prime(119954119954prime) sum 119890119890119894119894119954119954+119954119954prime∙119929119929119899119899119899119899119954119954prime120582120582prime119954119954120582120582119899119899120572120572 =
12sum sum 119876120582120582(119954119954 119905119905)119876120582120582prime(minus119954119954 119905119905)119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582
prime(minus119954119954)119954119954119954119954prime120582120582prime =119899119899120572120572
12sum 119876120582120582(119954119954 119905119905)119876120582120582prime
lowast (119954119954 119905119905) sum 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582prime(119954119954)
lowast119899119899120572120572 =119954119954120582120582120582120582prime
12sum 119876120582120582(119954119954 119905119905)119876120582120582lowast(119954119954 119905119905)119954119954120582120582 =
12sum 119876120582120582(119954119954 119905119905)119876120582120582(minus119954119954 119905119905)119954119954120582120582
ie
119879119879 = 12sum 119876120582120582(119954119954 119905119905)119876120582120582(minus119954119954 119905119905)119954119954120582120582 (24)
To derive Eq (24) we used Eqs (113) and (116) as well as the following equations
1119873119873sum 119890119890119894119894119954119954+119954119954prime∙119929119929119899119899119899119899 = 120575120575119954119954+119954119954prime120782120782 (25)
4
22 Potential energy term
Similarity the potential energy can be rewritten in terms of the normal coordinates as follows
119880119880 =12120601120601119899119899119899119899120572120572119899119899prime119899119899prime120572120572prime119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
120572120572120572120572prime119899119899119899119899prime119899119899119899119899prime=
12120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119899119899+119954119954prime∙119929119929119899119899prime
119873119873119872119872119899119899119872119872119899119899prime
[119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(119954119954prime 119905119905)] 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime120582120582prime (119954119954prime)
119954119954prime120582120582prime119954119954120582120582119899119899119899119899prime120572120572120572120572prime119899119899119899119899prime
=12120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894(119954119954+119954119954prime)∙119929119929119899119899119890119890minus119894119894119954119954prime∙119929119929119897119897
119873119873119872119872119899119899119872119872119899119899prime
[119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(119954119954prime 119905119905)] 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime120582120582prime (119954119954prime)
119954119954prime120582120582prime119954119954120582120582119899119899119897119897120572120572120572120572prime119899119899119899119899prime
=12120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890minus119894119894119954119954prime∙119929119929119897119897
119872119872119899119899119872119872119899119899prime[119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(119954119954prime 119905119905)] 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (119954119954prime)1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899
119899119899=120575120575119902119902+119902119902prime=0
119954119954prime120582120582prime119954119954120582120582119897119897120572120572120572120572prime119899119899119899119899prime
=12120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894119954119954119929119929119897119897
119872119872119899119899119872119872119899119899prime [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)] 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (minus119954119954)119954119954120582120582120582120582prime119897119897120572120572120572120572prime119899119899119899119899prime
=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]119890119890119899119899120572120572120582120582 (119954119954)
119899119899120572120572
119863119863119899119899120572120572119899119899prime120572120572prime(minus119954119954)119890119890119899119899prime120572120572prime
120582120582prime (minus119954119954)119899119899prime120572120572prime119954119954120582120582120582120582prime
=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]119890119890119899119899120572120572120582120582 (119954119954)
119899119899120572120572
1205961205961205821205822(minus119954119954)119890119890119899119899120572120572120582120582prime (minus119954119954)
119954119954120582120582120582120582prime
=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]1205961205961205821205822(119954119954)120575120575120582120582120582120582prime =119954119954120582120582120582120582prime
121205961205961205821205822(119954119954)119876119876120582120582(119954119954 119905119905)119876119876120582120582(minus119954119954 119905119905)
119954119954120582120582
ie
119880119880 = 12sum 1205961205961205821205822(119954119954)119876119876120582120582(119954119954 119905119905)119876119876120582120582(minus119954119954 119905119905)119954119954120582120582 (26)
where in above deriviation the orthogonality of its eigenvectores and the symmetry of its eigenvalues
are used Thus the Lagrangian reads as
ℒ = 119879119879 minus 119881119881 = 12sum 119876120582120582(minus119954119954 119905119905)119876120582120582(119954119954 119905119905) minus 120596120596120582120582
2(119954119954)119876119876120582120582(minus119954119954 119905119905)119876119876120582120582(119954119954 119905119905)119954119954120582120582 (27)
Using the relation 119875119875120582120582(119954119954 119905119905) = 120597120597ℒ part119876120582120582(119954119954 119905119905) the Hamiltonian of the system then can be written as
119867119867 = 119879119879 + 119881119881 = 12sum [119875119875120582120582(minus119954119954 119905119905)119875119875120582120582(119954119954 119905119905) + 120596120596120582120582
2(119954119954)119876119876120582120582(minus119954119954 119905119905)119876119876120582120582(119954119954 119905119905)]119954119954120582120582 (28)
Now we quantize H by asking the momenta and coordinates to be operators
119867119867 = 12sum 119875119875minus119954119954120582120582119875119875119954119954120582120582 + 120596120596119954119954120582120582
2 119876119876minus119954119954120582120582119876119876119954119954120582120582119954119954120582120582 (29)
here 119875119875119954119954120582120582 and 119876119876119954119954120582120582 obey the following commutation relations
119876119876119954119954120582120582119875119875119954119954prime120582120582prime = 119894119894ℏ120575120575119954119954119954119954prime120575120575120582120582120582120582prime
119876119876119954119954120582120582119876119876119954119954prime120582120582prime = 0 119875119875119954119954120582120582119875119875119954119954prime120582120582prime = 0 (210)
Similar to the case of ordinary quantum harmonic oscillators it is convenient to define ladder
operators for each mode as follows
119876119876119954119954120582120582 = ℏ
2120596120596119954119954120582120582119887119887119954119954120582120582 + 119887119887minus119954119954120582120582
dagger 119875119875119954119954120582120582 = 119894119894ℏ120596120596119954119954120582120582
2119887119887119954119954120582120582
dagger minus 119887119887minus119954119954120582120582 (211)
where 119887119887119954119954120582120582dagger and 119887119887119954119954120582120582 are Bosonic creation and annihilation operators for phonons with momentum
119954119954 branch index 120582120582 and frequency 120596120596119954119954120582120582 which obeys the Bosonic commutation relation
5
119887119887119954119954120582120582 119887119887119954119954prime120582120582prime
dagger = 120575120575119954119954119954119954prime120575120575120582120582120582120582prime
119887119887119954119954120582120582 119887119887119954119954prime120582120582prime = 0 119887119887119954119954120582120582dagger 119887119887119954119954prime120582120582prime
dagger = 0 (212)
Substituting Eqs (211) into (29) and using the properties Eq (210) and (212) we have
119867119867 =12119875119875minus119954119954120582120582119875119875119954119954120582120582 +120596120596119954119954120582120582
2 119876119876minus119954119954120582120582119876119876119954119954120582120582119954119954120582120582
=12minus
ℏ120596120596119954119954120582120582
2119887119887minus119954119954120582120582
dagger minus 119887119887119954119954120582120582119887119887119954119954120582120582dagger minus 119887119887minus119954119954120582120582+ 120596120596119954119954120582120582
2 ℏ2120596120596119954119954120582120582
119887119887minus119954119954120582120582 + 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 + 119887119887minus119954119954120582120582
dagger 119954119954120582120582
=ℏ4120596120596119954119954120582120582minus119887119887minus119954119954120582120582
dagger 119887119887119954119954120582120582dagger minus 119887119887minus119954119954120582120582
dagger 119887119887minus119954119954120582120582 minus 119887119887119954119954120582120582119887119887119954119954120582120582dagger + 119887119887119954119954120582120582119887119887minus119954119954120582120582
119954119954120582120582
+ 119887119887minus119954119954120582120582119887119887119954119954120582120582 + 119887119887minus119954119954120582120582119887119887minus119954119954120582120582dagger + 119887119887119954119954120582120582
dagger 119887119887119954119954120582120582 + 119887119887119954119954120582120582dagger 119887119887minus119954119954120582120582
dagger
=ℏ4120596120596119954119954120582120582119887119887minus119954119954120582120582
dagger 119887119887minus119954119954120582120582 + 119887119887minus119954119954120582120582119887119887minus119954119954120582120582dagger + 119887119887119954119954120582120582119887119887119954119954120582120582
dagger + 119887119887119954119954120582120582dagger 119887119887119954119954120582120582
119954119954120582120582
=ℏ41205961205961199541199541205821205822119887119887minus119954119954120582120582
dagger 119887119887minus119954119954120582120582 + 1+ 2119887119887119954119954120582120582dagger 119887119887119954119954120582120582 + 1
119954119954120582120582
= ℏ120596120596119954119954120582120582 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 +
12
119954119954120582120582
Therefore we finally get the quantized representation of non-interacting phonons
119867119867 = sum ℏ120596120596119954119954120582120582 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 + 1
2119954119954120582120582 (213)
The operator of atom displacements (Eq 114) is expressed in terms of the phonon operators by
119906119906119899119899119899119899120572120572 = sum ℏ
2119873119873119872119872119904119904120596120596119954119954120582120582119890119890119899119899120572120572120582120582 119890119890119894119894119954119954∙119929119929119899119899119887119887119954119954120582120582 + 119887119887minus119954119954120582120582
dagger 119954119954120582120582 (214)
These equations will be used below
3 Inter-phonon coupling within harmonic approximation 31 Hamiltonian with inter-phonon coupling
In this section we consider systems that consist of two (or more) covalently bonded units that are
weakly coupled between them Unlike previous studies that considered phonon-phonon couplings
resulting from anharmonicity effects [1] here all phonon mode considered are Harmonic and the
couplings arise from the division of the entire system into subunits The Hamiltonian of the whole
system can be written as
119867119867 = 1198671198671 + 1198671198672 + 11986711986712 (31)
To derive the expressions of 1198671198671 1198671198672 and 11986711986712 in Eq (31) we consider the Hamiltonian of the
whole system written as the function of the atomic displacements
119867119867 = 119879119879 + 119881119881 = sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)2119899119899119899119899120572120572 + 1
2sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime120572120572120572120572prime119899119899119899119899prime119899119899119899119899prime (32)
6
Letrsquos assume that subsystem I and II contain atoms with indeices ranging from 119904119904 = 12⋯ 1199031199032 and
119904119904 = 1199031199032
+ 1 1199031199032
+ 2⋯ 119903119903 respectively Then the kinetic energy term in Eq (32) can be rewritten as
119879119879 = sum sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)2
1199031199032119899119899=1 + sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)
2119903119903119899119899=1199031199032+1
119899119899120572120572 (33)
While the potential energy term is
119881119881 =12 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
1199031199032
119899119899119899119899prime=1
+ 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
119903119903
119899119899119899119899prime=1199031199032+1
+ 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime119899119899119899119899prime
+ 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
1199031199032
119899119899prime=1
119903119903
119899119899=1199031199032+1
= 11988111988111 + 11988111988122 + 11988111988112 + 11988111988121
Or equivalently
⎩⎪⎪⎨
⎪⎪⎧ 11988111988111 = 1
2sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
1199031199032119899119899119899119899prime=1120572120572120572120572prime119899119899119899119899prime
11988111988122 = 12sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
119903119903119899119899119899119899prime=1199031199032+1120572120572120572120572prime119899119899119899119899prime
11988111988112 = 12sum sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
119903119903119899119899prime=1199031199032+1
1199031199032119899119899=1120572120572120572120572prime119899119899119899119899prime
11988111988121 = 12sum sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
1199031199032119899119899prime=1
119903119903119899119899=1199031199032+1120572120572120572120572prime119899119899119899119899prime
(34)
32 Second quantization
We may now quantize this Hamiltonian and write it in the basis of the eigenstates of the coupled
subunits In second quantization the atomic displacement operators of the two subunits are given in
the following form
119906119906119899119899119899119899120572120572 =
⎩⎨
⎧ sum ℏ
2119873119873119872119872119904119904ω119954119954120582120582119890119890119899119899120572120572120582120582 119890119890119894119894119954119954∙119929119929119899119899119886119886119954119954120582120582 + 119886119886minus119954119954120582120582
dagger 119954119954120582120582 119904119904 isin 1 1199031199032
sum ℏ
2119873119873119872119872119904119904120596120596119954119954prime120582120582prime119890119899119899120572120572120582120582
prime119890119890119894119894119954119954prime∙119929119929119899119899 119886119886 119954119954prime120582120582prime + 119886119886minus119954119954prime120582120582primedagger 119954119954prime120582120582prime 119904119904 isin 119903119903
2+ 1 119903119903
(35)
Here we use different notations for the creation and annihilation operators (119886119886119954119954120582120582119886119886minus119954119954120582120582dagger and 119886119886119954119954120582120582119886119886minus119954119954120582120582
dagger )
eigenvalues (ω119954119954120582120582120596120596119954119954120582120582) and eigenvectors (119890119890119899119899120572120572120582120582 119890119899119899120572120572120582120582 ) for the two subunits Note that 119890119890119899119899120572120572120582120582 and 119890119899119899120572120572120582120582 are
of the dimensions of the whole system however their non-zero elements appear only on the relevant
subunits such that they obey the following relations sum 119890119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582
prime119899119899120572120572 = 120575120575120582120582120582120582prime sum 119890119899119899120572120572120582120582
lowast119890119899119899120572120572120582120582
prime119899119899120572120572 =
120575120575120582120582120582120582prime sum 119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582
prime119899119899120572120572 = sum 119890119890119899119899120572120572120582120582
lowast119890119899119899120572120572120582120582
prime119899119899120572120572 = 0 Defining the normal coordinates of the two subunits as
119876119876119954119954120582120582 equiv ℏ
2ω119954119954120582120582119886119886119954119954120582120582 + 119886119886minus119954119954120582120582
dagger 119876119876119954119954120582120582 equiv ℏ
2ω119954119954120582120582119886119886119954119954120582120582 + 119886119886minus119954119954120582120582
dagger (36)
and the corresponding momenta operators as
7
119875119875119954119954120582120582 = 119894119894ℏω119954119954120582120582
2119886119886119954119954120582120582
dagger minus 119886119886minus119954119954120582120582 119875119875119954119954120582120582 = 119894119894ℏω1199541199541205821205822
119886119886119954119954120582120582dagger minus 119886119886minus119954119954120582120582 (37)
Eq (35) reads
119906119906119899119899119899119899120572120572 = sum 119890119890119904119904119899119899120582120582 (119954119954)
119873119873119872119872119904119904119890119890119894119894119954119954∙119929119929119899119899119876119876119954119954120582120582119954119954120582120582 119904119904 isin 1 119903119903
2
sum 119890119904119904119899119899120582120582 (119954119954)119873119873119872119872119904119904
119890119890119894119894119954119954∙119929119929119899119899119876119876119954119954120582120582119954119954120582120582 119904119904 isin 1199031199032
+ 1 119903119903 (38)
The second quantized kinetic energy operator is then written as
119879119879 = 1198791198791 + 1198791198792 = 12sum 119875119875119954119954120582120582119875119875minus119954119954120582120582119954119954120582120582 + 1
2sum 119875119875119954119954prime120582120582prime119875119875minus119954119954prime120582120582prime119954119954prime120582120582prime (39)
Correspondingly the various potential energy terms are obtained by substituting Eq (38) in Eq (34)
11988111988111 =12 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119899119899119899119899prime
119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime120582120582prime (119954119954prime)
119872119872119899119899119872119872119899119899prime
1119873119873119890119890119894119894119954119954∙119929119929119899119899+119894119894119954119954prime∙119929119929119899119899prime119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954120582120582119954119954prime120582120582prime
1199031199032
119899119899119899119899prime=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954119954119954prime120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (119954119954prime)1119873119873 119890119890
119894119894(119954119954+119954119954prime)119929119929119899119899prime
119899119899prime120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime
119872119872119899119899119872119872119899119899prime119899119899
1199031199032
119899119899119899119899prime=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954119954119954prime120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (119954119954prime) 120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime
1199031199032
119899119899119899119899prime=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime
119954119954120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (minus119954119954)120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
1199031199032
119899119899119899119899prime=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime
119954119954120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954) 119863119863119899119899120572120572119899119899
prime120572120572prime(minus119954119954)119890119890119899119899prime120572120572prime120582120582prime (minus119954119954)
120572120572prime
1199031199032
119899119899prime=1
1199031199032
119899119899=1120572120572
=12 ω119954119954120582120582prime
2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582120582120582prime
119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582prime (119954119954)
lowast
1199031199032
119899119899=1
=120572120572
12ω119954119954120582120582
2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582
ie
11988111988111 = 12sum ω119954119954120582120582
2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582 (310)
Going from the second line to the third line we used the fact that the sum over 119899119899prime runs between plusmninfin
and the summand depends only on the difference between 119899119899 and 119899119899prime hence the sum is independent
of the value of the index 119899119899 Therefore we can replace the sum over 119899119899prime by a sum over 119897119897 equiv 119899119899 minus 119899119899prime
amd define 119929119929119897119897 equiv 119929119929119899119899 minus 119929119929119899119899prime Following the same procedure we can get the corresponding
expressions for the second diagonal term
11988111988122 = 12sum 1205961205961199541199541205821205822 119876119876119954119954120582120582119876119876minus119954119954120582120582119954119954120582120582 (311)
Similarly for the off-diagonal terms we get
8
11988111988112 =12 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119899119899119899119899prime
119903119903
119899119899prime=1199031199032+1
119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime
120582120582prime (119954119954prime)
119872119872119899119899119872119872119899119899prime
1119873119873119890119890119894119894119954119954∙119929119929119899119899+119894119894119954119954prime∙119929119929119899119899prime119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954120582120582119954119954prime120582120582prime
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954119954119954prime120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime
120582120582prime (119954119954prime)1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime
119872119872119899119899119872119872119899119899prime119899119899
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954119954119954prime120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime
120582120582prime (119954119954prime) 120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime
119954119954120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime
120582120582prime (minus119954119954)120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954
119890119899119899prime120572120572prime120582120582prime (119954119954)
lowast120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119890119890119899119899120572120572120582120582 (119954119954)119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954
Where we have defined
119881119881120582120582120582120582prime(119954119954) equiv sum sum sum 119890119899119899prime120572120572prime120582120582prime (119954119954)
lowast119881119881119899119899120572120572119899119899
prime120572120572prime(119954119954)119890119890119899119899120572120572120582120582 (119954119954)119903119903119899119899prime=1199031199032+1
1199031199032119899119899=1120572120572120572120572prime (312)
where
119881119881119899119899120572120572119899119899prime120572120572prime(119954119954) equiv sum 119890119890119894119894119954119954∙119929119929119897119897
119872119872119904119904119872119872119904119904prime119897119897 120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime (313)
Itrsquos easy to show that 119881119881120582120582120582120582prime(119954119954) has the following property
119881119881120582120582120582120582primelowast (119954119954) = 119881119881120582120582120582120582prime(minus119954119954) (314)
Following the same procedure we have
9
11988111988121 =12 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119899119899119899119899prime
119903119903
119899119899=1199031199032+1
119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (119954119954prime)
119872119872119899119899119872119872119899119899prime
1119873119873119890119890119894119894119954119954∙119929119929119899119899+119894119894119954119954prime∙119929119929119899119899prime119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954120582120582119954119954prime120582120582prime
1199031199032
119899119899prime=1120572120572120572120572prime
=12 120601120601120572120572prime120572120572
119899119899prime minus 119899119899119904119904prime 119904119904
119899119899prime119899119899
119903119903
119899119899prime=1199031199032+1
119890119899119899prime120572120572prime120582120582prime (119954119954prime)119890119890119899119899120572120572120582120582 (119954119954)
119872119872119899119899prime119872119872119899119899
1119873119873 119890119890
119894119894119954119954prime∙119929119929119899119899prime+119894119894119954119954∙119929119929119899119899119876119876119954119954prime120582120582prime119876119876119954119954120582120582119954119954prime120582120582prime119954119954120582120582
1199031199032
119899119899=1120572120572prime120572120572
=12 119876119876119954119954prime120582120582prime119876119876119954119954120582120582
119954119954119954119954prime120582120582120582120582prime119890119899119899prime120572120572prime120582120582prime (119954119954prime)119890119890119899119899120572120572120582120582 (119954119954)
1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime
119872119872119899119899119872119872119899119899prime119899119899
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954prime120582120582prime119876119876119954119954120582120582
119954119954119954119954prime120582120582120582120582prime119890119899119899prime120572120572prime120582120582prime (119954119954prime)119890119890119899119899120572120572120582120582 (119954119954) 120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954prime120582120582prime119876119876minus119954119954prime120582120582119890119899119899prime120572120572prime
120582120582prime (119954119954prime)119890119890119899119899120572120572120582120582 (minus119954119954prime)120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890minus119894119894119954119954prime∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897119954119954prime120582120582120582120582prime
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954prime120582120582prime119876119876119954119954prime120582120582
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954prime 119890119890119899119899120572120572120582120582 (119954119954prime)
lowast120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890minus119894119894119954119954prime∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119890119899119899prime120572120572prime120582120582prime (119954119954prime)
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582prime119876119876119954119954120582120582
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954
119890119890119899119899120572120572120582120582 (119954119954)lowast120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890minus119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119890119899119899prime120572120572prime120582120582prime (119954119954)
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger dagger
3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954
119890119899119899prime120572120572prime120582120582prime (119954119954)
lowast120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119890119890119899119899120572120572120582120582 (119954119954)119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
lowast
=
⎩⎨
⎧12 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954⎭⎬
⎫dagger
= 11988111988112dagger
Define
1198671198671 = 11987911987911 + 119881119881111198671198672 = 11987911987922 + 1198811198812211986711986712 = 11988111988112 + 11988111988121
(315)
we finally get the expressions of 1198671198671 1198671198672 and 11986711986712 in Eq (31) as follows
⎩⎪⎨
⎪⎧ 1198671198671 = sum sum ℏ120596120596119954119954120582120582 119886119886119954119954120582120582
dagger 119886119886119954119954120582120582 + 1231199031199032
120582120582=1119954119954
1198671198672 = sum sum ℏ120596120596119954119954120582120582prime 119886119886119954119954120582120582primedagger 119886119886119954119954120582120582prime + 1
23119903119903
120582120582prime=1+31199031199032119954119954
11986711986712 = 12sum sum sum 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903120582120582prime=31199031199032 +1
31199031199032120582120582=1 + h c 119954119954
(316)
Here hc means the Hermitian conjugate Since the indices of 119886119886119954119954120582120582 119886119886119954119954120582120582prime and 119876119876119954119954120582120582 119876119876119954119954120582120582prime belong to
the two system sections we can define an abbreviated notation 119886119886119954119954120582120582 and 119876119876119954119954120582120582 using the index 120582120582 to
identify which subsystem they belong to In this case the Hamiltonian of the coupled system can be
10
simplified as follows
119867119867 = 1198671198670 + 119867119867119862119862 (317)
where
1198671198670 = sum sum ℏ120596120596119954119954120582120582 119886119886119954119954120582120582
dagger 119886119886119954119954120582120582 + 123119903119903
120582120582=1119954119954
119867119867119862119862 = 12sum sum sum 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903120582120582prime=31199031199032 +1
31199031199032120582120582=1 + ℎ 119888119888 119954119954
(318)
4 Greenrsquos function 41 Dynamics of the ladder operators
In order to describe the dynamics of the ladder operators appearing in the Hamiltonian of Eq (317)
we express them in the Heisenberg picture as follows
119886119886119901119901(120591120591) = 119890119890119894119894ℏ119867119867119905119905119886119886119901119901119890119890
minus119894119894ℏ119867119867119905119905 = 119890119890
119867119867120591120591ℏ 119886119886119901119901119890119890
minus119867119867120591120591ℏ (41)
where we define the imaginary time 120591120591 equiv 119894119894119905119905 and introduce the notation 119953119953 equiv (119954119954 120582120582) and 119953119953 equiv (minus119954119954 120582120582)
The corresponding equation of motion for the ladder operators is given by
ℏ part119886119886119953119953(120591120591)
120597120597120591120591= 119867119867119886119886119953119953(120591120591) (42)
For the uncoupled system (119867119867119862119862 = 0) this gives
ℏpart119886119886119953119953(120591120591)120597120597120591120591
= 1198671198670119886119886119953119953(120591120591) = 1198671198670 1198901198901198671198670120591120591ℏ 119886119886119953119953119890119890
minus1198671198670120591120591ℏ = 1198671198670119890119890
1198671198670120591120591ℏ 119886119886119953119953119890119890
minus1198671198670120591120591ℏ minus 119890119890
1198671198670120591120591ℏ 119886119886119953119953119890119890
minus1198671198670120591120591ℏ 1198671198670
= 1198901198901198671198670120591120591ℏ 1198671198670119886119886119953119953 minus 1198861198861199531199531198671198670119890119890
minus1198671198670120591120591ℏ = 119890119890
1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890
minus1198671198670120591120591ℏ
ie
ℏ part119886119886119953119953(120591120591)
120597120597120591120591= 119890119890
1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890
minus1198671198670120591120591ℏ (43)
The commutator on the right-hand-side of Eq (43) can be evaluated as follows
1198671198670119886119886119953119953prime = sum ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953 + 1
2119953119953 119886119886119953119953prime = sum ℏ120596120596119953119953119886119886119953119953
dagger119886119886119953119953119886119886119953119953prime119953119953 = sum ℏ120596120596119953119953119886119886119953119953dagger119886119886119953119953119886119886119953119953prime minus 119886119886119953119953prime119886119886119953119953
dagger119886119886119953119953119953119953 =
sum ℏ120596120596119953119953119886119886119953119953dagger119886119886119953119953119886119886119953119953prime minus 120575120575119953119953prime119953119953 + 119886119886119953119953
dagger119886119886119953119953prime119886119886119953119953119953119953 = minusℏ120596120596119953119953prime119886119886119953119953prime + sum ℏ120596120596119953119953119886119886119953119953dagger119886119886119953119953119886119886119953119953prime minus 119886119886119953119953
dagger119886119886119953119953119886119886119953119953prime119953119953 = minusℏ120596120596119953119953prime119886119886119953119953prime
ie
1198671198670119886119886119953119953prime = minusℏ120596120596119953119953prime119886119886119953119953prime (44)
Therefore we have
ℏ part119886119886119953119953(120591120591)
120597120597120591120591= 119890119890
1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890
minus1198671198670120591120591ℏ = minusℏ120596120596119953119953119890119890
1198671198670120591120591ℏ 119886119886119953119953119890119890
minus1198671198670120591120591ℏ = minusℏ120596120596119953119953119886119886119953119953(120591120591) (45)
11
The solution of Eq (45) is 119886119886119953119953(120591120591) = 119886119886119953119953(0)119890119890minus120596120596119953119953120591120591 = 119886119886119953119953119890119890minus120596120596119953119953120591120591 For 119886119886119953119953dagger(120591120591) we have
1198671198670119886119886119953119953primedagger (0) = 1198671198670119886119886119953119953prime
dagger = ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953 +
12
119953119953
119886119886119953119953primedagger = ℏ120596120596119953119953119886119886119953119953
dagger119886119886119953119953119886119886119953119953primedagger
119953119953
= ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953119886119886119953119953prime
dagger minus 119886119886119953119953primedagger 119886119886119953119953
dagger119886119886119953119953119953119953
= ℏ120596120596119953119953 119886119886119953119953dagger 120575120575119953119953119953119953prime + 119886119886119953119953prime
dagger 119886119886119953119953 minus 119886119886119953119953primedagger 119886119886119953119953
dagger119886119886119953119953119953119953
= ℏ120596120596119953119953prime119886119886119953119953primedagger + ℏ120596120596119953119953 119886119886119953119953
dagger119886119886119953119953primedagger 119886119886119953119953 minus 119886119886119953119953prime
dagger 119886119886119953119953dagger119886119886119953119953
119953119953
= ℏ120596120596119953119953prime119886119886119953119953primedagger
In summary we have
119886119886119953119953(120591120591) = 119886119886119953119953(0)119890119890minus120596120596119953119953120591120591 = 119886119886119953119953119890119890minus120596120596119953119953120591120591
119886119886119953119953dagger(120591120591) = 119886119886119953119953
dagger(0)119890119890120596120596119953119953120591120591 = 119886119886119953119953dagger119890119890120596120596119953119953120591120591 (46)
42 Greenrsquos function
To describe thermal transport properties between different system sections we use the formalism of
thermal (or imaginary time) phonon Greenrsquos function [2] To this end we define the thermal Greenrsquos
function for phonons as
119866119866119953119953119953119953prime(120591120591 120591120591prime) = minuslang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime)rang = minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime) (47)
where 119879119879120591120591 is the time ordering operator and 120588120588119867119867 is the statistical operator for the grand canonical
ensemble (note that the chemical potential for phonons is zero)
120588120588119867119867 = 119890119890minus120573120573119867119867119885119885119867119867 (48)
where the partition function is given by 119885119885119867119867 equiv Tr119890119890minus120573120573119867119867 and 120573120573 = 1119896119896119861119861119879119879 with 119896119896119861119861 being
Boltzmannrsquos constant and 119879119879 the temperature For time independent Hamiltonians 119866119866119953119953119953119953prime(120591120591 120591120591prime)
depends only on 120591120591 minus 120591120591prime ie
119866119866119953119953119953119953prime(120591120591 120591120591prime) = 119866119866119953119953119953119953prime(120591120591 minus 120591120591prime 0) (49)
To show this we shall first assume that 120591120591 gt 120591120591prime such that
119866119866119953119953119953119953prime(120591120591 120591120591prime) = minuslang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime)rang = minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime) = minusTr 120588120588119867119867119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime)
= minusTr 120588120588119867119867119890119890119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591ℏ 119890119890
119867119867120591120591primeℏ 119886119886119953119953prime
dagger 119890119890minus119867119867120591120591primeℏ = minusTr 120588120588119867119867119890119890
minus119867119867120591120591primeℏ 119890119890
119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591minus120591120591prime
ℏ 119886119886119953119953primedagger
= minusTr 120588120588119867119867119890119890119867119867120591120591minus120591120591prime
ℏ 119886119886119953119953119890119890minus119867119867120591120591minus120591120591prime
ℏ 119886119886119953119953primedagger = minusTr 120588120588119867119867119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953prime
dagger (0)
= minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953primedagger (0) = minus lang119879119879120591120591119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953prime
dagger (0)rang = 119866119866119953119953119953119953prime(120591120591 minus 120591120591prime 0)
Where we have used the commutativity of 120588120588119867119867 and 119890119890plusmn119867119867120591120591prime
ℏ and the invariance of the trace operation
12
towards cyclic permutations Similarly for 120591120591 lt 120591120591prime we have
119866119866119953119953119953119953prime(120591120591 120591120591prime) = minuslang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime)rang = minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime) = minusTr 120588120588119867119867119886119886119953119953primedagger (120591120591prime)119886119886119953119953(120591120591)
= minusTr 120588120588119867119867119890119890119867119867120591120591primeℏ 119886119886119953119953prime
dagger 119890119890minus119867119867120591120591primeℏ 119890119890
119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591ℏ = minusTr 120588120588119867119867119886119886119953119953prime
dagger 119890119890119867119867120591120591minus120591120591prime
ℏ 119886119886119953119953119890119890minus119867119867120591120591ℏ 119890119890
119867119867120591120591primeℏ
= minusTr 120588120588119867119867119886119886119953119953primedagger 119890119890
119867119867120591120591minus120591120591primeℏ 119886119886119953119953119890119890
minus119867119867120591120591minus120591120591prime
ℏ = minusTr 120588120588119867119867119886119886119953119953primedagger (0)119886119886119953119953(120591120591 minus 120591120591prime)
= minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953primedagger (0)119886119886119953119953(120591120591 minus 120591120591prime) = minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953prime
dagger (0)
= minuslang119879119879120591120591119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953primedagger (0)rang = 119866119866119953119953119953119953prime(120591120591 minus 120591120591prime 0)
For simplicity we will introduce the notation 119866119866119953119953119953119953prime(120591120591 0) equiv 119866119866119953119953119953119953prime(120591120591) We further note that when using
imaginary time 119866119866119953119953119953119953prime(120591120591) is a periodic functions in the domain [minus120573120573ℏ120573120573ℏ] with a period of 120573120573ℏ (see
Page 236 of Ref [2])
119866119866119953119953119953119953prime(120591120591 + 120573120573ℏ 0) = 119866119866119953119953119953119953prime(120591120591 0) 120591120591 lt 0119866119866119953119953119953119953prime(120591120591 minus 120573120573ℏ 0) = 119866119866119953119953119953119953prime(120591120591 0) 120591120591 gt 0 (410)
To show this we shall again assume first that minus120573120573ℏ lt 120591120591 lt 0 to write that is
119866119866119953119953119953119953prime(120591120591 + 120573120573ℏ 0) = minus lang119879119879120591120591119886119886119953119953(120591120591 + 120573120573ℏ)119886119886119953119953primedagger (0)rang = minuslang119886119886119953119953(120591120591 + 120573120573ℏ)119886119886119953119953prime
dagger (0)rang
= minusTr 120588120588119867119867119890119890119867119867(120591120591+120573120573ℏ)
ℏ 119886119886119953119953119890119890minus119867119867(120591120591+120573120573ℏ)
ℏ 119886119886119953119953prime = minusTr 120588120588119867119867119890119890120573120573119867119867119890119890
119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591ℏ 119890119890minus120573120573119867119867119886119886119953119953prime
= minusTr120588120588119867119867119890119890120573120573119867119867119886119886119953119953(120591120591)119890119890minus120573120573119867119867119886119886119953119953prime(0) = minus
1119885119885119867119867
Tr119890119890minus120573120573119867119867119890119890120573120573119867119867119886119886119953119953(120591120591)119890119890minus120573120573119867119867119886119886119953119953prime(0)
= minus1119885119885119867119867
Tr119886119886119953119953(120591120591)119890119890minus120573120573119867119867119886119886119953119953prime(0) = minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119886119886119953119953prime(0)119886119886119953119953(120591120591)
= minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime(0) = minusTr120588120588119867119867119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime = 119866119866119953119953119953119953prime(120591120591 0)
Similarly for 120573120573ℏ gt 120591120591 gt 0 we have
119866119866119953119953119953119953prime(120591120591 minus 120573120573ℏ 0) = minus lang119879119879120591120591119886119886119953119953(120591120591 minus 120573120573ℏ)119886119886119953119953primedagger (0)rang = minuslang119886119886119953119953prime
dagger (0)119886119886119953119953(120591120591 minus 120573120573ℏ)rang
= minusTr 120588120588119867119867119886119886119953119953prime119890119890119867119867(120591120591minus120573120573ℏ)
ℏ 119886119886119953119953119890119890minus119867119867(120591120591minus120573120573ℏ)
ℏ = minusTr 120588120588119867119867119886119886119953119953prime119890119890minus120573120573119867119867119890119890
119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591ℏ 119890119890120573120573119867119867
= minusTr120588120588119867119867119886119886119953119953prime(0)119890119890minus120573120573119867119867119886119886119953119953(120591120591)119890119890120573120573119867119867 = minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119886119886119953119953prime(0)119890119890minus120573120573119867119867119886119886119953119953(120591120591)119890119890120573120573119867119867
= minus1119885119885119867119867
Tr119886119886119953119953prime(0)119890119890minus120573120573119867119867119886119886119953119953(120591120591) = minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119886119886119953119953(120591120591)119886119886119953119953prime(0)
= minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime(0) = minusTr120588120588119867119867119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime = 119866119866119953119953119953119953prime(120591120591 0)
13
Therefore 119866119866119953119953119953119953prime(120591120591) can be expanded as a Fourier series in the domain [0120573120573ℏ] as follows
119866119866119953119953119953119953prime(120591120591) = 1120573120573ℏsum 119890119890minus119894119894120596120596119899119899120591120591119866119866119953119953119953119953prime(119894119894120596120596119899119899)infinminusinfin (411)
where 120596120596119899119899 = 2120587120587119899119899120573120573ℏ
and the associated Fourier coefficient is given by
119866119866119953119953119953119953prime(119894119894120596120596119899119899) = int d120591120591119890119890119894119894120596120596119899119899120591120591119866119866119953119953119953119953prime(120591120591)120573120573ℏ0 120596120596119899119899 = 2119899119899120587120587
120573120573ℏ (412)
Having proven the translational time invariance and the periodicity of the Greenrsquos functios we can
now calculate it for the uncoupled system
119866119866119953119953119953119953prime0 (120591120591 120591120591prime) = minuslang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime)rang0 == minus lang119886119886119953119953(0)119890119890minus120596120596119953119953120591120591119886119886119953119953prime
dagger (0)119890119890120596120596119953119953prime120591120591primerang 120591120591 minus 120591120591prime gt 0
minus lang119886119886119953119953primedagger (0)119890119890120596120596119953119953prime120591120591
prime119886119886119953119953(0)119890119890minus120596120596119953119953120591120591rang 120591120591 minus 120591120591prime lt 0
= minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591
primelang119886119886119953119953(0)119886119886119953119953prime
dagger (0)rang 120591120591 minus 120591120591prime gt 0
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang 120591120591 minus 120591120591prime lt 0
= minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591
primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0) + 119886119886119953119953(0)119886119886119953119953primedagger (0)rang 120591120591 minus 120591120591prime gt 0
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang 120591120591 minus 120591120591prime lt 0
= minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591
primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang + 120575120575119953119953119953119953prime 120591120591 minus 120591120591prime gt 0
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang 120591120591 minus 120591120591prime lt 0
ie
119866119866119953119953119953119953prime0 (120591120591 120591120591prime) =
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang + 120575120575119953119953119953119953prime 120591120591 minus 120591120591prime gt 0
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang 120591120591 minus 120591120591prime lt 0 (413)
where we have used Eq (46) for 119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime) and Commutation relation 119886119886119953119953(0)119886119886119953119953prime
dagger (0) = 120575120575119953119953119953119953prime
To calculate lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang we first prove following equation
119890119890119860119860119861119861119890119890minus119860119860 = sum 1119899119899119860(119899119899)119861119861infin
119899119899=0 (414)
where 119860(119896119896)119861119861 equiv 119860 119860(119896119896minus1)119861119861 To prove this identity we define the operator 119891119891(119905119905) = 119890119890119905119905A119861119861119890119890minus119905119905119860119860
and taylor expand it around 119905119905 = 0
119891119891(119905119905) = 119891119891(0) + 119905119905119891119891prime(0) + 1199051199052
2119891119891primeprime(0) + ⋯ = sum 119905119905119899119899
119899119899119889119889119899119899119891119889119889119905119905119899119899
119905119905=0
infin119899119899=0 (415)
The corresponding derivatives are given by
⎩⎪⎨
⎪⎧ 119891119891prime(119905119905) = 119890119890119905119905119860119860119860119861119861119890119890minus119905119905119860119860 minus 119890119890119905119905119860119860119861119861119860119890119890minus119905119905119860119860 = 119890119890119905119905119860119860119860119861119861119890119890minus119905119905119860119860
119891119891primeprime(119905119905) = 119890119890119905119905119860119860119860119860119861119861 minus 119860119861119861119860119890119890minus119905119905119860119860 = 119890119890119905119905119860119860119860(2)119861119861119890119890minus119905119905119860119860
⋮119891119891(119899119899)(119905119905) = 119890119890119905119905119860119860119860(119899119899)119861119861119890119890minus119905119905119860119860
(416)
14
Substituting Eq (416) into Eq (415) we have
119890119890119905119905A119861119861119890119890minus119905119905119860119860 = sum 119905119905119899119899
119899119899119860(119899119899)119861119861infin
119899119899=0 (417)
Itrsquos clear that Eq (413) is a spectial case of Eq (416) with 119905119905 = 1 With this we can proceed as
follows
lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang =
11198851198851198671198670
Tr119890119890minus1205731205731198671198670119886119886119953119953primedagger (0)119886119886119953119953(0) =
11198851198851198671198670
Tr119890119890minus1205731205731198671198670119886119886119953119953primedagger (0)1198901198901205731205731198671198670119890119890minus1205731205731198671198670119886119886119953119953(0)
=11198851198851198671198670
Tr (minus120573120573)119899119899
1198991198991198671198670
(119899119899)119886119886119953119953primedagger (0)
infin
119899119899=0
119890119890minus1205731205731198671198670119886119886119953119953(0)
Note that 1198671198670119886119886119953119953primedagger (0) = ℏ120596120596119953119953prime119886119886119953119953prime
dagger (0) we have
1198671198670(119899119899)119886119886119953119953prime
dagger (0) = ℏ120596120596119953119953prime119899119899119886119886119953119953primedagger (0) (418)
such that
lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang =
11198851198851198671198670
Tr (minus120573120573)119899119899
1198991198991198671198670
(119899119899)119886119886119953119953primedagger (0)
infin
119899119899=0
119890119890minus1205731205731198671198670119886119886119953119953(0)
=11198851198851198671198670
Tr minus120573120573ℏ120596120596119953119953prime
119899119899
119899119899119886119886119953119953primedagger (0)
infin
119899119899=0
119890119890minus1205731205731198671198670119886119886119953119953(0)
= 119890119890minus120573120573ℏ120596120596119953119953prime11198851198851198671198670
Tr119890119890minus1205731205731198671198670119886119886119953119953(0)119886119886119953119953primedagger (0) = 119890119890minus120573120573ℏ120596120596119953119953primelang119886119886119953119953(0)119886119886119953119953prime
dagger (0)rang
= 119890119890minus120573120573ℏ120596120596119953119953primelang119886119886119953119953primedagger (0)119886119886119953119953(0)rang + 120575120575119953119953119953119953prime
therefore we obtain lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang 119890119890120573120573ℏ120596120596119953119953prime minus 1 = 120575120575119953119953119953119953prime For 119953119953prime ne 119953119953 and general 119890119890120573120573ℏ120596120596119953119953prime we have
lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang = 0 For 119953119953prime = 119953119953 we obtain lang119886119886119953119953
dagger(0)119886119886119953119953(0)rang = 119890119890120573120573ℏ120596120596119953119953 minus 1minus1
Hence we have
lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang = 120575120575119953119953prime119953119953
1119890119890120573120573ℏ120596120596119953119953minus1
equiv 120575120575119953119953prime119953119953119899119899119861119861120596120596119953119953 (419)
where 119899119899119861119861120596120596119953119953 is the Bose-Einstein distribution for phonons Substituting Eq (418) in Eq (413)
we obtain
119866119866119953119953119953119953prime0 (120591120591 120591120591prime) =
minus120575120575119953119953119953119953prime1 + 119899119899119861119861120596120596119953119953119890119890minus120596120596119953119953120591120591minus120591120591prime 120591120591 minus 120591120591prime gt 0
minus120575120575119953119953119953119953prime119899119899119861119861120596120596119953119953119890119890minus120596120596119953119953120591120591minus120591120591prime 120591120591 minus 120591120591prime lt 0 (420)
Note that only those Greenrsquos functions of the form 119866119866119953119953119953119953prime0 (120591120591 120591120591prime) = 1198661198661199531199530(120591120591 minus 120591120591prime)120575120575119953119953119953119953prime are non-zero due
to the orthogonality of normal modes Looking at the non-vanishing terms and setting 120591120591prime = 0
without loss of generality we can calculate the Fourier transform of 1198661198661199531199530(120591120591) from Eq (412)
1198661198661199531199530(119894119894120596120596119899119899) = int d1205911205911198901198901198941198941205961205961198991198991205911205911198661198661199531199530(120591120591)120573120573ℏ0 = minusint d120591120591119890119890119894119894120596120596119899119899120591120591120573120573ℏ
0 1 + 119899119899119861119861120596120596119953119953119890119890minus120596120596119953119953120591120591 = minus1+119899119899119861119861120596120596119953119953119894119894120596120596119899119899minus120596120596119953119953
119890119890119894119894120596120596119899119899minus120596120596119953119953120573120573ℏ minus
15
1 = minus1+ 1
119890119890120573120573ℏ120596120596119953119953minus1
119894119894120596120596119899119899minus120596120596119953119953119890119890119894119894
2120587120587119899119899120573120573ℏ 120573120573ℏ119890119890minus120573120573ℏ120596120596119953119953 minus 1 = minus 119890119890120573120573ℏ120596120596119953119953
119894119894120596120596119899119899minus120596120596119953119953
119890119890minus120573120573ℏ120596120596119953119953minus1
119890119890120573120573ℏ120596120596119953119953minus1= minus 1
119894119894120596120596119899119899minus120596120596119953119953
1minus119890119890120573120573ℏ120596120596119953119953
119890119890120573120573ℏ120596120596119953119953minus1= 1
119894119894120596120596119899119899minus120596120596119953119953
ie
1198661198661199531199530(119894119894120596120596119899119899) == 1119894119894120596120596119899119899minus120596120596119953119953
(421)
5 Fermirsquos golden rule 51 Interaction picture
Next we can proceed with calculating the Greenrsquos function of the coupled system To this end we
define the coupling Hamiltonian operator term in the interaction picture as
119867119867119862119862119868119868 (120591120591) = 1198901198901198671198670120591120591ℏ 119867119867119862119862119890119890
minus1198671198670120591120591ℏ (51)
The time evolution of 119867119867119862119862(120591120591) is then given by
ℏ part119867119867119862119862119868119868 (120591120591)120597120597120591120591
= 1198671198670119867119867119862119862119868119868 (120591120591) (52)
Note that the Greenrsquos function defined above was given in the Heisenberg picture To proceed we
need to transform it to the interaction picture
119866119866119953119953119953119953prime(120591120591 0) = minus lang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953primedagger (0)rang = minusTr 120588120588119867119867119879119879120591120591 119890119890
119867119867120591120591ℏ 119886119886119953119953(0)119890119890minus
119867119867120591120591ℏ 119886119886119953119953prime
dagger (0)
= minusTr 120588120588119867119867119879119879120591120591 119890119890119867119867120591120591ℏ 119890119890minus
1198671198670120591120591ℏ 119890119890
1198671198670120591120591ℏ 119886119886119953119953(0)119890119890minus
1198671198670120591120591ℏ 119890119890
1198671198670120591120591ℏ 119890119890minus
119867119867120591120591ℏ 119886119886119953119953prime
dagger (0)
= minusTr 120588120588119867119867119879119879120591120591 119880119880(0 120591120591)119886119886119953119953119868119868 (120591120591)119880119880(120591120591 0)119886119886119953119953primedagger (0)
where
119886119886119953119953119868119868 (120591120591) equiv 1198901198901198671198670120591120591ℏ 119886119886119953119953(0)119890119890minus
1198671198670120591120591ℏ (53)
is the operator in interaction picture and the operator 119880119880 is defined by
119880119880(1205911205911 1205911205912) equiv 11989011989011986711986701205911205911ℏ 119890119890minus
119867119867 (1205911205911minus1205911205912)ℏ 119890119890minus
11986711986701205911205912ℏ (54)
Note that while 119880119880 is not unitary it satisfies the following group property
119880119880(1205911205911 1205911205912)119880119880(1205911205912 1205911205913) = 119880119880(1205911205911 1205911205913) (55)
and the boundary condition 119880119880(1205911205911 1205911205911) = 1 In addition the 120591120591 derivative of 119880119880 is simply
ℏpart119880119880(120591120591 120591120591prime)
120597120597120591120591= 1198671198670119890119890
1198671198670120591120591ℏ 119890119890minus
119867119867120591120591minus120591120591primeℏ minus 119890119890
1198671198670120591120591ℏ 119867119867119890119890minus
119867119867120591120591minus120591120591primeℏ 119890119890minus
1198671198670120591120591primeℏ
= 1198901198901198671198670120591120591ℏ 1198671198670 minus 119867119867119890119890minus
1198671198670120591120591ℏ 119890119890
1198671198670120591120591ℏ 119890119890minus
119867119867120591120591minus120591120591primeℏ 119890119890minus
1198671198670120591120591primeℏ = 119890119890
1198671198670120591120591ℏ minus119867119867119862119862119890119890
minus1198671198670120591120591ℏ 119880119880(120591120591 120591120591prime)
= minus119867119867119862119862119868119868 (120591120591)119880119880(120591120591 120591120591prime)
16
ie
ℏ part119880119880120591120591120591120591prime120597120597120591120591
= minus119867119867119862119862119868119868 (120591120591)119880119880(120591120591 120591120591prime) (56)
The solution of Eq (56) is (see page 235 of Ref [2])
119880119880(120591120591 120591120591prime) = summinus1ℏ
119899119899
119899119899 int 1198891198891205911205911120591120591120591120591prime ⋯int 119889119889120591120591119899119899
120591120591120591120591prime 119879119879120591120591119867119867119862119862119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119899119899)infin
119899119899=0 (57)
The exact thermal Greens function now may be rewritten in the interaction picture as
119866119866119953119953119953119953prime(120591120591 0) = minus1119885119885119867119867
Tr 119890119890minus120573120573119867119867119879119879120591120591119880119880(0 120591120591)119886119886119953119953119868119868 (120591120591)119880119880(120591120591 0)119886119886119953119953primedagger (0)
= minusTr 119890119890minus120573120573119867119867119879119879120591120591 119880119880(0 120591120591)119886119886119953119953119868119868 (120591120591)119880119880(120591120591 0)119886119886119953119953prime
dagger (0)
Tr119890119890minus120573120573119867119867
= minusTr 119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)119879119879120591120591 119886119886119953119953119868119868 (120591120591)119880119880(0 120591120591)119880119880(120591120591 0)119886119886119953119953prime
dagger (0)
Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)
= minusTr 119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)119879119879120591120591 119886119886119953119953119868119868 (120591120591)119880119880(00)119886119886119953119953prime
dagger (0)
Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)=
= minusTr 119890119890minus1205731205731198671198670119879119879120591120591 119880119880(120573120573ℏ 0)119886119886119953119953119868119868 (120591120591)119886119886119953119953prime
dagger (0)
Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)
Where we used the fact that we are free to change the order of the operators within the time ordering
operation (see pages 241-242 of Ref [2])
52 Wickrsquos theorem
Thus the Greenrsquos function can be expanded as [2]
119866119866119953119953119953119953prime(120591120591 0) = minussum 1
119898119898minus1ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)119886119886119953119953119868119868 (120591120591)119886119886
119953119953primedagger (0)rang0infin
119898119898=0
sum 1119898119898minus
1ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)rang0infin
119898119898=0 (58)
where lang⋯ rang0 represents the ensemble average with respect to the non-interacting basis
Tr119890119890minus1205731205731198671198670(⋯ ) Or explicitly
119866119866119953119953119953119953prime(120591120591 0) = minus119866119866119953119953119953119953prime0 (120591120591)minus1ℏint 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953primedagger (0)rang0+
12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0+⋯
1minus1ℏint 1198891198891205911205911120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)rang0+12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0+⋯
(59)
For consiceness we introduce the following notation
119863119863119898119898 equiv 1119898119898minus 1
ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)rang0 (510)
To simplify the calculation in Eq (58) we adopt Wickrsquos theorem (see pages 237-241 of Ref
[2])which can be expressed as follows
lang119879119879120591120591119860119861119861119862119863119863 ⋯119884119884119885rang0 = lang119879119879120591120591119860119861119861rang0lang119879119879120591120591119862119863119863rang0⋯ lang119879119879120591120591119884119884119885rang0 + lang119879119879120591120591119860119862rang0lang119879119879120591120591119861119861119863119863rang0⋯ lang119879119879120591120591119883119883119885rang0 + ⋯ (511)
17
Here 119860119861119861 hellip 119884119884 119885 represent 119886119886119953119953(120591120591) or 119886119886119953119953dagger(120591120591) In Eq (511) each term corresponds to a particular
pairing of the operators 119860119861119861119862119863119863 ⋯119884119884119885 and all possible pairings are taken into account Here the only
non-vanishing propagators will have the form [see Eq (420)]
lang119879119879120591120591 119886119886119953119953primedagger (120591120591prime)119886119886119953119953(120591120591)rang0 = minuslang119879119879120591120591 119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime)rang0 = 119866119866119953119953119953119953prime0 (120591120591 120591120591prime) = 1198661198661199531199530(120591120591 minus 120591120591prime)120575120575119953119953119953119953prime (512)
Therefore the second term in the numerator of Eq (59) can be calculated as
1198681198682 = minus1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0
= minus1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 lang119879119879120591120591119886119886119953119953prime
dagger (0)119886119886119953119953(120591120591)rang0 minus1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
= 119866119866119953119953119953119953prime0 (120591120591 0) minus
1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 minus
1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
The first term in above equation is called disconnected part since the pairing is performed separately
on 119867119867119862119862(1205911205911) and 119886119886119953119953(120591120591)119886119886119953119953primedagger (0) All other terms have pairs that mix creation and annihilation operators
of the Hamiltonian with 119886119886119953119953(120591120591) or 119886119886119953119953primedagger (0) and are said to have connected party For simplicity we
include all these terms in the notation lang119879119879120591120591119867119867119862119862(1205911205911)119886119886119953119953(120591120591)119886119886119953119953primedagger (0)rang0119888119888 Furthermore we define 119866119866119953119953119953119953prime
(1) (120591120591) equiv
minus1ℏ int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862(1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 Then we have
1198681198683 = minus1ℏint 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 = 119866119866119953119953119953119953prime0 (120591120591 0)1198631198631 + 119866119866119953119953119953119953prime
(1) (120591120591) (513)
Using this method the third term in the numerator of Eq (59) can be calculated as
1198681198683 = 12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 = 12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0 lang119879119879120591120591119886119886119953119953prime
dagger (0)119886119886119953119953(120591120591)rang0 +
12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 + 12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 +
12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 = 119866119866119953119953119953119953prime0 (120591120591 0) 1
2ℏ2 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0 +
12ℏ2 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 + 12ℏ2 int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 +
12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
Here the last term is denoted as 119866119866119953119953119953119953prime(2) (120591120591)
Using Eq (513) the second and third terms on the right hand above equation can be simplified as
follows
18
12ℏ2
1198891198891205911205912120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 +1
2ℏ2 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
=12 minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0
minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
+12 minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0
minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
=12 minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0
minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
+12 minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0
minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
= minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0
minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 = 1198631198631119866119866119953119953119953119953prime(1) (120591120591)
Where we performed the following integration variables interchange 1205911205911 ⟷ 1205911205912 Thus we have
1198681198683 = 119866119866119953119953119953119953prime(2) (120591120591) + 119866119866119953119953119953119953prime
(1) (120591120591)1198631198631 + 119866119866119953119953119953119953prime0 (120591120591 0)1198631198632 (514)
Similarity we can calculate the fourth term of Eq (59) as
1198681198684 = minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 =
minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0 lang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 + 3 times
minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 + 3 times
minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 +
minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 =
119866119866119953119953119953119953prime0 (120591120591 0) minus1
6ℏ3 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0+
12ℏ2 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0
minus1ℏ int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888+
12ℏ2 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 minus1ℏ int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0+ 119866119866119953119953119953119953prime
(3) (120591120591) = 119866119866119953119953119953119953prime(3) (120591120591) +
119866119866119953119953119953119953prime(2) (120591120591)1198631198631 + 119866119866119953119953119953119953prime
(1) (120591120591)1198631198632 + 119866119866119953119953119953119953prime0 (120591120591 0)1198631198633
Higher order terms can be treated in the same manner (see pages 95-96 of Ref [2]) Then when
substituting 1198681198682 1198681198683 1198681198684 and all higher order terms into Eq (59) we can simplify the numerator as
follows
119866119866119953119953119953119953prime0 (120591120591) minus
1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0
+1
2ℏ2 1198891198891205911205911120573120573ℏ
0 1198891198891205911205912120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 + ⋯
= 119866119866119953119953119953119953prime0 (120591120591) + 119866119866119953119953119953119953prime
(1) (120591120591) + 119866119866119953119953119953119953prime(2) (120591120591) + ⋯ (1 + 1198631198631 + 1198631198632 + ⋯ )
Noting that the expression (1 + 1198631198631 + 1198631198632 + ⋯ ) is exactly canceled with the denominator the
Greenrsquos function can be simplified as
19
119866119866119953119953119953119953prime(120591120591) = 119866119866119953119953119953119953prime0 (120591120591) + 119866119866119953119953119953119953prime
(1) (120591120591) + 119866119866119953119953119953119953prime(2) (120591120591) + ⋯ (515)
where
119866119866119953119953119953119953prime
(1) (120591120591) = minus1ℏ int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888
119866119866119953119953119953119953prime(2) (120591120591) = 1
2ℏ2 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888 (516)
53 Calculation of Greenrsquos function
531 First order appriximation In Eq (516) we go back to the full notation 119886119886119954119954120582120582 to distinguish phonons of different branches
then 119866119866119953119953119953119953prime(1) (120591120591) is calculated as
119866119866119953119953119953119953prime(1) (120591120591) = minus
1ℏ
12 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591 119881119881119895119895119895119895prime(119948119948)119876119876119948119948119895119895(1205911205911)119876119876119948119948119895119895prime
dagger (1205911205911)3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ 119881119881119895119895119895119895primelowast (119948119948)119876119876119948119948119895119895prime(1205911205911)119876119876119948119948119895119895
dagger (1205911205911)3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= minus12ℏ
1198891198891205911205911120573120573ℏ
0lang119879119879120591120591
ℏ119881119881119895119895119895119895prime(119948119948)2120596120596119948119948119895119895120596120596119948119948119895119895prime
119886119886119948119948119895119895(1205911205911) + 119886119886minus119948119948119895119895dagger (1205911205911) 119886119886minus119948119948119895119895prime(1205911205911) + 119886119886119948119948119895119895prime
dagger (1205911205911)3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ ℏ119881119881119895119895119895119895prime
lowast (119948119948)
2120596120596119948119948119895119895120596120596119948119948119895119895prime119886119886119948119948119895119895prime(1205911205911) + 119886119886minus119948119948119895119895prime
dagger (1205911205911) 119886119886minus119948119948119895119895(1205911205911) + 119886119886119948119948119895119895dagger (1205911205911)
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
According to Wickrsquos theorem [2] the terms that contain 119886119886119948119948119895119895119886119886minus119948119948119895119895prime and 119886119886minus119948119948119895119895dagger 119886119886119948119948119895119895prime
dagger are equal to zero
thus the first term of above equation are calculated as
11989211989211 = minus14
119881119881119895119895119895119895prime(119948119948)
120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119886119886minus119948119948119895119895
dagger (1205911205911)119886119886minus119948119948119895119895prime(1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ lang119879119879120591120591119886119886119948119948119895119895(1205911205911)119886119886119948119948119895119895primedagger (1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888
= minus14
119881119881119895119895119895119895prime(119948119948)
120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119886119886minus119948119948119895119895
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus119948119948119895119895prime(1205911205911)rang0
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ lang119879119879120591120591119886119886119948119948119895119895primedagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)119886119886119948119948119895119895(1205911205911)rang0
= minus14
119881119881119895119895119895119895prime(119948119948)
120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
01198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954minus119948119948120575120575120582120582119895119895119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119954119954primeminus119948119948120575120575120582120582prime119895119895prime3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ 1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119948119948120575120575120582120582119895119895prime119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119954119954prime119948119948120575120575120582120582prime119895119895
20
Simplification of above equation gives
11989211989211 =
⎩⎪⎨
⎪⎧minus119881119881120582120582120582120582prime(minus119954119954)
4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954120582120582
0 (120591120591 1205911205911)119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903
2lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582(119954119954)
4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954120582120582
0 (120591120591 1205911205911)119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582prime le 3119903119903
2lt 120582120582 le 3119903119903
0 else
(517)
Similarity the second term of 119866119866119953119953119953119953prime(1) (120591120591) is calculated as
11989211989212 = minus14
minus119881119881119895119895119895119895primelowast (119948119948)
4120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119886119886119948119948119895119895prime(1205911205911)119886119886119948119948119895119895
dagger (1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ lang119879119879120591120591119886119886minus119948119948119895119895primedagger (1205911205911)119886119886minus119948119948119895119895(1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888
= minus14
minus119881119881119895119895119895119895primelowast (119948119948)
4120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)119886119886119948119948119895119895prime(1205911205911)rang0 lang119879119879120591120591119886119886119948119948119895119895dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus119948119948119895119895(1205911205911)rang0 lang119879119879120591120591119886119886minus119948119948119895119895prime
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
ie
11989211989212 =
⎩⎪⎨
⎪⎧
minus119881119881120582120582120582120582primelowast (119954119954)
4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954prime120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903
2lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582lowast (minus119954119954)
4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954prime120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119954119954prime 1 le 120582120582prime le 3119903119903
2lt 120582120582 le 3119903119903
0 else
(518)
Note that since 119881119881120582120582120582120582primelowast (119954119954) = 119881119881120582120582120582120582prime(minus119954119954) [Eq (314)] the first order approximation of Greenrsquos function
can be written as
119866119866119954119954120582120582119954119954prime120582120582prime(1) (120591120591) =
⎩⎪⎨
⎪⎧minus119881119881
120582120582120582120582prime(minus119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ
0 1198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582 le3119903119903
2lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582
(119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ
0 1198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582prime le3119903119903
2lt 120582120582 le 3119903119903
0 else
(519)
To derive the Greenrsquos function in frequency domain we use the expression for Fourier series then
we have
1198891198891205911205911120573120573ℏ
01198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0) = 1198891198891205911205911120573120573ℏ
0
1120573120573ℏ
119890119890minus119894119894120596120596119899119899(120591120591minus1205911205911)1198661198661199541199541205821205820 (119894119894120596120596119899119899)
119899119899
1120573120573ℏ
119890119890minus119894119894120596120596119899119899prime1205911205911119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899prime)
119899119899prime
=1
(120573120573ℏ)2119890119890minus119894119894120596120596119899119899120591120591 1198891198891205911205911120573120573ℏ
0119890119890119894119894120596120596119899119899minus120596120596119899119899prime1205911205911 119866119866119954119954120582120582
0 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899prime)
119899119899119899119899prime=
=1
(120573120573ℏ)2119890119890minus119894119894120596120596119899119899120591120591[120573120573ℏ120575120575119899119899119899119899prime]1198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899prime)119899119899119899119899prime
=1120573120573ℏ
119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899)119899119899
According to the definition of Fourier series [see Eq (412)] we get the Fourier coefficient of
21
119866119866119954119954120582120582119954119954prime120582120582prime(1) (120591120591)
119866119866119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) =
⎩⎪⎨
⎪⎧minus119881119881120582120582120582120582prime(minus119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582prime1198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime 1 le 120582120582 le 31199031199032
lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582(119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582prime1198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime 1 le 120582120582prime le 31199031199032
lt 120582120582 le 3119903119903
0 else
(520)
where 119866119866119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) = 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) times Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) times 119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899) and Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) is defined as
Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) =
⎩⎪⎨
⎪⎧minus119881119881120582120582120582120582prime(minus119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582prime120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903
2lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582(119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582prime120575120575119954119954119954119954prime 1 le 120582120582prime le 3119903119903
2lt 120582120582 le 3119903119903
0 else
(521)
532 second order approximation The second order approximation of the Greenrsquos function in Eq (516) 119866119866119954119954120582120582119954119954prime120582120582prime
(2) (120591120591) is calculated as
119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) =
18ℏ2 1198891198891205911205911
120573120573ℏ
0 1198891198891205911205912120573120573ℏ
0lang119879119879120591120591
⎣⎢⎢⎡ 11988111988111989511989511198951198951prime(119948119948120783120783)1198761198761199481199481207831207831198951198951(1205911205911)1198761198761199481199481207831207831198951198951prime
dagger (1205911205911)3119903119903
1198951198951prime=31199031199032 +1
31199031199032
1198951198951=1119948119948120783120783
+ 11988111988111989411989411198941198941primelowast (119948119948120783120783)1198761198761199481199481207831207831198941198941prime(1205911205911)1198761198761199481199481207831207831198941198941
dagger (1205911205911)3119903119903
1198941198941prime=31199031199032 +1
31199031199032
1198941198941=1119948119948120783120783 ⎦⎥⎥⎤
⎣⎢⎢⎡ 11988111988111989511989521198951198952prime(119948119948120784120784)1198761198761199481199481207841207841198951198952(1205911205912)1198761198761199481199481207841207841198951198952prime
dagger (1205911205912)3119903119903
1198951198952prime=31199031199032 +1
31199031199032
1198951198952=1119948119948120784120784
+ 11988111988111989411989421198941198942primelowast (119948119948120784120784)1198761198761199481199481207841207841198941198942prime(1205911205912)1198761198761199481199481207841207841198941198942
dagger (1205911205912)3119903119903
1198941198942prime=31199031199032 +1
31199031199032
1198941198942=1119948119948120784120784 ⎦⎥⎥⎤119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888 = 1198921198921 + 1198921198922 + 1198921198923 + 1198921198924
where
1198921198921 = 18ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0
sum sum 11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989511989521198951198952prime(119948119948120784120784)1le1198951198952le
31199031199032 lt1198951198952
primele311990311990311994811994812078412078411989511989521198951198952prime
1le1198951198951le31199031199032 lt1198951198951
primele311990311990311994811994812078312078311989511989511198951198951prime
times lang1198791198791205911205911198761198761199481199481207831207831198951198951(1205911205911)1198761198761199481199481207831207831198951198951primedagger (1205911205911)1198761198761199481199481207841207841198951198952(1205911205912)1198761198761199481199481207841207841198951198952prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(522)
1198921198922 = 18ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0
sum sum 11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942primelowast (119948119948120784120784)
1le1198941198942le31199031199032 lt1198941198942
primele311990311990311994811994812078412078411989411989421198941198942prime
1le1198951198951le31199031199032 lt1198951198951
primele311990311990311994811994812078312078311989511989511198951198951prime
times lang1198791198791205911205911198761198761199481199481207831207831198951198951(1205911205911)1198761198761199481199481207831207831198951198951primedagger (1205911205911)1198761198761199481199481207841207841198941198942prime (1205911205912)1198761198761199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(523)
1198921198923 = 18ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0
sum sum 11988111988111989411989411198941198941primelowast (119948119948120783120783)11988111988111989511989521198951198952prime(119948119948120784120784)
1le1198951198952le31199031199032 lt1198951198952
primele311990311990311994811994812078412078411989511989521198951198952prime
1le1198941198941le31199031199032 lt1198941198941
primele311990311990311994811994812078312078311989411989411198941198941prime
times lang1198791198791205911205911198761198761199481199481207831207831198941198941prime (1205911205911)1198761198761199481199481207831207831198941198941dagger (1205911205911)1198761198761199481199481207841207841198951198952(1205911205912)1198761198761199481199481207841207841198951198952prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(524)
22
1198921198924 = 18ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0
sum sum 11988111988111989411989411198941198941primelowast (119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)1le1198941198942le
31199031199032 lt1198941198942
primele311990311990311994811994812078412078411989411989421198941198942prime
1le1198941198941le31199031199032 lt1198941198941
primele311990311990311994811994812078312078311989411989411198941198941prime
times lang1198791198791205911205911198761198761199481199481207831207831198941198941prime (1205911205911)1198761198761199481199481207831207831198941198941dagger (1205911205911)1198761198761199481199481207841207841198941198942prime (1205911205912)1198761198761199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(525)
In what follows we will expand the term 1198921198922 The expansion of all other terms follows the same lines
and eventually results in the same expression Expanding the product 11987611987611994811994812078312078311989511989511198761198761199481199481207831207831198951198951primedagger 1198761198761199481199481207841207841198941198942prime1198761198761199481199481207841207841198941198942
dagger we can
rewrite Eq (523) as follows
1198921198922 = 132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)119881119881
11989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032 lt1198941198942
primele311990311990311994811994812078412078411989411989421198941198942prime
1le1198951198951le31199031199032 lt1198951198951
primele311990311990311994811994812078312078311989511989511198951198951prime
times 119868119868 (526)
Where
119868119868 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911) + 119886119886minus1199481199481207831207831198951198951dagger (1205911205911) 119886119886minus1199481199481207831207831198951198951prime(1205911205911) + 1198861198861199481199481207831207831198951198951prime
dagger (1205911205911) 1198861198861199481199481207841207841198941198942prime (1205911205912) + 119886119886minus1199481199481207841207841198941198942primedagger (1205911205912) 119886119886minus1199481199481207841207841198941198942(1205911205912) + 1198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0c
= lang119879119879120591120591 1198861198861199481199481207831207831198951198951119886119886minus1199481199481207831207831198951198951prime + 11988611988611994811994812078312078311989511989511198861198861199481199481207831207831198951198951primedagger + 119886119886minus1199481199481207831207831198951198951
dagger 119886119886minus1199481199481207831207831198951198951prime + 119886119886minus1199481199481207831207831198951198951dagger 1198861198861199481199481207831207831198951198951prime
dagger 12059112059111198861198861199481199481207841207841198941198942prime119886119886minus1199481199481207841207841198941198942 + 1198861198861199481199481207841207841198941198942prime1198861198861199481199481207841207841198941198942
dagger + 119886119886minus1199481199481207841207841198941198942primedagger 119886119886minus1199481199481207841207841198941198942
+ 119886119886minus1199481199481207841207841198941198942primedagger 1198861198861199481199481207841207841198941198942
dagger 1205911205912119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0c
= lang119879119879120591120591 11988611988611994811994812078312078311989511989511198861198861199481199481207831207831198951198951primedagger + 119886119886minus1199481199481207831207831198951198951
dagger 119886119886minus1199481199481207831207831198951198951prime12059112059111198861198861199481199481207841207841198941198942prime1198861198861199481199481207841207841198941198942
dagger + 119886119886minus1199481199481207841207841198941198942primedagger 119886119886minus11994811994812078412078411989411989421205911205912
119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
Where the symbol [hellip ]1205911205911 signifies that the operators in the brackets are given in the interaction
picture The last equality in above equation comes from the fact that the contractions of the product
of the operators are equal to zero when the number of creation and annihilation operators are not the
same (see Wickrsquos theorem in Ref [2]) Using Wickrsquos theorem [2] we are able to calculate the above
equation term by term as follows
⎩⎪⎨
⎪⎧ 1198681198681 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951prime
dagger (1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)1198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888
1198681198682 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
1198681198683 = lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)1198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
1198681198684 = lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(527)
We shall now calculate them term by term
23
1198681198681 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951primedagger (1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)1198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942
dagger (1205911205912)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
+ lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951prime
dagger (1205911205911)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0
+ lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)1198861198861199481199481207841207841198941198942prime(1205911205912)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0119888119888
= 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199481199481207841207841198951198952
0 (1205911205912 1205911205912)12057512057511989411989421198941198942prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582
+ 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 1198661198661199481199481207831207831198951198951
0 (1205911205911 1205911205911)12057512057511989511989511198951198951prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582
+ 1198661198661199481199481207831207831198951198951prime0 (1205911205912 1205911205911)1205751205751199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951prime ∙ 119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582
= 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582
+ 1198661198661199481199481207831207831198951198951prime0 (1205911205912 1205911205911)1205751205751199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951prime ∙ 119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582
The last equality results from the fact that 1198951198951 and 1198951198951prime are indices of different ranges (belonging to
different subsystems) and the same holds for 1198941198942 and 1198941198942prime Similarity
1198681198684 = lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0
= 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 119866119866minus11994811994812078412078411989411989420 (1205911205912 1205911205912)12057512057511989411989421198941198942prime
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 119866119866minus1199481199481207831207831198951198951prime 0 (1205911205911 1205911205912)1205751205751199481199481207831207831199481199481207841207841205751205751198951198951prime1198941198942prime
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205911 1205911205911)12057512057511989511989511198951198951prime
+ 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205912 1205911205911)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942
= 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 119866119866minus1199481199481207831207831198951198951prime 0 (1205911205911 1205911205912)1205751205751199481199481207831207831199481199481207841207841205751205751198951198951prime1198941198942prime
+ 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205912 1205911205911)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942
Where again the last equality results from the fact that 1198951198951 and 1198951198951prime are indices of different ranges
(belonging to different subsystems) and the same holds for 1198941198942 and 1198941198942prime Similarly we obtain 1198681198682 =
1198681198683 = 0 for the same reason as shown below
24
1198681198682 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951prime
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0
+ lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)1198861198861199481199481207831207831198951198951(1205911205911)rang0
+ lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0
= 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582 ∙ 119866119866minus11994811994812078412078411989411989420 (1205911205912 1205911205912)12057512057511989411989421198941198942prime
+ 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)120575120575minus11994811994812078412078411994811994812078312078312057512057511989511989511198941198942prime ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205911)12057512057511989511989511198951198951prime
+ 1198661198661199481199481207831207831198951198951prime0 (1205911205912 1205911205911)120575120575minus11994811994812078412078411994811994812078312078312057512057511989411989421198951198951prime ∙ 119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 = 0
1198681198683 = lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)1198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0
+ lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0
+ lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0
= 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 11986611986611994811994812078412078411989411989420 (1205911205912 1205911205912)12057512057511989411989421198941198942prime
+ 11986611986611994811994812078412078411989411989420 (1205911205911 1205911205912)120575120575minus1199481199481207831207831199481199481207841207841205751205751198951198951prime1198941198942 ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205911 1205911205911)12057512057511989511989511198951198951prime
+ 1198661198661199481199481207841207841198941198942prime0 (1205911205912 1205911205911)120575120575minus1199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951 ∙ 119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 = 0
The two non-vanishing terms (1198681198681 and 1198681198684) produce the following contributions to 1198921198922 of Eq (526)
11989211989221 = 132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime1198681198681 =
132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime1198661198661199481199481207831207831198951198951
0 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙
1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582 + 1
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime1198661198661199481199481207831207831198951198951prime
0 (1205911205912 1205911205911)1205751205751199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951prime ∙
119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 =
⎩⎪⎨
⎪⎧
120575120575119954119954119954119954prime
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast (119954119954prime)
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (1205911205911 1205911205912) ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
120575120575119954119954119954119954prime
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032+1 1198661198661199541199541198951198951prime
0 (1205911205912 1205911205911) ∙ 119866119866119954119954prime120582120582prime0 (1205911205911 0)119866119866119954119954120582120582
0 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032
0 else
and
25
11989211989224 = 132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime1198681198684 =
132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙
119866119866minus1199481199481207831207831198951198951prime 0 (1205911205911 1205911205912)1205751205751199481199481207831207831199481199481207841207841205751205751198951198951prime1198941198942prime + 1
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙
1198661198661199541199541205821205820 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 119866119866minus1199481199481207831207831198951198951
0 (1205911205912 1205911205911)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 =
⎩⎪⎨
⎪⎧120575120575119954119954119954119954prime
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032+1 1198661198661199541199541198951198951prime
0 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime0 (1205911205912 0)119866119866119954119954120582120582
0 (120591120591 1205911205911) 120582120582 120582120582prime isin 1 31199031199032
120575120575119954119954119954119954prime
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast (119954119954prime)
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
0 else
Here we used the following properties 120596120596minus119954119954120582120582 = 120596120596119954119954120582120582 and 1198811198811205821205821198951198951prime(minus119954119954) = 1198811198811205821205821198951198951primelowast (119954119954) [see Eqs (112) and
(314)]
The equivalence of 11989211989221 and 11989211989224 can be seen by changing the integration variables 1205911205912 harr 1205911205911
Substituting 11989211989221 and 11989211989224 into Eq (526) 1198921198922 is given by
1198921198922 =
⎩⎪⎨
⎪⎧
120575120575119954119954119954119954prime
16 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast (119954119954prime)
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime
0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
120575120575119954119954119954119954prime
16 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032+1 1198661198661199541199541198951198951prime
0 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime0 (1205911205911 0)119866119866119954119954120582120582
0 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032
0 else
(528)
By changing the indices and integration variables itrsquos straightforward to show that the other three
terms (119892119892111989211989231198921198924 ) are identical to 1198921198922 thus the final expression of 119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) [Eq (516)) can be
calculated as
119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) = 41198921198922 =
⎩⎪⎨
⎪⎧120575120575119954119954119954119954prime
4 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032
120575120575119954119954119954119954prime
4 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast 119954119954prime
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime
0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 31199031199032
+ 13119903119903
0 else
(529)
To calculate 119866119866119954119954120582120582119954119954prime120582120582prime(2) in frequency space we expand 119866119866119954119954120582120582119954119954prime120582120582prime
(2) (120591120591) in a Fourier series
119866119866119954119954120582120582119954119954prime120582120582prime
(2) (120591120591) = 1120573120573ℏsum 119890119890minus119894119894120596120596119899119899120591120591119866119866119954119954120582120582119954119954prime120582120582prime
(2) (119894119894120596120596119899119899)119899119899
119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = int d120591120591119890119890119894119894120596120596119899119899120591120591119866119866119954119954120582120582119954119954prime120582120582prime
(2) (120591120591)120573120573ℏ0
(530)
Using Eq (421) we get for 120582120582 120582120582prime isin 1 31199031199032
26
119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) =
120575120575119954119954119954119954prime4
1198891198891205911205911120573120573ℏ
0 1198891198891205911205912120573120573ℏ
0
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1198661198661199541199541198951198951prime0 (1205911205912 1205911205911)119866119866119954119954120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912)
=120575120575119954119954119954119954prime
4
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1198891198891205911205911120573120573ℏ
0 1198891198891205911205912120573120573ℏ
0
1120573120573ℏ
119890119890minus1198941198941205961205961198991198992(1205911205912minus1205911205911)1198661198661199541199541198951198951prime0 1198941198941205961205961198991198992
1198991198992
∙1120573120573ℏ
119890119890minus11989411989412059612059611989911989911205911205911119866119866119954119954120582120582prime0 1198941198941205961205961198991198991
1198991198991
∙1120573120573ℏ
119890119890minus119894119894120596120596119899119899(120591120591minus1205911205912)1198661198661199541199541205821205820 (119894119894120596120596119899119899)
119899119899
=120575120575119954119954119954119954prime
4
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1120573120573ℏ
119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954120582120582prime
0 11989411989412059612059611989911989911198661198661199541199541198951198951prime0 1198941198941205961205961198991198992
11989911989911989911989911198991198992⎩⎪⎨
⎪⎧ 1120573120573ℏ
1198891198891205911205911120573120573ℏ
0119890119890minus1198941198941205961205961198991198991minus12059612059611989911989921205911205911
times1120573120573ℏ
1198891198891205911205912120573120573ℏ
0119890119890minus1198941198941205961205961198991198992minus1205961205961198991198991205911205912
⎭⎪⎬
⎪⎫
=120575120575119954119954119954119954prime
4
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1120573120573ℏ
119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)
11989911989911989911989911198991198992
119866119866119954119954120582120582prime0 11989411989412059612059611989911989911198661198661199541199541198951198951prime
0 1198941198941205961205961198991198992120575120575119899119899111989911989921205751205751198991198991198991198992
=120575120575119954119954119954119954prime120573120573ℏ
119890119890minus119894119894120596120596119899119899120591120591
119899119899
1198661198661199541199541205821205820 (119894119894120596120596119899119899)
14
119881119881120582120582prime1198951198951prime(119954119954
prime)1198811198811205821205821198951198951primelowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899)119866119866119954119954120582120582prime
0 (119894119894120596120596119899119899)
Comparing above expression with Eq (530) the Fourier transform of 119866119866119954119954120582120582119954119954120582120582prime(2) (120591120591) reads as
119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) 120575120575119954119954119954119954prime
4sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 119866119866119954119954120582120582prime
0 (119894119894120596120596119899119899) (531)
where we have introduced the notation
Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) equiv
120575120575119954119954119954119954prime
4sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) (532)
and 119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954120582120582prime(2) (119894119894120596120596119899119899) ∙ 119866119866119954119954120582120582prime
0 (119894119894120596120596119899119899) Similarly for 120582120582 120582120582prime isin 31199031199032
+ 13119903119903 we
have
Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) =
120575120575119954119954119954119954prime
4sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast 119954119954prime
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (119894119894120596120596119899119899) (533)
and all together we have for Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899)
Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) =
⎩⎪⎪⎨
⎪⎪⎧120575120575119954119954119954119954prime
4sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 1 3119903119903
2
120575120575119954119954119954119954prime
4sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast 119954119954prime
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
0 else
(534)
27
533 Dysonrsquos equation Substituting Eqs (521) and (534) into Eq (516) and performing a Fourier transform we have
119866119866119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) = 119866119866119954119954120582120582119954119954prime120582120582prime0 (119894119894120596120596119899119899) + 119866119866119954119954120582120582119954119954prime120582120582prime
(1) (119894119894120596120596119899119899) + 119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) + ⋯
= 1198661198661199541199541205821205820 (119894119894120596120596119899119899)120575120575119954119954119954119954prime120575120575120582120582120582120582prime + 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) ∙ 119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899) + 1198661198661199541199541205821205820 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime
(2) (119894119894120596120596119899119899)
∙ 119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899) + ⋯ = 119866119866119954119954120582120582
0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime120575120575120582120582120582120582prime + 1198661198661199541199541205821205820 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) ∙ 119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899)
or
119866119866119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) = 1198661198661199541199541205821205820 (119894119894120596120596119899119899)120575120575119954119954119954119954prime120575120575120582120582120582120582prime + 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) ∙ 119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899) (535)
Eq (535) is the so-called Dysonrsquos equation (see Ref [2]) where
Σ119954119954120582120582119954119954120582120582prime(119894119894120596120596119899119899) = Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) + Σ119954119954120582120582119954119954prime120582120582prime
(2) (119894119894120596120596119899119899) + ⋯
is called self-energy and Σ119954119954120582120582119954119954120582120582prime(119899119899) (119894119894120596120596119899119899)119899119899 = 12⋯ is the self-energy of nth order approximation Up
to the second order approximation the self-energy is written as
Σ119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) =
⎩⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎧ minus
120575120575119954119954119954119954prime
2
119881119881120582120582120582120582primelowast (119954119954)
120596120596119954119954120582120582120596120596119954119954120582120582prime 1 le 120582120582 le 3119903119903
2lt 120582120582prime le 3119903119903
minus120575120575119954119954119954119954prime
2
119881119881120582120582prime120582120582(119954119954)
120596120596119954119954120582120582120596120596119954119954120582120582prime 1 le 120582120582prime le 3119903119903
2lt 120582120582 le 3119903119903
120575120575119954119954119954119954prime
4sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 1 3119903119903
2
120575120575119954119954119954119954prime
4sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast 119954119954prime
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
(536)
54 Fermirsquos golden Rule
To get the Fermirsquos golden Rule we need to calculate the retarded Greenrsquos function which can be
calculated from 119866119866119954119954120582120582119954119954120582120582prime(119894119894120596120596119899119899) by analytic continuation to the real axis via 119894119894120596120596119899119899 rarr 120596120596 + 119894119894119894119894 with an
infinitesimal positive 119894119894 [12]
119866119866119954119954120582120582119954119954120582120582prime119903119903 (120596120596) equiv 119866119866119954119954120582120582119954119954120582120582prime(119894119894120596120596119899119899 rarr 120596120596 + 119894119894119894119894) 119894119894 rarr 0 (537)
Similarity the retarded self-energy is defined as
Σ119954119954120582120582119954119954120582120582prime119903119903 (120596120596) equiv Σ119954119954120582120582119954119954120582120582prime(119894119894120596120596119899119899 rarr 120596120596 + 119894119894119894119894) 119894119894 rarr 0 (538)
The transition rate of phonons of branch 120582120582 at 119954119954 Γ119954119954120582120582(120596120596) is related to the retarded self energy
Σ119954119954120582120582119903119903 (120596120596) via [13]
Γ119954119954120582120582(120596120596) = minus2ImΣ119954119954120582120582119954119954120582120582119903119903 (120596120596) (539)
Using the expression for 1198661198661199541199541205821205820 (119894119894120596120596119899119899) [Eq (421)] and focused on the second order approximation [13]
we have
28
Σ119954119954120582120582119954119954120582120582119903119903 (119894119894120596120596119899119899) =
⎩⎪⎨
⎪⎧14sum
1198811198811205821205821198951198951prime(119954119954)
2
1205961205961199541199541198951198951prime120596120596119954119954120582120582
31199031199031198951198951prime=
31199031199032 +1
1
120596120596minus1205961205961199541199541198951198951prime +119894119894119894119894
120582120582 isin 1 31199031199032
14sum 1198811198811198951198951120582120582(119954119954)2
1205961205961199541199541198951198951120596120596119954119954120582120582
311990311990321198951198951=1
1120596120596minus1205961205961199541199541198951198951+119894119894119894119894
120582120582 isin 31199031199032
+ 13119903119903
(540)
Using the relation
Im 1
120596120596minus1205961205961199541199541198951198951prime +119894119894119894119894
= minus120587120587120575120575120596120596 minus 1205961205961199541199541198951198951prime (541)
The transition rate reads as
Γ119954119954120582120582(120596120596) =
⎩⎪⎨
⎪⎧1205871205872sum
1198811198811205821205821198951198951prime(119954119954)
2
1205961205961199541199541198951198951prime120596120596119954119954120582120582
31199031199031198951198951prime=
31199031199032 +1
120575120575120596120596 minus 1205961205961199541199541198951198951prime 120582120582 isin 1 31199031199032
1205871205872sum 1198811198811198951198951120582120582(119954119954)2
1205961205961199541199541198951198951120596120596119954119954120582120582
311990311990321198951198951=1
120575120575120596120596 minus 1205961205961199541199541198951198951 120582120582 isin 31199031199032
+ 13119903119903
(542)
Or equivalently
Γ119954119954120582120582(119864119864 = ℏ120596120596) =
⎩⎪⎨
⎪⎧120587120587ℏ3
2sum
1198811198811205821205821198951198951prime(119954119954)
2
1198641198641199541199541198951198951prime119864119864119954119954120582120582
31199031199031198951198951prime=
31199031199032 +1
120575120575119864119864 minus 1198641198641199541199541198951198951prime 120582120582 isin 1 31199031199032
120587120587ℏ3
2sum 1198811198811198951198951120582120582(119954119954)2
1198641198641199541199541198951198951119864119864119954119954120582120582
311990311990321198951198951=1
120575120575119864119864 minus 1198641198641199541199541198951198951 120582120582 isin 31199031199032
+ 13119903119903
(543)
Γ119954119954120582120582(119864119864) represents the probability per unit time of a transition with energy 119864119864 from phonon branch 120582120582
at wave number 119954119954 in subsystem I (II) to a set of phonon branches in subsystem II (I) with the same
wave number Here state 119954119954120582120582 in one subsystem is coupled to state 1199541199541198951198951prime in the other subsystem via
1198811198811205821205821198951198951prime(119954119954) Since the two expressions in Eq (543) are completely equivalent just representing
transitions of opposite directions we consider only the first one in what follows Assuming that
subsystem II sufficiently large such that its density of states is nearly continuous the sum over its
states in the above expression can be replaced by an integral over the energy sum (⋯ )119895119895 rarr
int(⋯ )120588120588119864119864119954119954d119864119864119954119954 giving
Γ119954119954120582120582(119864119864) = 120587120587ℏ3
2 intd119864119864119954119954prime120588120588119864119864119954119954prime 119881119881120582120582119895119895prime(119954119954)
2119864119864119954119954119895119895prime=119864119864119954119954prime
119864119864119954119954prime119864119864119954119954120582120582120575120575119864119864 minus 119864119864119954119954prime = 120587120587ℏ3
2120588120588(119864119864)
119881119881120582120582119895119895prime(119954119954)2119864119864119954119954119895119895prime=119864119864
119864119864119864119864119954119954120582120582 (544)
Eq (544) represents the transition rate of the phonons with energy 119864119864 from branch 120582120582 and wave
number 119954119954 in subsystem I to the continuous manifold of states in subsystem II The total transition
rate between the two subsystems is then given by the sum of transition rates from all states in
subsystem I weighted by their phonon populations to the manifold of states in subsystem II
Assuming that the two subsystems are weakly coupled such that the width of state 119954119954120582120582 in subsystem
I is small and the probability to leave it at an energy other than 119864119864119954119954120582120582 is negligible we can now write
29
total transition rate as
Γtot = 1119885119885119890119890minus120573120573119864119864119954119954120582120582Γ119954119954120582120582119864119864119954119954120582120582
119954119954120582120582
=120587120587ℏ3
2
119890119890minus120573120573119864119864119954119954120582120582
119885119885
120588120588119864119864119954119954120582120582 119881119881120582120582119895119895prime(119954119954)2119864119864119954119954119895119895prime=119864119864119954119954120582120582
1198641198641199541199541205821205822119954119954120582120582
=120587120587ℏ3
2
119890119890minus120573120573119864119864119954119954120582120582
119885119885
120588120588119864119864119954119954120582120582 119881119881120582120582120582120582+31199031199032(119954119954)
2
1198641198641199541199541205821205822119954119954120582120582
Finally we get
Γtot = Γ119954119954120582120582(119864119864) =120587120587ℏ3
2sum 119890119890minus120573120573119864119864119954119954120582120582
119885119885
120588120588119864119864119954119954120582120582119881119881120582120582120582120582+31199031199032(119954119954)
2
1198641198641199541199541205821205822119954119954120582120582 (545)
where 119885119885 = sum 119890119890minus120573120573119864119864119954119954120582120582119954119954120582120582 is the partition function Here we choose the index of phonon branches 120582120582
such that the phonon branch with lower energy has a smaller index ie 1198641198641199541199541205821205821 lt 1198641198641199541199541205821205822 for index 1205821205821 lt
1205821205822 Note that phonon branches 120582120582 and 1198951198951prime belong to the subsystem I (with index from 1 to 31199031199032) and
subsystem II (with index from 1 to 1 + 31199031199032 to 3119903119903) respectively we choose 119895119895prime = 120582120582 + 31199031199032
to make
sure that 119864119864119954119954119895119895prime = 119864119864119954119954120582120582 since phonon energy of two uncoupled system is identical Eq (545) is the
Fermirsquos golden rule for inter-phonon coupling
References
[1] Y Xu J-S Wang W Duan B-L Gu and B Li Nonequilibrium Greens function method for phonon-phonon interactions and ballistic-diffusive thermal transport Phys Rev B 78 224303 (2008) [2] A L Fetter and J D Walecka Quantum theory of many-particle systems (Courier Corporation 2012) [3] G E Stedman Fermis Golden RulemdashAn Exercise in Quantum Field Theory Am J Phys 39 205 (1971)
3
Figure 1 Schematic representation of the simulation setup Two identical AB-stacked graphite slabs
(gray and orange respectively) are twisted with respect to each other to create a stacking fault of misfit
angle θ A thermal bias is induced by applying Langevin thermostats to the two layers marked by
dashed red (Thot) and green (Tcold) rectangles The arrows indicate the direction of the vertical heat
flux Since periodic boundary conditions are applied also in the vertical direction two twisted
interfaces are shown across which heat flows in opposite directions
We start by studying the effect of the misfit angle on the cross-plane thermal conductivity of the
twisted graphite and h-BN stacks Figure 2 presents the dependence of the cross-plane thermal
conductivity of the entire stack on the misfit angle for model systems consisting of 8 (red circles) and
16 (black triangles) layers for (a) graphite and (b) h-BN A pronounced dependence of the cross-plane
thermal conductivity (120581120581CP) of the entire graphitic stack is clearly evident which above a misfit angle
of sim 5deg 120581120581CP drops by a factor of 3-4 with respect to the value obtained for the aligned contact
Similar misfit-angle dependence of 120581120581CP is obtained for twisted bilayer graphene (tBLG) using the
transient MD simulation approach (see Sec 2 of the SI) We note that this sharp drop for graphite is
steeper and that the overall reduction is higher than those previously obtained using Lennard-Jones
interlayer potentials in finite model systems [1617] The corresponding cross-plane thermal
conductivity of the commensurate h-BN stack is found to be approximately double that of graphite
for the same number of layers Notably it reduces more gradually with the twist angle and saturates
at sim 15deg with an overall two-three fold reduction
The thermal conductivity of both graphite and h-BN stacks is found to increase when doubling their
thickness To identify the source of this thickness dependence we plot in Figure 2(c-d) the interfacial
thermal resistance (ITR) (see Sec 12 of the SI for the definition) associated with the twisted junction
formed between the contacting graphene or h-BN layers of the two optimally-stacked slabs Note that
unlike 120581120581CP which measures the conductivity of the entire stack the ITR corresponds to the heat
transport resistance of the two adjacent layers forming the twisted interface Two important
4
observations can be made (i) the ITR is weakly dependent on the stack thickness indicating that the
thickness dependence arises from the conductivity of the optimally-stacked interfacing slabs
Specifically in the thickness range considered the heat conductivity grows with slab thickness due to
reduction of phonon-phonon interactions and increased contribution of long wave-length phonons
below the mean-free path [28-30] (ii) the ITR strongly depends on the twist angle demonstrating a
~10-fold (4-fold) increase when the twist angle at the graphene (h-BN) interface is varied from 0deg
to 15deg This clearly indicates that the twist angle can be utilized to control the cross-plane thermal
conductivity of hexagonal two-dimensional (2D) materials and to effectively thermally isolate the top
layers from the underlying substrate
Figure 2 Twist-angle dependence of the cross-plane thermal conductivity of the entire stack (a b)
and the interfacial thermal resistance (c d) of the twisted contact formed between the optimally-
stacked slabs of graphite (a c) and bulk h-BN (b d) respectively Red circles and black triangles
correspond to the results obtained using 8 and 16 layer models respectively
The strong dependence of the cross-plane thermal conductivity of graphene and h-BN on the stacking
fault twist angle is related to the degree of coupling between the phonon modes of the two contacting
layers at the twisted interface Note that the term ldquocouplingrdquo used herein is not related to the standard
notion of phonon-phonon couplings due to anharmonic effects Instead we regard to the off-diagonal
terms of the Hessian when represented in the basis of the harmonic phonon modes of the isolated
5
layers To demonstrate this we write the dynamical matrix (the mass-reduced Fourier transform of
the force constant matrix) in block form as follows
120625120625(119954119954) = 12062512062511(119954119954) 12062512062512(119954119954)12062512062521(119954119954) 12062512062522(119954119954) (1)
where 12062512062511(119954119954) and 12062512062522(119954119954) are the block matrices relating to the first and second layer and 12062512062512(119954119954)
and 12062512062521(119954119954) = 12062512062512dagger (119954119954) all evaluated at wave-vector 119954119954 The interlayer phonon-phonon couplings are
obtained by diagonalizing separately 12062512062511(119954119954) and 12062512062522(119954119954) such that 12062512062511(119954119954) =
1199321199321dagger(119954119954)12062512062511(119954119954)1199321199321(119954119954) and 12062512062522(119954119954) = 1199321199322
dagger(119954119954)12062512062522(119954119954)1199321199322(119954119954) are diagonal matrices containing the
frequencies (120596120596119894119894) of the phonon modes of the two layers and 1199321199321(119954119954) and 1199321199322(119954119954) are unitary
matrices of the corresponding eigenvectors We now construct a global block diagonal transformation
matrix of the form
119932119932(119954119954) = 1199321199321(119954119954) 120782120782120782120782 1199321199322(119954119954) (2)
and transform the full dynamical matrix as follows
119932119932dagger(119954119954)120625120625(119954119954)119932119932(119954119954) = 12062512062511(119954119954) 12062512062512(119954119954)12062512062521(119954119954) 12062512062522(119954119954)
(3)
where 12062512062512(119954119954) = 1199321199321dagger(119954119954)12062512062512(119954119954)1199321199322(119954119954) and 12062512062521(119954119954) = 12062512062512
dagger (119954119954) are the interlayer phonon-phonon
coupling blocks Naturally when the two layers are infinitely separated these coupling blocks vanish
and the diagonal blocks converge to those of the isolated layers
The overall coupling between the two layers can be obtained from the individual phonon-phonon
coupling matrix elements via Fermirsquos golden rule [31] which reads as (see Sec 4 of SI for a detailed
derivation)
Γtot = 120587120587ℏ3
2sum 119890119890minus120573120573119864119864119954119954120582120582
119885119885
120588120588119864119864119954119954120582120582119881119881120582120582120582120582+31199031199032(119954119954)
2
1198641198641199541199541205821205822119954119954120582120582 (4)
where 119885119885 = sum 119890119890minus120573120573119864119864119954119954120582120582119954119954120582120582 is the partition function 119864119864119954119954120582120582 is the energy of phonons at branch 120582120582 with
wave number 119954119954 120588120588119864119864119954119954120582120582 is the density-of-states (DOS) at 119864119864119954119954120582120582 and 119881119881120582120582120582120582+31199031199032(119954119954)
2 is the coupling
matrix element between branches of phonons of similar energy in the two layers whose number of
atoms in one unit cell is 119903119903
Using Eq (4) we can rationalize the misfit angle dependence of the heat flux across the twisted
interface from the calculated inter-phonon coupling To that end we performed room temperature
(300K) simulations (technical details can be found in Sec 3 of SI) for tBLG with different misfit
angles using the Greenrsquos function molecular dynamics (GFMD) developed by Kong et al [32] as
6
implemented in LAMMPS [33] The simulations allow us to evaluate the dynamical matrix from
which the phonon-phonon couplings can be extracted (see details in Sec 3 of SI) and the overall heat
transfer rate calculated Figure 3 shows the resulting heat transfer rate (normalized to is value for the
aligned contact (120579120579 = 0) ) as a function of the misfit angle compared to the interfacial thermal
conductivity defined as the inverse of the ITR presented in Figure 2(c) ITC equiv 1ITR The
remarkable agreement between the calculated interfacial thermal conductivity and Fermirsquos golden
rule results indicate that the dependence of the interlayer phonon-phonon couplings on the misfit
angle is responsible for the strong angle dependence of the interfacial conductivity Notably the sharp
heat conductivity drop at misfit angles in the range of 0deg-5deg as well as the small conductivity for larger
misfit angles are well captured by Fermirsquos golden rule
Figure 3 Comparison between Fermirsquos golden rule results (open blue squares) for the interfacial heat-
transfer rate of a tBLG and the calculated interfacial thermal conductivity at various misfit angles
ITC simulation results are presented for both 8 layers (open red circles) and 16 layers (open black
triangles) showing similar behavior For comparison purposes all data sets are normalized to their
value obtained for the aligned contact
To correlate our results with experimentally measured thermal conductivities that are often obtained
for thick samples we repeated our calculations for increasing stack thicknesses at fixed misfit angles
Figure 4 presents results for the calculated heat conductivity of (a) graphite and (b) h-BN stacks either
aligned (open red circles) or twisted by 120579120579 = 3016deg (open black diamond symbols) as a function of
number of layers in the stack As discussed above for both systems the misoriented stack exhibits
lower heat conductivity compared to the aligned system however its thickness dependence is
considerably stronger This can be attributed to the significantly higher interface resistance of the
twisted interface that when plugged in Eq (S2) of the SI for the overall conductivity induces stronger
7
thickness dependence
Comparing our calculated heat conductivities for the aligned contact (open red circles) to available
experimental data for ~35 nm thick graphite slabs [34] (dashed green line) we find that at the thickest
model system considered of 104 graphene layers (~34 nm thick) the calculated value of 085plusmn005
W m sdot Kfrasl is in remarkable agreement with the measured value of ~07 W m sdot Kfrasl Furthermore
experimental values for bulk graphite [5] indicate that the thermal conductivity continues to grow up
to ~68 W m sdot Kfrasl (black dash-dotted line) which is consistent with the general trend of the calculated
heat conductivity that does not saturate for the thickest model system considered These results
strongly enforce the validity of our force-field and model systems to model the heat conductivity of
twisted layered materials interfaces Available experimental results for the heat conductivity of bulk
h-BN are marked by the dashed-dotted black and dashed-green lines in Figure 4(b) In line with our
findings for the graphitic interface our calculated finite slab heat conductivities for the aligned
interface (open red circles) continue to grow with the number of layers and are consistently below the
bulk value
Figure 4 Thickness dependence of the thermal conductivity 120581120581CP of aligned (open red circles) and
twisted by 3016deg (open black diamond symbols) graphite (a) and h-BN (b) stacks Blue squares
represent results obtained using the isotropic Lennard-Jones potential for the aligned contacts The
green dashed and black dash-dotted lines represent experimental results measured for graphite (a)
(Refs [534]) and bulk h-BN (b) (Refs [1415]) Note that both axis scales are logarithmic Error bar
estimation procedure is discussed in Sec 1 of the SI
Another important factor that may affect the interlayer thermal transport properties of 2D material
stacks is the average temperature of the system which was taken to be ~300K in all abovementioned
simulations To evaluate the sensitivity of our results towards this parameter we repeated the heat
conductivity and interfacial resistance calculations of optimally stacked graphite and h-BN stacks for
an average temperature of 400 K The results presented in see Sec 4 of the SI indicate that 120581120581CP and
8
ITR weakly depend on the average temperature over the entire thickness range considered This
conclusion is consistent with recent experimental findings [11] We may attribute this to the fact that
the thickness of the graphite and h-BN stack model considered is much smaller than their phonon
mean-free path (~200 nm for graphite [1011133536] and ~100 nm for bulk h-BN [15]) such that
the phonon transport is dominated by phonon-boundary scattering which weakly depends on
temperature in the range considered Therefore temperature dependent Umklapp processes have only
marginal contribution to our results [11]
Finally we note that previous calculations of the heat conductivity of twisted graphitic interfaces
relied on Lennard-Jones (LJ) potentials describing the interlayer interactions [8-1013] To
demonstrate the importance of using registry-dependent interlayer potentials we have repeated our
calculations of the heat conductivity of graphitic slabs with the REBO intralayer potential augmented
by LJ interlayer interactions [37] (120576120576 = 284 meV120590120590 = 34 Å ) We find that the calculated heat
conductivities obtained using the LJ interlayer potential are consistently higher than those obtained
by our ILP and that the difference between them grows with the model system thickness Notably the
heat conductivity obtained using the LJ potential for a graphitic slab of thickness ~34 nm is 154
W m sdot Kfrasl overestimating the experimental value by more than a factor of 2
The excellent agreement of our ILP calculations with experimental data of nanoscale graphitic stacks
therefore demonstrates the reliability of our predictions for the strong interfacial misfit angle
dependence of cross-layer thermal conductivity in graphite and h-BN The observed sharp
conductivity decrease of twisted graphitic interfaces at misfit angles lt 5deg opens the way to control
the thermal evacuation rate and thermal isolation of active layers in graphene-based electronic and
mechanical devices The revealed underlying mechanism suggests that design rules can be obtained
by carefully tailoring the phonon-phonon couplings across the twisted interface While the misfit
angle dependence of h-BN is found to be weaker than that of graphite the overall thermal
conductivity of the former is found to be higher This may be utilized to achieve higher conductivity
and controllability in twisted heterogeneous junctions of layered materials
ASSOCIATED CONTENT
Supporting Information
The supporting Information section includes a description of the methodology thermal conductivity
of twisted bilayer graphene theory for calculating the phonon coupling of twisted bilayer graphene
9
derivation of Fermirsquos golden rule and temperature dependence of the cross-plane thermal
conductivity
AUTHOR INFORMATION
Corresponding Author
E-mail urbakhtauextauacil
Notes
The authors declare no competing financial interest
ACKNOWLEDGMENTS
The authors would like to thank Prof Abraham Nitzan Dr Guy Cohen and Dr Yiming Pan for
helpful discussions WO acknowledges the financial support from a fellowship program for
outstanding postdoctoral researchers from China and India in Israeli Universities and the support from
the National Natural Science Foundation of China (Nos 11890673 and 11890674) HQ
acknowledges the financial support from the National Natural Science Foundation of China (No
11890674) MU acknowledges the financial support of the Israel Science Foundation Grant
No 114118 and the ISF-NSFC joint grant 319119 OH is grateful for the generous financial
support of the Israel Science Foundation under grant no 158617 and the Naomi Foundation for
generous financial support via the 2017 Kadar Award This work is supported in part by COST Action
MP1303
10
References
[1] A A Balandin Thermal properties of graphene and nanostructured carbon materials Nat Mater 10 569 (2011) [2] S Ghosh I Calizo D Teweldebrhan E P Pokatilov D L Nika A A Balandin W Bao F Miao and C N Lau Extremely high thermal conductivity of graphene Prospects for thermal management applications in nanoelectronic circuits Appl Phys Lett 92 151911 (2008) [3] A A Balandin S Ghosh W Bao I Calizo D Teweldebrhan F Miao and C N Lau Superior thermal conductivity of single-layer graphene Nano Lett 8 902 (2008) [4] J H Seol et al Two-Dimensional Phonon Transport in Supported Graphene Science 328 213 (2010) [5] R Taylor The thermal conductivity of pyrolytic graphite J Philosophical Magazine 13 157 (1966) [6] S Ghosh W Bao D L Nika S Subrina E P Pokatilov C N Lau and A A Balandin Dimensional crossover of thermal transport in few-layer graphene Nat Mater 9 555 (2010) [7] Z Wang R Xie C T Bui D Liu X Ni B Li and J T L Thong Thermal Transport in Suspended and Supported Few-Layer Graphene Nano Lett 11 113 (2011) [8] Z Wei Z Ni K Bi M Chen and Y Chen Interfacial thermal resistance in multilayer graphene structures Phys Lett A 375 1195 (2011) [9] Y Ni Y Chalopin and S Volz Significant thickness dependence of the thermal resistance between few-layer graphenes Appl Phys Lett 103 061906 (2013) [10] J Chen J H Walther and P Koumoutsakos Strain Engineering of Kapitza Resistance in Few-Layer Graphene Nano Lett 14 819 (2014) [11] Q Fu J Yang Y Chen D Li and D Xu Experimental evidence of very long intrinsic phonon mean free path along the c-axis of graphite Appl Phys Lett 106 031905 (2015) [12] G Fugallo A Cepellotti L Paulatto M Lazzeri N Marzari and F Mauri Thermal Conductivity of Graphene and Graphite Collective Excitations and Mean Free Paths Nano Lett 14 6109 (2014) [13] Z Wei J Yang W Chen K Bi D Li and Y Chen Phonon mean free path of graphite along the c-axis Appl Phys Lett 104 081903 (2014) [14] E K Sichel R E Miller M S Abrahams and C J Buiocchi Heat capacity and thermal conductivity of hexagonal pyrolytic boron nitride Phys Rev B 13 4607 (1976) [15] P Jiang X Qian R Yang and L Lindsay Anisotropic thermal transport in bulk hexagonal boron nitride Phys Rev Mater 2 064005 (2018) [16] M-H Wang Y-E Xie and Y-P Chen Thermal transport in twisted few-layer graphene Chinese Physics B 26 116503 (2017) [17] X Nie L Zhao S Deng Y Zhang and Z Du How interlayer twist angles affect in-plane and cross-plane thermal conduction of multilayer graphene A non-equilibrium molecular dynamics study Int J Heat Mass Tran 137 161 (2019) [18] A N Kolmogorov and V H Crespi Registry-dependent interlayer potential for graphitic systems Phys Rev B 71 235415 (2005) [19] M Reguzzoni A Fasolino E Molinari and M C Righi Potential energy surface for graphene on graphene Ab initio derivation analytical description and microscopic interpretation Phys Rev B 86 (2012) [20] M M van Wijk A Schuring M I Katsnelson and A Fasolino Moire Patterns as a Probe of Interplanar Interactions for Graphene on h-BN Phys Rev Lett 113 135504 (2014) [21] D Mandelli W Ouyang M Urbakh and O Hod The Princess and the Nanoscale Pea Long-Range Penetration of Surface Distortions into Layered Materials Stacks ACS Nano 13 7603 (2019) [22] E Koren E Loumlrtscher C Rawlings A W Knoll and U Duerig Adhesion and friction in mesoscopic graphite contacts Science 348 679 (2015) [23] E Koren I Leven E Loumlrtscher A Knoll O Hod and U Duerig Coherent commensurate electronic states at the interface between misoriented graphene layers Nat Nanotechnol 11 752 (2016)
11
[24] W Ouyang I Azuri D Mandelli A Tkatchenko L Kronik M Urbakh and O Hod Mechanical and Tribological Properties of Layered Materials under High Pressure Assessing the Importance of Many-Body Dispersion Effects J Chem Theory Comput 16 666 (2020) [25] W Ouyang D Mandelli M Urbakh and O Hod Nanoserpents Graphene Nanoribbon Motion on Two-Dimensional Hexagonal Materials Nano Lett 18 6009 (2018) [26] I Leven T Maaravi I Azuri L Kronik and O Hod Interlayer Potential for Grapheneh-BN Heterostructures J Chem Theory Comput 12 2896 (2016) [27] T Maaravi I Leven I Azuri L Kronik and O Hod Interlayer Potential for Homogeneous Graphene and Hexagonal Boron Nitride Systems Reparametrization for Many-Body Dispersion Effects J Phys Chem C 121 22826 (2017) [28] L H Liang and B Li Size-dependent thermal conductivity of nanoscale semiconducting systems Phys Rev B 73 153303 (2006) [29] P K Schelling S R Phillpot and P Keblinski Comparison of atomic-level simulation methods for computing thermal conductivity Phys Rev B 65 144306 (2002) [30] J Shiomi Nonequilirium molecular dynamics methods for lattice heat conduction calculations Annual Review of Heat Transfer 17 (2014) [31] G E Stedman Fermis Golden RulemdashAn Exercise in Quantum Field Theory Am J Phys 39 205 (1971) [32] L T Kong G Bartels C Campantildeaacute C Denniston and M H Muumlser Implementation of Greens function molecular dynamics An extension to LAMMPS Comput Phys Commun 180 1004 (2009) [33] S Plimpton Fast Parallel Algorithms for Short-Range Molecular Dynamics J Comput Phys 117 1 (1995) [34] M Harb et al The c-axis thermal conductivity of graphite film of nanometer thickness measured by time resolved X-ray diffraction Appl Phys Lett 101 233108 (2012) [35] H Zhang X Chen Y-D Jho and A J Minnich Temperature-Dependent Mean Free Path Spectra of Thermal Phonons Along the c-Axis of Graphite Nano Lett 16 1643 (2016) [36] D P Sellan E S Landry J E Turney A J H McGaughey and C H Amon Size effects in molecular dynamics thermal conductivity predictions Phys Rev B 81 214305 (2010) [37] S J Stuart A B Tutein and J A Harrison A reactive potential for hydrocarbons with intermolecular interactions J Chem Phys 112 6472 (2000)
S1
Supporting information for ldquoControllable Thermal Conductivity of
Twisted Graphene and Hexagonal Boron Nitride Interfacesrdquo
Wengen Ouyang1 Huasong Qin2 Michael Urbakh1 and Oded Hod1
1Department of Physical Chemistry School of Chemistry The Raymond and Beverly
Sackler Faculty of Exact Sciences and The Sackler Center for Computational
Molecular and Materials Science Tel Aviv University Tel Aviv 6997801 Israel
2State Key Laboratory for Strength and Vibration of Mechanical Structures School of
Aerospace Xirsquoan Jiaotong University Xirsquoan 710049 China
The Supporting information includes following sections
1 Methodology
2 Thermal conductivity of twisted bilayer graphene
3 Theory for calculating the phonon coupling of twisted bilayer graphene
4 Derivation of Fermirsquos golden rule
5 Temperature dependence of cross-plane thermal conductivity
S2
1 Methodology
11 Model system
The initial interlayer distance across the layered stack was set equal to 34 Aring and 33 Aring for graphite
and bulk h-BN respectively Periodic boundary conditions were applied in all directions It should be
noted that the lattice structure is rigorously periodic only at some specific twist angles the values of
which are listed in Table S1 in section 3 below While the cross-sectional area for each misfit angle
θ is different all systems considered have a contact area exceeding 12 nm2 which was shown to
provide converged results with respect to unit-cell dimensions [1] The intralayer interactions within
each graphene and h-BN layer were modeled via the second generation REBO potential [2] and
Tersoff potential [3] respectively The interlayer interactions between the layers of graphite and bulk
h-BN were described via our dedicated interlayer potential (ILP) [4] which is implemented in the
LAMMPS [5] suite of codes [6]
Fig S1 Schematic representation of the simulation setup (a) and steady-state temperature profile (b) respectively In panel (a) two identical AB-stacked graphite slabs (gray and orange respectively) are twisted with respect to each other to create a stacking fault of misfit angle θ A thermal bias is induced by applying Langevin thermostats to the two layers marked by dashed red (Thot) and green (Tcold) rectangles The arrows indicate the direction of the vertical heat flux Since periodic boundary conditions are applied also in the vertical direction two twisted interfaces are shown across which heat flows in opposite directions The steady-state temperature profiles are illustrated in panel (b) where N is the total number of layers in the model system and RAB dAB and Rθ and dθ mark the interfacial Kapitza resistance [89] and interlayer distance for contacting graphene layers with AB-stacking and misfit angle θ respectively The red lines in panel (b) mark the temperature variation across the twisted interface where the vertical axis corresponds to the position of the various layers along the stack and the horizontal axis marks the temperature of the various layers
S3
12 Simulation Protocol
All MD simulations were performed with the LAMMPS simulation package [5] The velocity-Verlet
algorithm with a time-step of 05 fs was used to propagate the equations of motion A Noseacute-Hoover
thermostat with a time constant of 025 ps was used for constant temperature simulations To maintain
a specified hydrostatic pressure the three translational vectors of the simulation cell were adjusted
independently by a Noseacute-Hoover barostat with a time constant of 10 ps [7] To relax the box we first
equilibrated the systems in the NPT ensemble at a temperature of T = 300 K and zero pressure for
250 ps (see Fig S2) After equilibration Langevin thermostats with damping coefficients 10 ps-1
were applied to the bottom and middle layer of the graphene stack (see Fig S1) with target
temperature Thot = 375 K (hot reservoir) and Tcold = 225 K (cold reservoir) respectively Then the
system was allowed to reach steady-state over a subsequent simulation period of 750 ps (see Fig S2)
during which the dynamics of all non-thermostated layers followed the NVE ensemble For the larger
model systems the length of the NPT and Langevin stages was doubled (for the 32 and 48 layers
systems) or tripled (for the 104 layers graphitic system) to ensure convergence of the obtained steady-
state Once steady-state was obtained the last 500 ps were used to calculate the thermal conductivity
of the twisted graphite and bulk h-BN The statistical errors were estimated using ten different data
sets each calculated over a time interval of 50 ps
Fig S2 Time evolution of the temperature of thermostated layers for 16 layers twisted graphite with misfit angle (a) θ = 0deg (b) θ = 509deg (c) θ = 1518deg and (d) θ = 3016deg Note that the thermal fluctuations increase with increasing the misfit angle due to the growing interfacial thermal resistance that enhances phonon back scattering at the twisted junction
S4
13 Calculation of the interfacial thermal resistance
According to Fourierrsquos law the cross-plane thermal conductivity (120581120581CP) of a twisted graphitic interface
of misfit angle θ can be calculated as
120581120581CP = 119876119860119860Δ119879119879Δ119911119911
(S1)
where 119876 is the heat flux A is the cross-section area and Δ119879119879Δ119911119911 is the temperature gradient along
the direction of heat flux (perpendicular to the basal plane in our case) Fig S1(b) shows a schematic
temperature profile along the z-direction where the vertical axis corresponds to the position of the
various layers along the stack and the horizontal axis marks the temperature of the various layers The
actual temperature profiles extracted from the NEMD simulations for twisted graphite with different
number of layers can be found in Fig S3 For Bernal-stacked graphite (ie θ = 0deg red circles in Fig
S3) only the linear region of the temperature profile was used to calculate 120581120581CP and the points
corresponding to the layers where the thermostats were applied were omitted (marked with green
triangle in Fig S3) The 120581120581CP of the system was calculated using Eq (S1) by averaging over the two
linear regions of the temperature profiles For the twisted case (θ ne 0deg) we found a sudden
temperature decrease Δ119879119879120579120579 at the position of the twisted interface (see black squares in Fig S3) 120581120581CP
in this case was calculated using the temperature gradient calculated for the same layer range as that
for θ = 0deg To characterize the thermal properties of the twisted interface the concept of interfacial
thermal resistance (ITR) ie Kapitza resistance [89] was introduced According the definition of
the Kapitza resistance [8] 119877119877 = 119860119860Δ119879119879119876 and noticing that Δ119879119879tot = (1198731198732 minus 3)Δ119879119879AB + Δ119879119879120579120579 and
Δ119911119911 = (1198731198732 minus 3)119889119889AB + 119889119889120579120579 Eq (1) can be rewritten considering two-resistors in series as [see Fig
S1(b)]
(1198731198732 minus 3)119877119877AB + 119877119877120579120579 = [(1198731198732 minus 3)119889119889AB + 119889119889120579120579] 120581120581CPfrasl (S2)
Here 119877119877AB 119889119889AB Δ119879119879AB and 119877119877120579120579 119889119889120579120579 Δ119879119879120579120579 are the ITR interlayer distance and temperature
difference for adjacent AB-stacked and twisted graphene layers respectively For the aligned contact
(120579120579 = 0deg) the ITR can be simply calculated as 119877119877AB = 119889119889AB120581120581AB Once 119877119877AB is known 119877119877120579120579 can be
calculated from Eq (S2) We note that the sharp temperature drop at the twisted interface indicates
that 119877119877120579120579 should be much larger than 119877119877AB
S5
Fig S3 Temperature profiles for graphitic stacks consisting of (a) 8 and (b) 16 layers The red circles and black squares represent the temperature profiles for the aligned (120579120579 = 0deg) and twisted (120579120579 =3016deg) junctions respectively Green triangles represent data points that were omitted in the 120581120581CP calculation
2 Thermal conductivity of twisted bilayer graphene
As comparison we also calculated the interfacial thermal conductivity (ITC) and ITR of twisted
bilayer graphene (tBLG) with the transient MD simulation approach [15-17] since the NEMD
simulation protocol used in the main text becomes invalid in this case In this protocol the system
was first equilibrated within the NPT ensemble at T = 200 K and zero pressure for 100 ps which was
followed by a 100 ps NVT ensemble equilibration stage and a 100 ps of NVE ensemble equilibration
stage After the system reached steady-state an ultrafast heat impulse was imposed on the top layer
of the t-BLG for 50 fs to increase the temperature of the top layer from 200 K to 400 K while that of
bottom layer of tBLG remained unchanged After the external heat source was removed thermal
energy flowed from the top layer to the bottom layer due to the temperature difference and the
temperature of both layers approached 300 K when quasi-steady-state was reached During the
thermal relaxation time interval (500 ps) the temperature and energy of the system sections were
recorded The ITR could then be extracted using the following equation [15-17]
part119864119864t120597120597120597120597
= 119860119860119877119877119879119879bot(119905119905) minus 119879119879top(119905119905) (S3)
where 119864119864t is the total energy of the top graphene layer R is the ITR of the tBLG A is the interfacial
cross-section area and 119879119879bot and 119879119879top are the instantaneous temperatures measured for the bottom
and top layers of the tBLG respectively Note that in Eq S3 we assume a linear dependence of the
heat flux on the temperature difference between the layers The ITC of the tBLG is simply defined as
ITC equiv 119889119889ITR where d is the average interlayer distance
S6
The ITC and ITR of tBLG as functions of misfit angle calculated with the transient MD simulation
protocol are illustrated in Fig S4 demonstrating similar misfit-angle dependence as that for the
NEMD protocol with Langevin thermostat exercised to obtain the results presented in the main text
This further validates the reliability of the simulation protocol adopted in the main text which is more
suitable to treat thick slabs and allows to obtain a true stead-state
Fig S4 Misfit-angle dependence of (a) ITC and (b) ITR for a twisted bilayer graphene obtained using the transient MD simulation approach
3 Theory for calculating the phonon-phonon coupling of twisted bilayer graphene
31 Brillouin Zone of supercell in tBLG
For tBLG the lattice structure is rigorously periodic only at some specific misfit angles θ where the
lattice vector 1199231199231 = 11989911989911199381199381 + 11989911989921199381199382 in the bottom layer equals the vector 1199231199232 = 11989811989811199381199381 + 11989811989821199381199382 in the
top layer with certain integers m1 m2 and n1 n2 Here 1199381199381 = 119886119886(10) and 1199381199382 = 11988611988612radic32 are
the primitive lattice vectors of the bottom layer and a is the lattice constant of monolayer graphene
Thus the exact superlattice period is then given by [18]
119871119871 = |11989911989911199381199381 + 11989911989921199381199382| = 11988611988611989911989912 + 11989911989922 + 11989911989911198991198992 = |1198991198991minus1198991198992|1198861198862 sin(1205791205792) (S4)
where θ is the angle between two lattice vectors 1199231199231 and 1199231199232 In the simulations below we always
rotated the supercell such that its lattice vector is 1199231199231 = 119871119871(10) and 1199231199232 = 11987111987112radic32 In this case
the corresponding reciprocal lattice vector of the moireacute superlattice satisfies the relation 119918119918119894119894 ∙ 119923119923119895119895 =
2120587120587120575120575119894119894119895119895 such that
1199181199181 = 4120587120587radic3119871119871
radic32
minus12 1199181199182 = 4120587120587
radic3119871119871(01) (S5)
S7
Both the lattice vectors and the corresponding reciprocal lattice vectors of the superlattice of the tBLG
are presented in Fig S5 Table S1 reports the parameters used to construct rhombus periodic
supercells of different misfit angles that can be duplicated to construct a rectangular periodic supercell
Table S1 The parameters used to construct periodic supercells of various misfit angles
120579120579 (deg) 119860119860 (nm2) 1198991198991 1198991198992 1198981198981 1198981198982
0 60383688 24 0 24 0
0696407 709613169 48 47 47 48
1121311 273718420 30 29 29 30
2000628 85648391 17 16 16 17
3006558 152322047 23 21 21 23
4048894 189013524 26 23 23 26
5085849 53255058 14 12 12 14
7926470 49376245 14 11 11 14
9998709 86277388 19 14 14 19
15178179 31554670 15 4 4 15
19932013 42876612 15 8 8 15
25039660 13942761 9 4 4 9
30158276 18974735 11 4 4 11
32204228 21805221 12 4 4 12
32 Special points for Brillouin Zone integration
The calculation of the sum over wave vector q in Eq (4) in the main text can be transformed to an
integral using the relation sum (⋯ )119954119954 = 1119881119881119887119887int (⋯ )BZ d119954119954 where 119881119881119887119887 = (2120587120587)3119881119881 is the volume of the
Brillouin Zone (BZ) and V is volume of the real-space unit-cell The calculation of integral is usually
inefficient since it requires calculating the value of the function over a large set of k points in the first
BZ To calculate such integrations more efficiently simple k-point meshes can be replaced by a
carefully selected set of special points in the BZ 119954119954119894119894 [19-22] over which the function is evaluated
The integral can then be estimated via
119868119868 = 1119881119881119887119887int 119891119891(119954119954)BZ d119954119954 asymp 1
119873119873sum 119908119908119894119894119891119891(119954119954119894119894)119894119894 (S6)
S8
where 119881119881119887119887 is the volume of the BZ 119908119908119894119894 is the weight of the ith data point and N normalizes the
weighting factors to unity 119873119873 = sum 119908119908119894119894119894119894 The set of selected 119954119954119894119894 forms a grid in the irreducible
Brillouin zone (IBZ) as is illustrated by the red points in Fig S5(b) The coordinates of these points
for a hexagonal lattice are presented in Eq (S7)
Fig S5 (a) Twisted bilayer graphene of misfit angle θ = 509deg L1 and L2 are the superlattice vectors (b) The corresponding first Brillouin Zone of (a) G1 and G2 are the reciprocal lattice vectors of the superlattice The triangle ΔΓMK represents the irreducible Brillouin zone Red circles mark the position of the special points used to evaluate the integral over the first Brillouin Zone
⎩⎪⎨
⎪⎧ 1199541199541 = 1
9 1205971205973 1199541199542 = 2
9 1205971205973 1199541199543 = 1
18 512059712059718
1199541199544 = 518
512059712059718 1199541199545 = 1
9 21205971205979 1199541199546 = 2
9 21205971205979
1199541199547 = 118
312059712059718 1199541199548 = 1
9 1205971205979 1199541199549 = 1
18 12059712059718
(S7)
Here 119905119905 = radic3 and the unit of the coordinates is 2120587120587119871119871 The weighting factors 119908119908119894119894 are [19]
119908119908124689 = 112
119908119908357 = 16 (S8)
Using Eqs (S6-S8) Eq (4) in the main text can be evaluated as follows
Γtot = 120587120587ℏ3
2sum 119908119908119896119896 sum
119890119890minus120573120573119864119864119954119954119948119948120582120582
119885119885
120588120588119864119864119954119954119948119948120582120582119881119881120582120582120582120582+31199031199032(119954119954119948119948)
2
1198641198641199541199541199481199481205821205822120582120582
9119896119896=1 (S9)
This equation was used to calculate the transition rate presented in Fig 3 in the main text
4 Derivation of Fermirsquos golden rule
The derivation of Fermirsquos golden rule is provided in a separate file (Fermirsquos Golden Rule for
phononspdf) due to its extent
S9
5 Temperature dependence of interfacial thermal conductivity
In the main text the target temperatures of the Langevin thermostats for the bottom and middle layers
of graphene and h-BN were set to 225 K and 375 K respectively After reaching the steady-state the
average temperature of the system was found to be ~300 K To check the effect of average temperature
on our results we calculated the cross-plane thermal conductivity (120581120581CP ) and the corresponding
interfacial thermal resistance (ITR) at a different temperature gradient (325 K ndash 475 K) resulting the
average steady-state temperature of ~400K The protocol described in Section 1 above was used to
perform these calculations as well Both ILP and Lennard-Jones (LJ) potential were tested for
graphite whereas for the bulk h-BN simulations only the ILP was used The results for graphite and
bulk h-BN are illustrated in Fig S6 and Fig S7 respectively For the ILP we find that the overall
values of 120581120581CP (ITR) decrease (increase) slightly with increasing average temperature which is
consistent with a recent experiment [10] The LJ potential calculations as well exhibit very week
dependence on average temperature within the range studied Altogether the layer dependences of
both quantities remain mostly insensitive to the average temperature The reason is that the thickness
of the graphite and h-BN model systems was chosen to be much smaller than their phonon mean free
path (~200 nm for graphite [110-13] and ~100 nm for bulk h-BN [14]) such that phonon transport is
dominated by phonon-boundary scattering and the Umklapp process only makes marginal
contribution [10]
S10
Fig S6 Layer dependence of 120581120581CP (a c) and ITR (b d) for Bernal-stacked graphite at average steady-state temperatures of 300 K (red circles) and 400 K (black squares) The left and right columns correspond to the 120581120581CP and ITR calculated with ILP and LJ potential respectively
Fig S7 Layer dependence of 120581120581CP (a) and ITR (b) for AArsquo-stacked h-BN at average temperatures of 300 K (red circles) and 400 K (blue squares)
S11
References [1] Z Wei J Yang W Chen K Bi D Li and Y Chen Phonon mean free path of graphite along the c-axis Appl Phys Lett 104 081903 (2014) [2] D W Brenner O A Shenderova J A Harrison S J Stuart B Ni and S B Sinnott A second-generation reactive empirical bond order (REBO) potential energy expression for hydrocarbons J Phys Condens Matter 14 783 (2002) [3] A Kınacı J B Haskins C Sevik and T Ccedilağın Thermal conductivity of BN-C nanostructures Phys Rev B 86 (2012) [4] W Ouyang I Azuri D Mandelli A Tkatchenko L Kronik M Urbakh and O Hod Mechanical and Tribological Properties of Layered Materials under High Pressure Assessing the Importance of Many-Body Dispersion Effects J Chem Theory Comput 16 666 (2020) [5] S Plimpton Fast Parallel Algorithms for Short-Range Molecular Dynamics J Comput Phys 117 1 (1995) [6] W Ouyang D Mandelli M Urbakh and O Hod Nanoserpents Graphene Nanoribbon Motion on Two-Dimensional Hexagonal Materials Nano Lett 18 6009 (2018) [7] W Shinoda M Shiga and M Mikami Rapid estimation of elastic constants by molecular dynamics simulation under constant stress Phys Rev B 69 134103 (2004) [8] G L Pollack Kapitza Resistance Rev Mod Phys 41 48 (1969) [9] H-S Yang G R Bai L J Thompson and J A Eastman Interfacial thermal resistance in nanocrystalline yttria-stabilized zirconia Acta Mater 50 2309 (2002) [10] Q Fu J Yang Y Chen D Li and D Xu Experimental evidence of very long intrinsic phonon mean free path along the c-axis of graphite Appl Phys Lett 106 031905 (2015) [11] J Chen J H Walther and P Koumoutsakos Strain Engineering of Kapitza Resistance in Few-Layer Graphene Nano Lett 14 819 (2014) [12] H Zhang X Chen Y-D Jho and A J Minnich Temperature-Dependent Mean Free Path Spectra of Thermal Phonons Along the c-Axis of Graphite Nano Lett 16 1643 (2016) [13] D P Sellan E S Landry J E Turney A J H McGaughey and C H Amon Size effects in molecular dynamics thermal conductivity predictions Phys Rev B 81 214305 (2010) [14] P Jiang X Qian R Yang and L Lindsay Anisotropic thermal transport in bulk hexagonal boron nitride Phys Rev Mater 2 064005 (2018) [15] J Zhang Y Hong and Y Yue Thermal transport across graphene and single layer hexagonal boron nitride J Appl Phys 117 134307 (2015) [16] B Liu F Meng C D Reddy J A Baimova N Srikanth S V Dmitriev and K Zhou Thermal transport in a graphenendashMoS2 bilayer heterostructure a molecular dynamics study RSC Advances 5 29193 (2015) [17] Z Ding Q-X Pei J-W Jiang W Huang and Y-W Zhang Interfacial thermal conductance in grapheneMoS2 heterostructures Carbon 96 888 (2016) [18] N N T Nam and M Koshino Lattice relaxation and energy band modulation in twisted bilayer graphene Phys Rev B 96 075311 (2017)
S12
[19] R Ramiacutearez and M C Boumlhm Simple geometric generation of special points in brillouin-zone integrations Two-dimensional bravais lattices Int J Quantum Chem 30 391 (1986) [20]S L Cunningham Special points in the two-dimensional Brillouin zone Phys Rev B 10 4988 (1974) [21] H J Monkhorst and J D Pack Special points for Brillouin-zone integrations Phys Rev B 13 5188 (1976) [22] D J Chadi Special points for Brillouin-zone integrations Phys Rev B 16 1746 (1977)
1
Fermirsquos golden rule for phonons
1 Basic theory for phonons 11 Basic notations
Let us consider a 3D crystal with a total of 119873119873 = 119873119873111987311987321198731198733 unit cells and periodic boundary conditions
To be specific let 119938119938119894119894 119894119894 = 123 be the lattice vectors that define the unit cell We index unit cells
with n = (n1n2n3) where each ni = 12 ∙∙∙ Ni and their locations are 119929119929119899119899 = sum 1198991198991198941198941199381199381198941198943119894119894=1 Assume that
there are r atoms in each unit cell which are indexed with 119904119904 = 1⋯ 119903119903 The mass and the equilibrium
distance of the sth atom are notated as 119872119872s and 119929119929s0 respectively Then the location of the sth atom in
the nth unit cell at time t can be expressed as
119955119955119899119899119899119899(119905119905) = 119929119929119899119899 + 119929119929s0 + 119958119958119899119899119899119899(119905119905) (11)
where 119958119958119899119899119899119899(119905119905) is its displacement from its equilibrium position
The Lagrangian for this classical problem can be written as
ℒ = sum sum 119872119872119904119904|119955119899119899119904119904|2(119905119905)2
119903119903119899119899=1
119873119873119899119899=1 minus 119881119881 (12)
where the second term is the sum of interactions between all pairs of atoms
Under the harmonic approximation ie expanding the total potential energy 119881119881 around the
equilibrium positions The Lagrangian can be simplified as
ℒ = sum sum 119872119872119904119904|119958119899119899119904119904|2(119905119905)2
119903119903119899119899=1
119873119873119899119899=1 minus 1
2sum sum sum 120601120601120572120572120572120572prime
119899119899119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime120572120572120572120572prime119899119899119899119899prime119899119899119899119899prime (13)
where 119906119906119899119899119899119899120572120572 120572120572 = 123 are the Cartesian coordinates of the displacement 119958119958119899119899119899119899(119905119905) and
120601120601120572120572120572120572prime 119899119899119899119899prime119904119904 119904119904prime = 1205971205972119881119881
120597120597119903119903119899119899119904119904119899119899119903119903119899119899prime119904119904prime119899119899primeeq
= 120601120601120572120572prime120572120572 119899119899prime119899119899119904119904prime 119904119904 = 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime (14)
Note that the first order term vanishes because we are expanding around the equilibrium positions
120601120601120572120572120572120572prime 119899119899119899119899prime119904119904 119904119904prime represents the component of the force acted on the sth atom in the nth unit cell along 120572120572
direction when the atom 119904119904prime in the unit cell 119899119899prime moves a unit displacement along 120572120572prime direction The
symmetries of 120601120601120572120572120572120572prime 119899119899 119899119899prime119904119904 119904119904prime appearing in Eq (14) arise from the intechangability of the second
derivative and the translational invariance of the interactions
12 Dynamical matrix
The equation of motion of the sth atom in the nth unit cell can be derived using the EulerndashLagrange
equation as follows
2
119872119872119899119899119906119899119899119899119899120572120572 = minussum 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119899119899prime119899119899prime120572120572prime 119906119906119899119899prime119899119899prime120572120572prime (15)
If we displace all atoms equally ie shifting 119906119906119899119899prime119899119899prime120572120572prime to 119906119906119899119899prime119899119899prime120572120572prime + 120575120575 the total force on the sth atom in
the nth unit cell does not change From the above equation we have
sum 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119899119899prime120572120572prime = 0 (16)
We are looking for normal modes (because any general solution can be written as a linear combination
of them) these are solutions where all atoms oscillate with the same frequency Moreover because
of the lattice structure we expect solutions to reflect this periodicity So we guess solutions of the
form
119906119906119899119899119899119899120572120572(119905119905) = 1119872119872119904119904
119890119890119899119899120572120572119890119890119894119894(119954119954∙119929119929119899119899minus120596120596119905119905) (17)
where 119890119890119899119899120572120572 are real-space solutions that will be determined later and 119954119954 is the wave-vector in
reciprocal space Substituting Eq (17) into Eq (15) we can get
1205961205962119890119890119899119899120572120572(119954119954) = sum 119863119863119899119899120572120572119899119899prime120572120572prime(119954119954)119890119890119899119899prime120572120572prime(119954119954)119899119899prime120572120572prime (18)
where
119863119863119899119899120572120572119899119899prime120572120572prime(119954119954) = sum 1
119872119872119904119904119872119872119904119904prime120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119890119890
minus119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime119899119899prime = sum 1
119872119872119904119904119872119872119904119904prime120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime 119890119890
minus119894119894119954119954∙119929119929119897119897119897119897 (19)
is called dynamical matrix (dimension 3119903119903 times 3119903119903) Note that we have defined the relative cell distance
vector 119929119929119897119897 equiv 119929119929119899119899 minus 119929119929119899119899prime and number index 119897119897 equiv 119899119899 minus 119899119899prime where the infinite sum over 119899119899prime can be
replaced by the sum over 119897119897 for any value of the index 119899119899 Note that dynamical matrix is Hermitian
symmetric (ie 119863119863119899119899120572120572119899119899prime120572120572prime(119954119954)
lowast= 119863119863119899119899prime120572120572prime
119899119899120572120572 (minus119954119954)) because ϕ is symmetric so all the eigenvalues of Eq (18)
(120596120596120582120582(119954119954) 120582120582 = 12⋯ 3119903119903) are real for each 119954119954 in the Brillouin zone (BZ) which is determined by
det1205961205961205821205822(119954119954)120575120575119899119899120572120572120575120575119899119899prime120572120572prime minus 119863119863119899119899120572120572119899119899prime120572120572prime(119954119954) = 0 (110)
Taking the conjugate of this equation we have
0 = det 1205961205961205821205822(119954119954)lowast120575120575119899119899120572120572120575120575119899119899prime120572120572prime minus 119863119863119899119899120572120572119899119899
prime120572120572prime(119954119954)lowast = det1205961205961205821205822(119954119954)120575120575119899119899120572120572120575120575119899119899prime120572120572prime minus 119863119863119899119899120572120572119899119899
prime120572120572prime(minus119954119954) (111)
while replacing 119954119954 by minus119954119954 in Eq (110) we have det1205961205961205821205822(minus119954119954)120575120575119899119899120572120572120575120575119899119899prime120572120572prime minus 119863119863119899119899120572120572119899119899prime120572120572prime(minus119954119954) = 0
Itrsquos clear that 1205961205961205821205822(119954119954) and 1205961205961205821205822(minus119954119954) obey the same equation thus we have
120596120596120582120582(minus119954119954) = 120596120596120582120582(119954119954) (112)
The corresponding eigenvectors are orthonormal
sum 119890119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582
prime119899119899120572120572 = 120575120575120582120582120582120582prime (113)
The complex conjugate of Eq (18) gives
1205961205962119890119890119899119899120572120572lowast (119954119954) = sum 119863119863119899119899120572120572119899119899prime120572120572prime(119954119954)
lowast119890119890119899119899prime120572120572primelowast (119954119954)119899119899prime120572120572prime = sum 119863119863119899119899prime120572120572prime
119899119899120572120572 (minus119954119954)119890119890119899119899prime120572120572primelowast (119954119954)119899119899prime120572120572prime (114)
3
While replacing 119954119954 by minus119954119954 in Eq (18) we get the following equation
1205961205962119890119890119899119899120572120572(minus119954119954) = sum 119863119863119899119899120572120572119899119899prime120572120572prime(minus119954119954)119890119890119899119899prime120572120572prime(minus119954119954)119899119899prime120572120572prime (115)
From Eq (114) and Eq (115) we see that eigenvectors 119890119890119899119899120572120572120582120582 (minus119954119954)lowast and 119890119890119899119899120572120572120582120582 (119954119954) obey the same
eigenvalue equation Since the eigenvectors are normalized we get the following property
119890119890119899119899120572120572120582120582 (minus119954119954)lowast
= 119890119890119899119899120572120572120582120582 (119954119954) (116)
2 Second quantization The general solution is a linear combination of all these normal modes thus we have
119906119906119899119899119899119899120572120572(119905119905) = sum 119862119862120582120582(119954119954)119873119873119872119872119904119904
119890119890119899119899120572120572120582120582 (119954119954)119890119890minus119894119894120596120596119954119954120582120582119905119905119890119890119894119894119954119954∙119929119929119899119899119954119954120582120582 = sum 119876119876120582120582(119954119954119905119905)119873119873119872119872119904119904
119890119890119899119899120572120572120582120582 (119954119954)119890119890119894119894119954119954∙119929119929119899119899119954119954120582120582 (21)
where the we define the normal coordinates as 119876119876120582120582(119954119954 119905119905) equiv 119862119862120582120582(119954119954)119890119890minus119894119894120596120596120582120582(119954119954)119905119905 in the eigenvectors
representation To ensure that the displacements 119906119906119899119899119899119899120572120572(119905119905) are real (namely 119906119906119899119899119899119899120572120572lowast (119905119905) = 119906119906119899119899119899119899120572120572(119905119905)) the
following relation on 119876119876120582120582(119954119954 119905119905) and 119890119890119899119899120572120572120582120582 is enforced
119876119876120582120582(119954119954 119905119905)119890119890119899119899120572120572120582120582 (119954119954)lowast
= 119876119876120582120582(minus119954119954 119905119905)119890119890119899119899120572120572120582120582 (minus119954119954) (22)
where we used the fact that the sum over 119954119954 runs symmetrically over both negative and positive
values Using Eq (116) we have
[119876119876120582120582(119954119954 119905119905)]lowast = 119876119876120582120582(minus119954119954 119905119905) (23)
21 Kinetic energy term
Using Eq (21) the kinetic energy of the system can be expressed as
119879119879 = 12sum 1198721198721198991198991199061198991198991198991198991205721205722 (119905119905)119899119899119899119899120572120572 = 1
2sum 119872119872119904119904
119873119873119872119872119904119904sum sum 119876120582120582(119954119954 119905119905)119876120582120582prime(119954119954prime 119905119905)119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582
prime(119954119954prime) sum 119890119890119894119894119954119954+119954119954prime∙119929119929119899119899119899119899119954119954prime120582120582prime119954119954120582120582119899119899120572120572 =
12sum sum 119876120582120582(119954119954 119905119905)119876120582120582prime(minus119954119954 119905119905)119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582
prime(minus119954119954)119954119954119954119954prime120582120582prime =119899119899120572120572
12sum 119876120582120582(119954119954 119905119905)119876120582120582prime
lowast (119954119954 119905119905) sum 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582prime(119954119954)
lowast119899119899120572120572 =119954119954120582120582120582120582prime
12sum 119876120582120582(119954119954 119905119905)119876120582120582lowast(119954119954 119905119905)119954119954120582120582 =
12sum 119876120582120582(119954119954 119905119905)119876120582120582(minus119954119954 119905119905)119954119954120582120582
ie
119879119879 = 12sum 119876120582120582(119954119954 119905119905)119876120582120582(minus119954119954 119905119905)119954119954120582120582 (24)
To derive Eq (24) we used Eqs (113) and (116) as well as the following equations
1119873119873sum 119890119890119894119894119954119954+119954119954prime∙119929119929119899119899119899119899 = 120575120575119954119954+119954119954prime120782120782 (25)
4
22 Potential energy term
Similarity the potential energy can be rewritten in terms of the normal coordinates as follows
119880119880 =12120601120601119899119899119899119899120572120572119899119899prime119899119899prime120572120572prime119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
120572120572120572120572prime119899119899119899119899prime119899119899119899119899prime=
12120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119899119899+119954119954prime∙119929119929119899119899prime
119873119873119872119872119899119899119872119872119899119899prime
[119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(119954119954prime 119905119905)] 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime120582120582prime (119954119954prime)
119954119954prime120582120582prime119954119954120582120582119899119899119899119899prime120572120572120572120572prime119899119899119899119899prime
=12120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894(119954119954+119954119954prime)∙119929119929119899119899119890119890minus119894119894119954119954prime∙119929119929119897119897
119873119873119872119872119899119899119872119872119899119899prime
[119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(119954119954prime 119905119905)] 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime120582120582prime (119954119954prime)
119954119954prime120582120582prime119954119954120582120582119899119899119897119897120572120572120572120572prime119899119899119899119899prime
=12120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890minus119894119894119954119954prime∙119929119929119897119897
119872119872119899119899119872119872119899119899prime[119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(119954119954prime 119905119905)] 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (119954119954prime)1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899
119899119899=120575120575119902119902+119902119902prime=0
119954119954prime120582120582prime119954119954120582120582119897119897120572120572120572120572prime119899119899119899119899prime
=12120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894119954119954119929119929119897119897
119872119872119899119899119872119872119899119899prime [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)] 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (minus119954119954)119954119954120582120582120582120582prime119897119897120572120572120572120572prime119899119899119899119899prime
=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]119890119890119899119899120572120572120582120582 (119954119954)
119899119899120572120572
119863119863119899119899120572120572119899119899prime120572120572prime(minus119954119954)119890119890119899119899prime120572120572prime
120582120582prime (minus119954119954)119899119899prime120572120572prime119954119954120582120582120582120582prime
=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]119890119890119899119899120572120572120582120582 (119954119954)
119899119899120572120572
1205961205961205821205822(minus119954119954)119890119890119899119899120572120572120582120582prime (minus119954119954)
119954119954120582120582120582120582prime
=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]1205961205961205821205822(119954119954)120575120575120582120582120582120582prime =119954119954120582120582120582120582prime
121205961205961205821205822(119954119954)119876119876120582120582(119954119954 119905119905)119876119876120582120582(minus119954119954 119905119905)
119954119954120582120582
ie
119880119880 = 12sum 1205961205961205821205822(119954119954)119876119876120582120582(119954119954 119905119905)119876119876120582120582(minus119954119954 119905119905)119954119954120582120582 (26)
where in above deriviation the orthogonality of its eigenvectores and the symmetry of its eigenvalues
are used Thus the Lagrangian reads as
ℒ = 119879119879 minus 119881119881 = 12sum 119876120582120582(minus119954119954 119905119905)119876120582120582(119954119954 119905119905) minus 120596120596120582120582
2(119954119954)119876119876120582120582(minus119954119954 119905119905)119876119876120582120582(119954119954 119905119905)119954119954120582120582 (27)
Using the relation 119875119875120582120582(119954119954 119905119905) = 120597120597ℒ part119876120582120582(119954119954 119905119905) the Hamiltonian of the system then can be written as
119867119867 = 119879119879 + 119881119881 = 12sum [119875119875120582120582(minus119954119954 119905119905)119875119875120582120582(119954119954 119905119905) + 120596120596120582120582
2(119954119954)119876119876120582120582(minus119954119954 119905119905)119876119876120582120582(119954119954 119905119905)]119954119954120582120582 (28)
Now we quantize H by asking the momenta and coordinates to be operators
119867119867 = 12sum 119875119875minus119954119954120582120582119875119875119954119954120582120582 + 120596120596119954119954120582120582
2 119876119876minus119954119954120582120582119876119876119954119954120582120582119954119954120582120582 (29)
here 119875119875119954119954120582120582 and 119876119876119954119954120582120582 obey the following commutation relations
119876119876119954119954120582120582119875119875119954119954prime120582120582prime = 119894119894ℏ120575120575119954119954119954119954prime120575120575120582120582120582120582prime
119876119876119954119954120582120582119876119876119954119954prime120582120582prime = 0 119875119875119954119954120582120582119875119875119954119954prime120582120582prime = 0 (210)
Similar to the case of ordinary quantum harmonic oscillators it is convenient to define ladder
operators for each mode as follows
119876119876119954119954120582120582 = ℏ
2120596120596119954119954120582120582119887119887119954119954120582120582 + 119887119887minus119954119954120582120582
dagger 119875119875119954119954120582120582 = 119894119894ℏ120596120596119954119954120582120582
2119887119887119954119954120582120582
dagger minus 119887119887minus119954119954120582120582 (211)
where 119887119887119954119954120582120582dagger and 119887119887119954119954120582120582 are Bosonic creation and annihilation operators for phonons with momentum
119954119954 branch index 120582120582 and frequency 120596120596119954119954120582120582 which obeys the Bosonic commutation relation
5
119887119887119954119954120582120582 119887119887119954119954prime120582120582prime
dagger = 120575120575119954119954119954119954prime120575120575120582120582120582120582prime
119887119887119954119954120582120582 119887119887119954119954prime120582120582prime = 0 119887119887119954119954120582120582dagger 119887119887119954119954prime120582120582prime
dagger = 0 (212)
Substituting Eqs (211) into (29) and using the properties Eq (210) and (212) we have
119867119867 =12119875119875minus119954119954120582120582119875119875119954119954120582120582 +120596120596119954119954120582120582
2 119876119876minus119954119954120582120582119876119876119954119954120582120582119954119954120582120582
=12minus
ℏ120596120596119954119954120582120582
2119887119887minus119954119954120582120582
dagger minus 119887119887119954119954120582120582119887119887119954119954120582120582dagger minus 119887119887minus119954119954120582120582+ 120596120596119954119954120582120582
2 ℏ2120596120596119954119954120582120582
119887119887minus119954119954120582120582 + 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 + 119887119887minus119954119954120582120582
dagger 119954119954120582120582
=ℏ4120596120596119954119954120582120582minus119887119887minus119954119954120582120582
dagger 119887119887119954119954120582120582dagger minus 119887119887minus119954119954120582120582
dagger 119887119887minus119954119954120582120582 minus 119887119887119954119954120582120582119887119887119954119954120582120582dagger + 119887119887119954119954120582120582119887119887minus119954119954120582120582
119954119954120582120582
+ 119887119887minus119954119954120582120582119887119887119954119954120582120582 + 119887119887minus119954119954120582120582119887119887minus119954119954120582120582dagger + 119887119887119954119954120582120582
dagger 119887119887119954119954120582120582 + 119887119887119954119954120582120582dagger 119887119887minus119954119954120582120582
dagger
=ℏ4120596120596119954119954120582120582119887119887minus119954119954120582120582
dagger 119887119887minus119954119954120582120582 + 119887119887minus119954119954120582120582119887119887minus119954119954120582120582dagger + 119887119887119954119954120582120582119887119887119954119954120582120582
dagger + 119887119887119954119954120582120582dagger 119887119887119954119954120582120582
119954119954120582120582
=ℏ41205961205961199541199541205821205822119887119887minus119954119954120582120582
dagger 119887119887minus119954119954120582120582 + 1+ 2119887119887119954119954120582120582dagger 119887119887119954119954120582120582 + 1
119954119954120582120582
= ℏ120596120596119954119954120582120582 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 +
12
119954119954120582120582
Therefore we finally get the quantized representation of non-interacting phonons
119867119867 = sum ℏ120596120596119954119954120582120582 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 + 1
2119954119954120582120582 (213)
The operator of atom displacements (Eq 114) is expressed in terms of the phonon operators by
119906119906119899119899119899119899120572120572 = sum ℏ
2119873119873119872119872119904119904120596120596119954119954120582120582119890119890119899119899120572120572120582120582 119890119890119894119894119954119954∙119929119929119899119899119887119887119954119954120582120582 + 119887119887minus119954119954120582120582
dagger 119954119954120582120582 (214)
These equations will be used below
3 Inter-phonon coupling within harmonic approximation 31 Hamiltonian with inter-phonon coupling
In this section we consider systems that consist of two (or more) covalently bonded units that are
weakly coupled between them Unlike previous studies that considered phonon-phonon couplings
resulting from anharmonicity effects [1] here all phonon mode considered are Harmonic and the
couplings arise from the division of the entire system into subunits The Hamiltonian of the whole
system can be written as
119867119867 = 1198671198671 + 1198671198672 + 11986711986712 (31)
To derive the expressions of 1198671198671 1198671198672 and 11986711986712 in Eq (31) we consider the Hamiltonian of the
whole system written as the function of the atomic displacements
119867119867 = 119879119879 + 119881119881 = sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)2119899119899119899119899120572120572 + 1
2sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime120572120572120572120572prime119899119899119899119899prime119899119899119899119899prime (32)
6
Letrsquos assume that subsystem I and II contain atoms with indeices ranging from 119904119904 = 12⋯ 1199031199032 and
119904119904 = 1199031199032
+ 1 1199031199032
+ 2⋯ 119903119903 respectively Then the kinetic energy term in Eq (32) can be rewritten as
119879119879 = sum sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)2
1199031199032119899119899=1 + sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)
2119903119903119899119899=1199031199032+1
119899119899120572120572 (33)
While the potential energy term is
119881119881 =12 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
1199031199032
119899119899119899119899prime=1
+ 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
119903119903
119899119899119899119899prime=1199031199032+1
+ 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime119899119899119899119899prime
+ 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
1199031199032
119899119899prime=1
119903119903
119899119899=1199031199032+1
= 11988111988111 + 11988111988122 + 11988111988112 + 11988111988121
Or equivalently
⎩⎪⎪⎨
⎪⎪⎧ 11988111988111 = 1
2sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
1199031199032119899119899119899119899prime=1120572120572120572120572prime119899119899119899119899prime
11988111988122 = 12sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
119903119903119899119899119899119899prime=1199031199032+1120572120572120572120572prime119899119899119899119899prime
11988111988112 = 12sum sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
119903119903119899119899prime=1199031199032+1
1199031199032119899119899=1120572120572120572120572prime119899119899119899119899prime
11988111988121 = 12sum sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
1199031199032119899119899prime=1
119903119903119899119899=1199031199032+1120572120572120572120572prime119899119899119899119899prime
(34)
32 Second quantization
We may now quantize this Hamiltonian and write it in the basis of the eigenstates of the coupled
subunits In second quantization the atomic displacement operators of the two subunits are given in
the following form
119906119906119899119899119899119899120572120572 =
⎩⎨
⎧ sum ℏ
2119873119873119872119872119904119904ω119954119954120582120582119890119890119899119899120572120572120582120582 119890119890119894119894119954119954∙119929119929119899119899119886119886119954119954120582120582 + 119886119886minus119954119954120582120582
dagger 119954119954120582120582 119904119904 isin 1 1199031199032
sum ℏ
2119873119873119872119872119904119904120596120596119954119954prime120582120582prime119890119899119899120572120572120582120582
prime119890119890119894119894119954119954prime∙119929119929119899119899 119886119886 119954119954prime120582120582prime + 119886119886minus119954119954prime120582120582primedagger 119954119954prime120582120582prime 119904119904 isin 119903119903
2+ 1 119903119903
(35)
Here we use different notations for the creation and annihilation operators (119886119886119954119954120582120582119886119886minus119954119954120582120582dagger and 119886119886119954119954120582120582119886119886minus119954119954120582120582
dagger )
eigenvalues (ω119954119954120582120582120596120596119954119954120582120582) and eigenvectors (119890119890119899119899120572120572120582120582 119890119899119899120572120572120582120582 ) for the two subunits Note that 119890119890119899119899120572120572120582120582 and 119890119899119899120572120572120582120582 are
of the dimensions of the whole system however their non-zero elements appear only on the relevant
subunits such that they obey the following relations sum 119890119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582
prime119899119899120572120572 = 120575120575120582120582120582120582prime sum 119890119899119899120572120572120582120582
lowast119890119899119899120572120572120582120582
prime119899119899120572120572 =
120575120575120582120582120582120582prime sum 119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582
prime119899119899120572120572 = sum 119890119890119899119899120572120572120582120582
lowast119890119899119899120572120572120582120582
prime119899119899120572120572 = 0 Defining the normal coordinates of the two subunits as
119876119876119954119954120582120582 equiv ℏ
2ω119954119954120582120582119886119886119954119954120582120582 + 119886119886minus119954119954120582120582
dagger 119876119876119954119954120582120582 equiv ℏ
2ω119954119954120582120582119886119886119954119954120582120582 + 119886119886minus119954119954120582120582
dagger (36)
and the corresponding momenta operators as
7
119875119875119954119954120582120582 = 119894119894ℏω119954119954120582120582
2119886119886119954119954120582120582
dagger minus 119886119886minus119954119954120582120582 119875119875119954119954120582120582 = 119894119894ℏω1199541199541205821205822
119886119886119954119954120582120582dagger minus 119886119886minus119954119954120582120582 (37)
Eq (35) reads
119906119906119899119899119899119899120572120572 = sum 119890119890119904119904119899119899120582120582 (119954119954)
119873119873119872119872119904119904119890119890119894119894119954119954∙119929119929119899119899119876119876119954119954120582120582119954119954120582120582 119904119904 isin 1 119903119903
2
sum 119890119904119904119899119899120582120582 (119954119954)119873119873119872119872119904119904
119890119890119894119894119954119954∙119929119929119899119899119876119876119954119954120582120582119954119954120582120582 119904119904 isin 1199031199032
+ 1 119903119903 (38)
The second quantized kinetic energy operator is then written as
119879119879 = 1198791198791 + 1198791198792 = 12sum 119875119875119954119954120582120582119875119875minus119954119954120582120582119954119954120582120582 + 1
2sum 119875119875119954119954prime120582120582prime119875119875minus119954119954prime120582120582prime119954119954prime120582120582prime (39)
Correspondingly the various potential energy terms are obtained by substituting Eq (38) in Eq (34)
11988111988111 =12 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119899119899119899119899prime
119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime120582120582prime (119954119954prime)
119872119872119899119899119872119872119899119899prime
1119873119873119890119890119894119894119954119954∙119929119929119899119899+119894119894119954119954prime∙119929119929119899119899prime119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954120582120582119954119954prime120582120582prime
1199031199032
119899119899119899119899prime=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954119954119954prime120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (119954119954prime)1119873119873 119890119890
119894119894(119954119954+119954119954prime)119929119929119899119899prime
119899119899prime120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime
119872119872119899119899119872119872119899119899prime119899119899
1199031199032
119899119899119899119899prime=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954119954119954prime120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (119954119954prime) 120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime
1199031199032
119899119899119899119899prime=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime
119954119954120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (minus119954119954)120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
1199031199032
119899119899119899119899prime=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime
119954119954120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954) 119863119863119899119899120572120572119899119899
prime120572120572prime(minus119954119954)119890119890119899119899prime120572120572prime120582120582prime (minus119954119954)
120572120572prime
1199031199032
119899119899prime=1
1199031199032
119899119899=1120572120572
=12 ω119954119954120582120582prime
2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582120582120582prime
119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582prime (119954119954)
lowast
1199031199032
119899119899=1
=120572120572
12ω119954119954120582120582
2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582
ie
11988111988111 = 12sum ω119954119954120582120582
2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582 (310)
Going from the second line to the third line we used the fact that the sum over 119899119899prime runs between plusmninfin
and the summand depends only on the difference between 119899119899 and 119899119899prime hence the sum is independent
of the value of the index 119899119899 Therefore we can replace the sum over 119899119899prime by a sum over 119897119897 equiv 119899119899 minus 119899119899prime
amd define 119929119929119897119897 equiv 119929119929119899119899 minus 119929119929119899119899prime Following the same procedure we can get the corresponding
expressions for the second diagonal term
11988111988122 = 12sum 1205961205961199541199541205821205822 119876119876119954119954120582120582119876119876minus119954119954120582120582119954119954120582120582 (311)
Similarly for the off-diagonal terms we get
8
11988111988112 =12 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119899119899119899119899prime
119903119903
119899119899prime=1199031199032+1
119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime
120582120582prime (119954119954prime)
119872119872119899119899119872119872119899119899prime
1119873119873119890119890119894119894119954119954∙119929119929119899119899+119894119894119954119954prime∙119929119929119899119899prime119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954120582120582119954119954prime120582120582prime
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954119954119954prime120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime
120582120582prime (119954119954prime)1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime
119872119872119899119899119872119872119899119899prime119899119899
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954119954119954prime120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime
120582120582prime (119954119954prime) 120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime
119954119954120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime
120582120582prime (minus119954119954)120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954
119890119899119899prime120572120572prime120582120582prime (119954119954)
lowast120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119890119890119899119899120572120572120582120582 (119954119954)119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954
Where we have defined
119881119881120582120582120582120582prime(119954119954) equiv sum sum sum 119890119899119899prime120572120572prime120582120582prime (119954119954)
lowast119881119881119899119899120572120572119899119899
prime120572120572prime(119954119954)119890119890119899119899120572120572120582120582 (119954119954)119903119903119899119899prime=1199031199032+1
1199031199032119899119899=1120572120572120572120572prime (312)
where
119881119881119899119899120572120572119899119899prime120572120572prime(119954119954) equiv sum 119890119890119894119894119954119954∙119929119929119897119897
119872119872119904119904119872119872119904119904prime119897119897 120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime (313)
Itrsquos easy to show that 119881119881120582120582120582120582prime(119954119954) has the following property
119881119881120582120582120582120582primelowast (119954119954) = 119881119881120582120582120582120582prime(minus119954119954) (314)
Following the same procedure we have
9
11988111988121 =12 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119899119899119899119899prime
119903119903
119899119899=1199031199032+1
119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (119954119954prime)
119872119872119899119899119872119872119899119899prime
1119873119873119890119890119894119894119954119954∙119929119929119899119899+119894119894119954119954prime∙119929119929119899119899prime119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954120582120582119954119954prime120582120582prime
1199031199032
119899119899prime=1120572120572120572120572prime
=12 120601120601120572120572prime120572120572
119899119899prime minus 119899119899119904119904prime 119904119904
119899119899prime119899119899
119903119903
119899119899prime=1199031199032+1
119890119899119899prime120572120572prime120582120582prime (119954119954prime)119890119890119899119899120572120572120582120582 (119954119954)
119872119872119899119899prime119872119872119899119899
1119873119873 119890119890
119894119894119954119954prime∙119929119929119899119899prime+119894119894119954119954∙119929119929119899119899119876119876119954119954prime120582120582prime119876119876119954119954120582120582119954119954prime120582120582prime119954119954120582120582
1199031199032
119899119899=1120572120572prime120572120572
=12 119876119876119954119954prime120582120582prime119876119876119954119954120582120582
119954119954119954119954prime120582120582120582120582prime119890119899119899prime120572120572prime120582120582prime (119954119954prime)119890119890119899119899120572120572120582120582 (119954119954)
1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime
119872119872119899119899119872119872119899119899prime119899119899
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954prime120582120582prime119876119876119954119954120582120582
119954119954119954119954prime120582120582120582120582prime119890119899119899prime120572120572prime120582120582prime (119954119954prime)119890119890119899119899120572120572120582120582 (119954119954) 120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954prime120582120582prime119876119876minus119954119954prime120582120582119890119899119899prime120572120572prime
120582120582prime (119954119954prime)119890119890119899119899120572120572120582120582 (minus119954119954prime)120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890minus119894119894119954119954prime∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897119954119954prime120582120582120582120582prime
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954prime120582120582prime119876119876119954119954prime120582120582
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954prime 119890119890119899119899120572120572120582120582 (119954119954prime)
lowast120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890minus119894119894119954119954prime∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119890119899119899prime120572120572prime120582120582prime (119954119954prime)
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582prime119876119876119954119954120582120582
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954
119890119890119899119899120572120572120582120582 (119954119954)lowast120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890minus119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119890119899119899prime120572120572prime120582120582prime (119954119954)
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger dagger
3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954
119890119899119899prime120572120572prime120582120582prime (119954119954)
lowast120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119890119890119899119899120572120572120582120582 (119954119954)119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
lowast
=
⎩⎨
⎧12 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954⎭⎬
⎫dagger
= 11988111988112dagger
Define
1198671198671 = 11987911987911 + 119881119881111198671198672 = 11987911987922 + 1198811198812211986711986712 = 11988111988112 + 11988111988121
(315)
we finally get the expressions of 1198671198671 1198671198672 and 11986711986712 in Eq (31) as follows
⎩⎪⎨
⎪⎧ 1198671198671 = sum sum ℏ120596120596119954119954120582120582 119886119886119954119954120582120582
dagger 119886119886119954119954120582120582 + 1231199031199032
120582120582=1119954119954
1198671198672 = sum sum ℏ120596120596119954119954120582120582prime 119886119886119954119954120582120582primedagger 119886119886119954119954120582120582prime + 1
23119903119903
120582120582prime=1+31199031199032119954119954
11986711986712 = 12sum sum sum 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903120582120582prime=31199031199032 +1
31199031199032120582120582=1 + h c 119954119954
(316)
Here hc means the Hermitian conjugate Since the indices of 119886119886119954119954120582120582 119886119886119954119954120582120582prime and 119876119876119954119954120582120582 119876119876119954119954120582120582prime belong to
the two system sections we can define an abbreviated notation 119886119886119954119954120582120582 and 119876119876119954119954120582120582 using the index 120582120582 to
identify which subsystem they belong to In this case the Hamiltonian of the coupled system can be
10
simplified as follows
119867119867 = 1198671198670 + 119867119867119862119862 (317)
where
1198671198670 = sum sum ℏ120596120596119954119954120582120582 119886119886119954119954120582120582
dagger 119886119886119954119954120582120582 + 123119903119903
120582120582=1119954119954
119867119867119862119862 = 12sum sum sum 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903120582120582prime=31199031199032 +1
31199031199032120582120582=1 + ℎ 119888119888 119954119954
(318)
4 Greenrsquos function 41 Dynamics of the ladder operators
In order to describe the dynamics of the ladder operators appearing in the Hamiltonian of Eq (317)
we express them in the Heisenberg picture as follows
119886119886119901119901(120591120591) = 119890119890119894119894ℏ119867119867119905119905119886119886119901119901119890119890
minus119894119894ℏ119867119867119905119905 = 119890119890
119867119867120591120591ℏ 119886119886119901119901119890119890
minus119867119867120591120591ℏ (41)
where we define the imaginary time 120591120591 equiv 119894119894119905119905 and introduce the notation 119953119953 equiv (119954119954 120582120582) and 119953119953 equiv (minus119954119954 120582120582)
The corresponding equation of motion for the ladder operators is given by
ℏ part119886119886119953119953(120591120591)
120597120597120591120591= 119867119867119886119886119953119953(120591120591) (42)
For the uncoupled system (119867119867119862119862 = 0) this gives
ℏpart119886119886119953119953(120591120591)120597120597120591120591
= 1198671198670119886119886119953119953(120591120591) = 1198671198670 1198901198901198671198670120591120591ℏ 119886119886119953119953119890119890
minus1198671198670120591120591ℏ = 1198671198670119890119890
1198671198670120591120591ℏ 119886119886119953119953119890119890
minus1198671198670120591120591ℏ minus 119890119890
1198671198670120591120591ℏ 119886119886119953119953119890119890
minus1198671198670120591120591ℏ 1198671198670
= 1198901198901198671198670120591120591ℏ 1198671198670119886119886119953119953 minus 1198861198861199531199531198671198670119890119890
minus1198671198670120591120591ℏ = 119890119890
1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890
minus1198671198670120591120591ℏ
ie
ℏ part119886119886119953119953(120591120591)
120597120597120591120591= 119890119890
1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890
minus1198671198670120591120591ℏ (43)
The commutator on the right-hand-side of Eq (43) can be evaluated as follows
1198671198670119886119886119953119953prime = sum ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953 + 1
2119953119953 119886119886119953119953prime = sum ℏ120596120596119953119953119886119886119953119953
dagger119886119886119953119953119886119886119953119953prime119953119953 = sum ℏ120596120596119953119953119886119886119953119953dagger119886119886119953119953119886119886119953119953prime minus 119886119886119953119953prime119886119886119953119953
dagger119886119886119953119953119953119953 =
sum ℏ120596120596119953119953119886119886119953119953dagger119886119886119953119953119886119886119953119953prime minus 120575120575119953119953prime119953119953 + 119886119886119953119953
dagger119886119886119953119953prime119886119886119953119953119953119953 = minusℏ120596120596119953119953prime119886119886119953119953prime + sum ℏ120596120596119953119953119886119886119953119953dagger119886119886119953119953119886119886119953119953prime minus 119886119886119953119953
dagger119886119886119953119953119886119886119953119953prime119953119953 = minusℏ120596120596119953119953prime119886119886119953119953prime
ie
1198671198670119886119886119953119953prime = minusℏ120596120596119953119953prime119886119886119953119953prime (44)
Therefore we have
ℏ part119886119886119953119953(120591120591)
120597120597120591120591= 119890119890
1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890
minus1198671198670120591120591ℏ = minusℏ120596120596119953119953119890119890
1198671198670120591120591ℏ 119886119886119953119953119890119890
minus1198671198670120591120591ℏ = minusℏ120596120596119953119953119886119886119953119953(120591120591) (45)
11
The solution of Eq (45) is 119886119886119953119953(120591120591) = 119886119886119953119953(0)119890119890minus120596120596119953119953120591120591 = 119886119886119953119953119890119890minus120596120596119953119953120591120591 For 119886119886119953119953dagger(120591120591) we have
1198671198670119886119886119953119953primedagger (0) = 1198671198670119886119886119953119953prime
dagger = ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953 +
12
119953119953
119886119886119953119953primedagger = ℏ120596120596119953119953119886119886119953119953
dagger119886119886119953119953119886119886119953119953primedagger
119953119953
= ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953119886119886119953119953prime
dagger minus 119886119886119953119953primedagger 119886119886119953119953
dagger119886119886119953119953119953119953
= ℏ120596120596119953119953 119886119886119953119953dagger 120575120575119953119953119953119953prime + 119886119886119953119953prime
dagger 119886119886119953119953 minus 119886119886119953119953primedagger 119886119886119953119953
dagger119886119886119953119953119953119953
= ℏ120596120596119953119953prime119886119886119953119953primedagger + ℏ120596120596119953119953 119886119886119953119953
dagger119886119886119953119953primedagger 119886119886119953119953 minus 119886119886119953119953prime
dagger 119886119886119953119953dagger119886119886119953119953
119953119953
= ℏ120596120596119953119953prime119886119886119953119953primedagger
In summary we have
119886119886119953119953(120591120591) = 119886119886119953119953(0)119890119890minus120596120596119953119953120591120591 = 119886119886119953119953119890119890minus120596120596119953119953120591120591
119886119886119953119953dagger(120591120591) = 119886119886119953119953
dagger(0)119890119890120596120596119953119953120591120591 = 119886119886119953119953dagger119890119890120596120596119953119953120591120591 (46)
42 Greenrsquos function
To describe thermal transport properties between different system sections we use the formalism of
thermal (or imaginary time) phonon Greenrsquos function [2] To this end we define the thermal Greenrsquos
function for phonons as
119866119866119953119953119953119953prime(120591120591 120591120591prime) = minuslang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime)rang = minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime) (47)
where 119879119879120591120591 is the time ordering operator and 120588120588119867119867 is the statistical operator for the grand canonical
ensemble (note that the chemical potential for phonons is zero)
120588120588119867119867 = 119890119890minus120573120573119867119867119885119885119867119867 (48)
where the partition function is given by 119885119885119867119867 equiv Tr119890119890minus120573120573119867119867 and 120573120573 = 1119896119896119861119861119879119879 with 119896119896119861119861 being
Boltzmannrsquos constant and 119879119879 the temperature For time independent Hamiltonians 119866119866119953119953119953119953prime(120591120591 120591120591prime)
depends only on 120591120591 minus 120591120591prime ie
119866119866119953119953119953119953prime(120591120591 120591120591prime) = 119866119866119953119953119953119953prime(120591120591 minus 120591120591prime 0) (49)
To show this we shall first assume that 120591120591 gt 120591120591prime such that
119866119866119953119953119953119953prime(120591120591 120591120591prime) = minuslang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime)rang = minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime) = minusTr 120588120588119867119867119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime)
= minusTr 120588120588119867119867119890119890119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591ℏ 119890119890
119867119867120591120591primeℏ 119886119886119953119953prime
dagger 119890119890minus119867119867120591120591primeℏ = minusTr 120588120588119867119867119890119890
minus119867119867120591120591primeℏ 119890119890
119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591minus120591120591prime
ℏ 119886119886119953119953primedagger
= minusTr 120588120588119867119867119890119890119867119867120591120591minus120591120591prime
ℏ 119886119886119953119953119890119890minus119867119867120591120591minus120591120591prime
ℏ 119886119886119953119953primedagger = minusTr 120588120588119867119867119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953prime
dagger (0)
= minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953primedagger (0) = minus lang119879119879120591120591119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953prime
dagger (0)rang = 119866119866119953119953119953119953prime(120591120591 minus 120591120591prime 0)
Where we have used the commutativity of 120588120588119867119867 and 119890119890plusmn119867119867120591120591prime
ℏ and the invariance of the trace operation
12
towards cyclic permutations Similarly for 120591120591 lt 120591120591prime we have
119866119866119953119953119953119953prime(120591120591 120591120591prime) = minuslang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime)rang = minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime) = minusTr 120588120588119867119867119886119886119953119953primedagger (120591120591prime)119886119886119953119953(120591120591)
= minusTr 120588120588119867119867119890119890119867119867120591120591primeℏ 119886119886119953119953prime
dagger 119890119890minus119867119867120591120591primeℏ 119890119890
119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591ℏ = minusTr 120588120588119867119867119886119886119953119953prime
dagger 119890119890119867119867120591120591minus120591120591prime
ℏ 119886119886119953119953119890119890minus119867119867120591120591ℏ 119890119890
119867119867120591120591primeℏ
= minusTr 120588120588119867119867119886119886119953119953primedagger 119890119890
119867119867120591120591minus120591120591primeℏ 119886119886119953119953119890119890
minus119867119867120591120591minus120591120591prime
ℏ = minusTr 120588120588119867119867119886119886119953119953primedagger (0)119886119886119953119953(120591120591 minus 120591120591prime)
= minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953primedagger (0)119886119886119953119953(120591120591 minus 120591120591prime) = minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953prime
dagger (0)
= minuslang119879119879120591120591119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953primedagger (0)rang = 119866119866119953119953119953119953prime(120591120591 minus 120591120591prime 0)
For simplicity we will introduce the notation 119866119866119953119953119953119953prime(120591120591 0) equiv 119866119866119953119953119953119953prime(120591120591) We further note that when using
imaginary time 119866119866119953119953119953119953prime(120591120591) is a periodic functions in the domain [minus120573120573ℏ120573120573ℏ] with a period of 120573120573ℏ (see
Page 236 of Ref [2])
119866119866119953119953119953119953prime(120591120591 + 120573120573ℏ 0) = 119866119866119953119953119953119953prime(120591120591 0) 120591120591 lt 0119866119866119953119953119953119953prime(120591120591 minus 120573120573ℏ 0) = 119866119866119953119953119953119953prime(120591120591 0) 120591120591 gt 0 (410)
To show this we shall again assume first that minus120573120573ℏ lt 120591120591 lt 0 to write that is
119866119866119953119953119953119953prime(120591120591 + 120573120573ℏ 0) = minus lang119879119879120591120591119886119886119953119953(120591120591 + 120573120573ℏ)119886119886119953119953primedagger (0)rang = minuslang119886119886119953119953(120591120591 + 120573120573ℏ)119886119886119953119953prime
dagger (0)rang
= minusTr 120588120588119867119867119890119890119867119867(120591120591+120573120573ℏ)
ℏ 119886119886119953119953119890119890minus119867119867(120591120591+120573120573ℏ)
ℏ 119886119886119953119953prime = minusTr 120588120588119867119867119890119890120573120573119867119867119890119890
119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591ℏ 119890119890minus120573120573119867119867119886119886119953119953prime
= minusTr120588120588119867119867119890119890120573120573119867119867119886119886119953119953(120591120591)119890119890minus120573120573119867119867119886119886119953119953prime(0) = minus
1119885119885119867119867
Tr119890119890minus120573120573119867119867119890119890120573120573119867119867119886119886119953119953(120591120591)119890119890minus120573120573119867119867119886119886119953119953prime(0)
= minus1119885119885119867119867
Tr119886119886119953119953(120591120591)119890119890minus120573120573119867119867119886119886119953119953prime(0) = minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119886119886119953119953prime(0)119886119886119953119953(120591120591)
= minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime(0) = minusTr120588120588119867119867119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime = 119866119866119953119953119953119953prime(120591120591 0)
Similarly for 120573120573ℏ gt 120591120591 gt 0 we have
119866119866119953119953119953119953prime(120591120591 minus 120573120573ℏ 0) = minus lang119879119879120591120591119886119886119953119953(120591120591 minus 120573120573ℏ)119886119886119953119953primedagger (0)rang = minuslang119886119886119953119953prime
dagger (0)119886119886119953119953(120591120591 minus 120573120573ℏ)rang
= minusTr 120588120588119867119867119886119886119953119953prime119890119890119867119867(120591120591minus120573120573ℏ)
ℏ 119886119886119953119953119890119890minus119867119867(120591120591minus120573120573ℏ)
ℏ = minusTr 120588120588119867119867119886119886119953119953prime119890119890minus120573120573119867119867119890119890
119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591ℏ 119890119890120573120573119867119867
= minusTr120588120588119867119867119886119886119953119953prime(0)119890119890minus120573120573119867119867119886119886119953119953(120591120591)119890119890120573120573119867119867 = minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119886119886119953119953prime(0)119890119890minus120573120573119867119867119886119886119953119953(120591120591)119890119890120573120573119867119867
= minus1119885119885119867119867
Tr119886119886119953119953prime(0)119890119890minus120573120573119867119867119886119886119953119953(120591120591) = minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119886119886119953119953(120591120591)119886119886119953119953prime(0)
= minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime(0) = minusTr120588120588119867119867119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime = 119866119866119953119953119953119953prime(120591120591 0)
13
Therefore 119866119866119953119953119953119953prime(120591120591) can be expanded as a Fourier series in the domain [0120573120573ℏ] as follows
119866119866119953119953119953119953prime(120591120591) = 1120573120573ℏsum 119890119890minus119894119894120596120596119899119899120591120591119866119866119953119953119953119953prime(119894119894120596120596119899119899)infinminusinfin (411)
where 120596120596119899119899 = 2120587120587119899119899120573120573ℏ
and the associated Fourier coefficient is given by
119866119866119953119953119953119953prime(119894119894120596120596119899119899) = int d120591120591119890119890119894119894120596120596119899119899120591120591119866119866119953119953119953119953prime(120591120591)120573120573ℏ0 120596120596119899119899 = 2119899119899120587120587
120573120573ℏ (412)
Having proven the translational time invariance and the periodicity of the Greenrsquos functios we can
now calculate it for the uncoupled system
119866119866119953119953119953119953prime0 (120591120591 120591120591prime) = minuslang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime)rang0 == minus lang119886119886119953119953(0)119890119890minus120596120596119953119953120591120591119886119886119953119953prime
dagger (0)119890119890120596120596119953119953prime120591120591primerang 120591120591 minus 120591120591prime gt 0
minus lang119886119886119953119953primedagger (0)119890119890120596120596119953119953prime120591120591
prime119886119886119953119953(0)119890119890minus120596120596119953119953120591120591rang 120591120591 minus 120591120591prime lt 0
= minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591
primelang119886119886119953119953(0)119886119886119953119953prime
dagger (0)rang 120591120591 minus 120591120591prime gt 0
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang 120591120591 minus 120591120591prime lt 0
= minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591
primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0) + 119886119886119953119953(0)119886119886119953119953primedagger (0)rang 120591120591 minus 120591120591prime gt 0
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang 120591120591 minus 120591120591prime lt 0
= minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591
primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang + 120575120575119953119953119953119953prime 120591120591 minus 120591120591prime gt 0
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang 120591120591 minus 120591120591prime lt 0
ie
119866119866119953119953119953119953prime0 (120591120591 120591120591prime) =
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang + 120575120575119953119953119953119953prime 120591120591 minus 120591120591prime gt 0
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang 120591120591 minus 120591120591prime lt 0 (413)
where we have used Eq (46) for 119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime) and Commutation relation 119886119886119953119953(0)119886119886119953119953prime
dagger (0) = 120575120575119953119953119953119953prime
To calculate lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang we first prove following equation
119890119890119860119860119861119861119890119890minus119860119860 = sum 1119899119899119860(119899119899)119861119861infin
119899119899=0 (414)
where 119860(119896119896)119861119861 equiv 119860 119860(119896119896minus1)119861119861 To prove this identity we define the operator 119891119891(119905119905) = 119890119890119905119905A119861119861119890119890minus119905119905119860119860
and taylor expand it around 119905119905 = 0
119891119891(119905119905) = 119891119891(0) + 119905119905119891119891prime(0) + 1199051199052
2119891119891primeprime(0) + ⋯ = sum 119905119905119899119899
119899119899119889119889119899119899119891119889119889119905119905119899119899
119905119905=0
infin119899119899=0 (415)
The corresponding derivatives are given by
⎩⎪⎨
⎪⎧ 119891119891prime(119905119905) = 119890119890119905119905119860119860119860119861119861119890119890minus119905119905119860119860 minus 119890119890119905119905119860119860119861119861119860119890119890minus119905119905119860119860 = 119890119890119905119905119860119860119860119861119861119890119890minus119905119905119860119860
119891119891primeprime(119905119905) = 119890119890119905119905119860119860119860119860119861119861 minus 119860119861119861119860119890119890minus119905119905119860119860 = 119890119890119905119905119860119860119860(2)119861119861119890119890minus119905119905119860119860
⋮119891119891(119899119899)(119905119905) = 119890119890119905119905119860119860119860(119899119899)119861119861119890119890minus119905119905119860119860
(416)
14
Substituting Eq (416) into Eq (415) we have
119890119890119905119905A119861119861119890119890minus119905119905119860119860 = sum 119905119905119899119899
119899119899119860(119899119899)119861119861infin
119899119899=0 (417)
Itrsquos clear that Eq (413) is a spectial case of Eq (416) with 119905119905 = 1 With this we can proceed as
follows
lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang =
11198851198851198671198670
Tr119890119890minus1205731205731198671198670119886119886119953119953primedagger (0)119886119886119953119953(0) =
11198851198851198671198670
Tr119890119890minus1205731205731198671198670119886119886119953119953primedagger (0)1198901198901205731205731198671198670119890119890minus1205731205731198671198670119886119886119953119953(0)
=11198851198851198671198670
Tr (minus120573120573)119899119899
1198991198991198671198670
(119899119899)119886119886119953119953primedagger (0)
infin
119899119899=0
119890119890minus1205731205731198671198670119886119886119953119953(0)
Note that 1198671198670119886119886119953119953primedagger (0) = ℏ120596120596119953119953prime119886119886119953119953prime
dagger (0) we have
1198671198670(119899119899)119886119886119953119953prime
dagger (0) = ℏ120596120596119953119953prime119899119899119886119886119953119953primedagger (0) (418)
such that
lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang =
11198851198851198671198670
Tr (minus120573120573)119899119899
1198991198991198671198670
(119899119899)119886119886119953119953primedagger (0)
infin
119899119899=0
119890119890minus1205731205731198671198670119886119886119953119953(0)
=11198851198851198671198670
Tr minus120573120573ℏ120596120596119953119953prime
119899119899
119899119899119886119886119953119953primedagger (0)
infin
119899119899=0
119890119890minus1205731205731198671198670119886119886119953119953(0)
= 119890119890minus120573120573ℏ120596120596119953119953prime11198851198851198671198670
Tr119890119890minus1205731205731198671198670119886119886119953119953(0)119886119886119953119953primedagger (0) = 119890119890minus120573120573ℏ120596120596119953119953primelang119886119886119953119953(0)119886119886119953119953prime
dagger (0)rang
= 119890119890minus120573120573ℏ120596120596119953119953primelang119886119886119953119953primedagger (0)119886119886119953119953(0)rang + 120575120575119953119953119953119953prime
therefore we obtain lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang 119890119890120573120573ℏ120596120596119953119953prime minus 1 = 120575120575119953119953119953119953prime For 119953119953prime ne 119953119953 and general 119890119890120573120573ℏ120596120596119953119953prime we have
lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang = 0 For 119953119953prime = 119953119953 we obtain lang119886119886119953119953
dagger(0)119886119886119953119953(0)rang = 119890119890120573120573ℏ120596120596119953119953 minus 1minus1
Hence we have
lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang = 120575120575119953119953prime119953119953
1119890119890120573120573ℏ120596120596119953119953minus1
equiv 120575120575119953119953prime119953119953119899119899119861119861120596120596119953119953 (419)
where 119899119899119861119861120596120596119953119953 is the Bose-Einstein distribution for phonons Substituting Eq (418) in Eq (413)
we obtain
119866119866119953119953119953119953prime0 (120591120591 120591120591prime) =
minus120575120575119953119953119953119953prime1 + 119899119899119861119861120596120596119953119953119890119890minus120596120596119953119953120591120591minus120591120591prime 120591120591 minus 120591120591prime gt 0
minus120575120575119953119953119953119953prime119899119899119861119861120596120596119953119953119890119890minus120596120596119953119953120591120591minus120591120591prime 120591120591 minus 120591120591prime lt 0 (420)
Note that only those Greenrsquos functions of the form 119866119866119953119953119953119953prime0 (120591120591 120591120591prime) = 1198661198661199531199530(120591120591 minus 120591120591prime)120575120575119953119953119953119953prime are non-zero due
to the orthogonality of normal modes Looking at the non-vanishing terms and setting 120591120591prime = 0
without loss of generality we can calculate the Fourier transform of 1198661198661199531199530(120591120591) from Eq (412)
1198661198661199531199530(119894119894120596120596119899119899) = int d1205911205911198901198901198941198941205961205961198991198991205911205911198661198661199531199530(120591120591)120573120573ℏ0 = minusint d120591120591119890119890119894119894120596120596119899119899120591120591120573120573ℏ
0 1 + 119899119899119861119861120596120596119953119953119890119890minus120596120596119953119953120591120591 = minus1+119899119899119861119861120596120596119953119953119894119894120596120596119899119899minus120596120596119953119953
119890119890119894119894120596120596119899119899minus120596120596119953119953120573120573ℏ minus
15
1 = minus1+ 1
119890119890120573120573ℏ120596120596119953119953minus1
119894119894120596120596119899119899minus120596120596119953119953119890119890119894119894
2120587120587119899119899120573120573ℏ 120573120573ℏ119890119890minus120573120573ℏ120596120596119953119953 minus 1 = minus 119890119890120573120573ℏ120596120596119953119953
119894119894120596120596119899119899minus120596120596119953119953
119890119890minus120573120573ℏ120596120596119953119953minus1
119890119890120573120573ℏ120596120596119953119953minus1= minus 1
119894119894120596120596119899119899minus120596120596119953119953
1minus119890119890120573120573ℏ120596120596119953119953
119890119890120573120573ℏ120596120596119953119953minus1= 1
119894119894120596120596119899119899minus120596120596119953119953
ie
1198661198661199531199530(119894119894120596120596119899119899) == 1119894119894120596120596119899119899minus120596120596119953119953
(421)
5 Fermirsquos golden rule 51 Interaction picture
Next we can proceed with calculating the Greenrsquos function of the coupled system To this end we
define the coupling Hamiltonian operator term in the interaction picture as
119867119867119862119862119868119868 (120591120591) = 1198901198901198671198670120591120591ℏ 119867119867119862119862119890119890
minus1198671198670120591120591ℏ (51)
The time evolution of 119867119867119862119862(120591120591) is then given by
ℏ part119867119867119862119862119868119868 (120591120591)120597120597120591120591
= 1198671198670119867119867119862119862119868119868 (120591120591) (52)
Note that the Greenrsquos function defined above was given in the Heisenberg picture To proceed we
need to transform it to the interaction picture
119866119866119953119953119953119953prime(120591120591 0) = minus lang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953primedagger (0)rang = minusTr 120588120588119867119867119879119879120591120591 119890119890
119867119867120591120591ℏ 119886119886119953119953(0)119890119890minus
119867119867120591120591ℏ 119886119886119953119953prime
dagger (0)
= minusTr 120588120588119867119867119879119879120591120591 119890119890119867119867120591120591ℏ 119890119890minus
1198671198670120591120591ℏ 119890119890
1198671198670120591120591ℏ 119886119886119953119953(0)119890119890minus
1198671198670120591120591ℏ 119890119890
1198671198670120591120591ℏ 119890119890minus
119867119867120591120591ℏ 119886119886119953119953prime
dagger (0)
= minusTr 120588120588119867119867119879119879120591120591 119880119880(0 120591120591)119886119886119953119953119868119868 (120591120591)119880119880(120591120591 0)119886119886119953119953primedagger (0)
where
119886119886119953119953119868119868 (120591120591) equiv 1198901198901198671198670120591120591ℏ 119886119886119953119953(0)119890119890minus
1198671198670120591120591ℏ (53)
is the operator in interaction picture and the operator 119880119880 is defined by
119880119880(1205911205911 1205911205912) equiv 11989011989011986711986701205911205911ℏ 119890119890minus
119867119867 (1205911205911minus1205911205912)ℏ 119890119890minus
11986711986701205911205912ℏ (54)
Note that while 119880119880 is not unitary it satisfies the following group property
119880119880(1205911205911 1205911205912)119880119880(1205911205912 1205911205913) = 119880119880(1205911205911 1205911205913) (55)
and the boundary condition 119880119880(1205911205911 1205911205911) = 1 In addition the 120591120591 derivative of 119880119880 is simply
ℏpart119880119880(120591120591 120591120591prime)
120597120597120591120591= 1198671198670119890119890
1198671198670120591120591ℏ 119890119890minus
119867119867120591120591minus120591120591primeℏ minus 119890119890
1198671198670120591120591ℏ 119867119867119890119890minus
119867119867120591120591minus120591120591primeℏ 119890119890minus
1198671198670120591120591primeℏ
= 1198901198901198671198670120591120591ℏ 1198671198670 minus 119867119867119890119890minus
1198671198670120591120591ℏ 119890119890
1198671198670120591120591ℏ 119890119890minus
119867119867120591120591minus120591120591primeℏ 119890119890minus
1198671198670120591120591primeℏ = 119890119890
1198671198670120591120591ℏ minus119867119867119862119862119890119890
minus1198671198670120591120591ℏ 119880119880(120591120591 120591120591prime)
= minus119867119867119862119862119868119868 (120591120591)119880119880(120591120591 120591120591prime)
16
ie
ℏ part119880119880120591120591120591120591prime120597120597120591120591
= minus119867119867119862119862119868119868 (120591120591)119880119880(120591120591 120591120591prime) (56)
The solution of Eq (56) is (see page 235 of Ref [2])
119880119880(120591120591 120591120591prime) = summinus1ℏ
119899119899
119899119899 int 1198891198891205911205911120591120591120591120591prime ⋯int 119889119889120591120591119899119899
120591120591120591120591prime 119879119879120591120591119867119867119862119862119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119899119899)infin
119899119899=0 (57)
The exact thermal Greens function now may be rewritten in the interaction picture as
119866119866119953119953119953119953prime(120591120591 0) = minus1119885119885119867119867
Tr 119890119890minus120573120573119867119867119879119879120591120591119880119880(0 120591120591)119886119886119953119953119868119868 (120591120591)119880119880(120591120591 0)119886119886119953119953primedagger (0)
= minusTr 119890119890minus120573120573119867119867119879119879120591120591 119880119880(0 120591120591)119886119886119953119953119868119868 (120591120591)119880119880(120591120591 0)119886119886119953119953prime
dagger (0)
Tr119890119890minus120573120573119867119867
= minusTr 119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)119879119879120591120591 119886119886119953119953119868119868 (120591120591)119880119880(0 120591120591)119880119880(120591120591 0)119886119886119953119953prime
dagger (0)
Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)
= minusTr 119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)119879119879120591120591 119886119886119953119953119868119868 (120591120591)119880119880(00)119886119886119953119953prime
dagger (0)
Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)=
= minusTr 119890119890minus1205731205731198671198670119879119879120591120591 119880119880(120573120573ℏ 0)119886119886119953119953119868119868 (120591120591)119886119886119953119953prime
dagger (0)
Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)
Where we used the fact that we are free to change the order of the operators within the time ordering
operation (see pages 241-242 of Ref [2])
52 Wickrsquos theorem
Thus the Greenrsquos function can be expanded as [2]
119866119866119953119953119953119953prime(120591120591 0) = minussum 1
119898119898minus1ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)119886119886119953119953119868119868 (120591120591)119886119886
119953119953primedagger (0)rang0infin
119898119898=0
sum 1119898119898minus
1ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)rang0infin
119898119898=0 (58)
where lang⋯ rang0 represents the ensemble average with respect to the non-interacting basis
Tr119890119890minus1205731205731198671198670(⋯ ) Or explicitly
119866119866119953119953119953119953prime(120591120591 0) = minus119866119866119953119953119953119953prime0 (120591120591)minus1ℏint 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953primedagger (0)rang0+
12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0+⋯
1minus1ℏint 1198891198891205911205911120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)rang0+12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0+⋯
(59)
For consiceness we introduce the following notation
119863119863119898119898 equiv 1119898119898minus 1
ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)rang0 (510)
To simplify the calculation in Eq (58) we adopt Wickrsquos theorem (see pages 237-241 of Ref
[2])which can be expressed as follows
lang119879119879120591120591119860119861119861119862119863119863 ⋯119884119884119885rang0 = lang119879119879120591120591119860119861119861rang0lang119879119879120591120591119862119863119863rang0⋯ lang119879119879120591120591119884119884119885rang0 + lang119879119879120591120591119860119862rang0lang119879119879120591120591119861119861119863119863rang0⋯ lang119879119879120591120591119883119883119885rang0 + ⋯ (511)
17
Here 119860119861119861 hellip 119884119884 119885 represent 119886119886119953119953(120591120591) or 119886119886119953119953dagger(120591120591) In Eq (511) each term corresponds to a particular
pairing of the operators 119860119861119861119862119863119863 ⋯119884119884119885 and all possible pairings are taken into account Here the only
non-vanishing propagators will have the form [see Eq (420)]
lang119879119879120591120591 119886119886119953119953primedagger (120591120591prime)119886119886119953119953(120591120591)rang0 = minuslang119879119879120591120591 119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime)rang0 = 119866119866119953119953119953119953prime0 (120591120591 120591120591prime) = 1198661198661199531199530(120591120591 minus 120591120591prime)120575120575119953119953119953119953prime (512)
Therefore the second term in the numerator of Eq (59) can be calculated as
1198681198682 = minus1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0
= minus1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 lang119879119879120591120591119886119886119953119953prime
dagger (0)119886119886119953119953(120591120591)rang0 minus1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
= 119866119866119953119953119953119953prime0 (120591120591 0) minus
1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 minus
1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
The first term in above equation is called disconnected part since the pairing is performed separately
on 119867119867119862119862(1205911205911) and 119886119886119953119953(120591120591)119886119886119953119953primedagger (0) All other terms have pairs that mix creation and annihilation operators
of the Hamiltonian with 119886119886119953119953(120591120591) or 119886119886119953119953primedagger (0) and are said to have connected party For simplicity we
include all these terms in the notation lang119879119879120591120591119867119867119862119862(1205911205911)119886119886119953119953(120591120591)119886119886119953119953primedagger (0)rang0119888119888 Furthermore we define 119866119866119953119953119953119953prime
(1) (120591120591) equiv
minus1ℏ int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862(1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 Then we have
1198681198683 = minus1ℏint 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 = 119866119866119953119953119953119953prime0 (120591120591 0)1198631198631 + 119866119866119953119953119953119953prime
(1) (120591120591) (513)
Using this method the third term in the numerator of Eq (59) can be calculated as
1198681198683 = 12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 = 12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0 lang119879119879120591120591119886119886119953119953prime
dagger (0)119886119886119953119953(120591120591)rang0 +
12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 + 12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 +
12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 = 119866119866119953119953119953119953prime0 (120591120591 0) 1
2ℏ2 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0 +
12ℏ2 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 + 12ℏ2 int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 +
12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
Here the last term is denoted as 119866119866119953119953119953119953prime(2) (120591120591)
Using Eq (513) the second and third terms on the right hand above equation can be simplified as
follows
18
12ℏ2
1198891198891205911205912120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 +1
2ℏ2 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
=12 minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0
minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
+12 minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0
minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
=12 minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0
minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
+12 minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0
minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
= minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0
minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 = 1198631198631119866119866119953119953119953119953prime(1) (120591120591)
Where we performed the following integration variables interchange 1205911205911 ⟷ 1205911205912 Thus we have
1198681198683 = 119866119866119953119953119953119953prime(2) (120591120591) + 119866119866119953119953119953119953prime
(1) (120591120591)1198631198631 + 119866119866119953119953119953119953prime0 (120591120591 0)1198631198632 (514)
Similarity we can calculate the fourth term of Eq (59) as
1198681198684 = minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 =
minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0 lang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 + 3 times
minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 + 3 times
minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 +
minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 =
119866119866119953119953119953119953prime0 (120591120591 0) minus1
6ℏ3 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0+
12ℏ2 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0
minus1ℏ int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888+
12ℏ2 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 minus1ℏ int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0+ 119866119866119953119953119953119953prime
(3) (120591120591) = 119866119866119953119953119953119953prime(3) (120591120591) +
119866119866119953119953119953119953prime(2) (120591120591)1198631198631 + 119866119866119953119953119953119953prime
(1) (120591120591)1198631198632 + 119866119866119953119953119953119953prime0 (120591120591 0)1198631198633
Higher order terms can be treated in the same manner (see pages 95-96 of Ref [2]) Then when
substituting 1198681198682 1198681198683 1198681198684 and all higher order terms into Eq (59) we can simplify the numerator as
follows
119866119866119953119953119953119953prime0 (120591120591) minus
1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0
+1
2ℏ2 1198891198891205911205911120573120573ℏ
0 1198891198891205911205912120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 + ⋯
= 119866119866119953119953119953119953prime0 (120591120591) + 119866119866119953119953119953119953prime
(1) (120591120591) + 119866119866119953119953119953119953prime(2) (120591120591) + ⋯ (1 + 1198631198631 + 1198631198632 + ⋯ )
Noting that the expression (1 + 1198631198631 + 1198631198632 + ⋯ ) is exactly canceled with the denominator the
Greenrsquos function can be simplified as
19
119866119866119953119953119953119953prime(120591120591) = 119866119866119953119953119953119953prime0 (120591120591) + 119866119866119953119953119953119953prime
(1) (120591120591) + 119866119866119953119953119953119953prime(2) (120591120591) + ⋯ (515)
where
119866119866119953119953119953119953prime
(1) (120591120591) = minus1ℏ int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888
119866119866119953119953119953119953prime(2) (120591120591) = 1
2ℏ2 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888 (516)
53 Calculation of Greenrsquos function
531 First order appriximation In Eq (516) we go back to the full notation 119886119886119954119954120582120582 to distinguish phonons of different branches
then 119866119866119953119953119953119953prime(1) (120591120591) is calculated as
119866119866119953119953119953119953prime(1) (120591120591) = minus
1ℏ
12 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591 119881119881119895119895119895119895prime(119948119948)119876119876119948119948119895119895(1205911205911)119876119876119948119948119895119895prime
dagger (1205911205911)3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ 119881119881119895119895119895119895primelowast (119948119948)119876119876119948119948119895119895prime(1205911205911)119876119876119948119948119895119895
dagger (1205911205911)3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= minus12ℏ
1198891198891205911205911120573120573ℏ
0lang119879119879120591120591
ℏ119881119881119895119895119895119895prime(119948119948)2120596120596119948119948119895119895120596120596119948119948119895119895prime
119886119886119948119948119895119895(1205911205911) + 119886119886minus119948119948119895119895dagger (1205911205911) 119886119886minus119948119948119895119895prime(1205911205911) + 119886119886119948119948119895119895prime
dagger (1205911205911)3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ ℏ119881119881119895119895119895119895prime
lowast (119948119948)
2120596120596119948119948119895119895120596120596119948119948119895119895prime119886119886119948119948119895119895prime(1205911205911) + 119886119886minus119948119948119895119895prime
dagger (1205911205911) 119886119886minus119948119948119895119895(1205911205911) + 119886119886119948119948119895119895dagger (1205911205911)
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
According to Wickrsquos theorem [2] the terms that contain 119886119886119948119948119895119895119886119886minus119948119948119895119895prime and 119886119886minus119948119948119895119895dagger 119886119886119948119948119895119895prime
dagger are equal to zero
thus the first term of above equation are calculated as
11989211989211 = minus14
119881119881119895119895119895119895prime(119948119948)
120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119886119886minus119948119948119895119895
dagger (1205911205911)119886119886minus119948119948119895119895prime(1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ lang119879119879120591120591119886119886119948119948119895119895(1205911205911)119886119886119948119948119895119895primedagger (1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888
= minus14
119881119881119895119895119895119895prime(119948119948)
120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119886119886minus119948119948119895119895
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus119948119948119895119895prime(1205911205911)rang0
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ lang119879119879120591120591119886119886119948119948119895119895primedagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)119886119886119948119948119895119895(1205911205911)rang0
= minus14
119881119881119895119895119895119895prime(119948119948)
120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
01198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954minus119948119948120575120575120582120582119895119895119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119954119954primeminus119948119948120575120575120582120582prime119895119895prime3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ 1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119948119948120575120575120582120582119895119895prime119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119954119954prime119948119948120575120575120582120582prime119895119895
20
Simplification of above equation gives
11989211989211 =
⎩⎪⎨
⎪⎧minus119881119881120582120582120582120582prime(minus119954119954)
4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954120582120582
0 (120591120591 1205911205911)119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903
2lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582(119954119954)
4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954120582120582
0 (120591120591 1205911205911)119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582prime le 3119903119903
2lt 120582120582 le 3119903119903
0 else
(517)
Similarity the second term of 119866119866119953119953119953119953prime(1) (120591120591) is calculated as
11989211989212 = minus14
minus119881119881119895119895119895119895primelowast (119948119948)
4120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119886119886119948119948119895119895prime(1205911205911)119886119886119948119948119895119895
dagger (1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ lang119879119879120591120591119886119886minus119948119948119895119895primedagger (1205911205911)119886119886minus119948119948119895119895(1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888
= minus14
minus119881119881119895119895119895119895primelowast (119948119948)
4120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)119886119886119948119948119895119895prime(1205911205911)rang0 lang119879119879120591120591119886119886119948119948119895119895dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus119948119948119895119895(1205911205911)rang0 lang119879119879120591120591119886119886minus119948119948119895119895prime
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
ie
11989211989212 =
⎩⎪⎨
⎪⎧
minus119881119881120582120582120582120582primelowast (119954119954)
4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954prime120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903
2lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582lowast (minus119954119954)
4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954prime120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119954119954prime 1 le 120582120582prime le 3119903119903
2lt 120582120582 le 3119903119903
0 else
(518)
Note that since 119881119881120582120582120582120582primelowast (119954119954) = 119881119881120582120582120582120582prime(minus119954119954) [Eq (314)] the first order approximation of Greenrsquos function
can be written as
119866119866119954119954120582120582119954119954prime120582120582prime(1) (120591120591) =
⎩⎪⎨
⎪⎧minus119881119881
120582120582120582120582prime(minus119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ
0 1198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582 le3119903119903
2lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582
(119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ
0 1198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582prime le3119903119903
2lt 120582120582 le 3119903119903
0 else
(519)
To derive the Greenrsquos function in frequency domain we use the expression for Fourier series then
we have
1198891198891205911205911120573120573ℏ
01198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0) = 1198891198891205911205911120573120573ℏ
0
1120573120573ℏ
119890119890minus119894119894120596120596119899119899(120591120591minus1205911205911)1198661198661199541199541205821205820 (119894119894120596120596119899119899)
119899119899
1120573120573ℏ
119890119890minus119894119894120596120596119899119899prime1205911205911119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899prime)
119899119899prime
=1
(120573120573ℏ)2119890119890minus119894119894120596120596119899119899120591120591 1198891198891205911205911120573120573ℏ
0119890119890119894119894120596120596119899119899minus120596120596119899119899prime1205911205911 119866119866119954119954120582120582
0 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899prime)
119899119899119899119899prime=
=1
(120573120573ℏ)2119890119890minus119894119894120596120596119899119899120591120591[120573120573ℏ120575120575119899119899119899119899prime]1198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899prime)119899119899119899119899prime
=1120573120573ℏ
119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899)119899119899
According to the definition of Fourier series [see Eq (412)] we get the Fourier coefficient of
21
119866119866119954119954120582120582119954119954prime120582120582prime(1) (120591120591)
119866119866119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) =
⎩⎪⎨
⎪⎧minus119881119881120582120582120582120582prime(minus119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582prime1198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime 1 le 120582120582 le 31199031199032
lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582(119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582prime1198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime 1 le 120582120582prime le 31199031199032
lt 120582120582 le 3119903119903
0 else
(520)
where 119866119866119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) = 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) times Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) times 119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899) and Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) is defined as
Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) =
⎩⎪⎨
⎪⎧minus119881119881120582120582120582120582prime(minus119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582prime120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903
2lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582(119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582prime120575120575119954119954119954119954prime 1 le 120582120582prime le 3119903119903
2lt 120582120582 le 3119903119903
0 else
(521)
532 second order approximation The second order approximation of the Greenrsquos function in Eq (516) 119866119866119954119954120582120582119954119954prime120582120582prime
(2) (120591120591) is calculated as
119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) =
18ℏ2 1198891198891205911205911
120573120573ℏ
0 1198891198891205911205912120573120573ℏ
0lang119879119879120591120591
⎣⎢⎢⎡ 11988111988111989511989511198951198951prime(119948119948120783120783)1198761198761199481199481207831207831198951198951(1205911205911)1198761198761199481199481207831207831198951198951prime
dagger (1205911205911)3119903119903
1198951198951prime=31199031199032 +1
31199031199032
1198951198951=1119948119948120783120783
+ 11988111988111989411989411198941198941primelowast (119948119948120783120783)1198761198761199481199481207831207831198941198941prime(1205911205911)1198761198761199481199481207831207831198941198941
dagger (1205911205911)3119903119903
1198941198941prime=31199031199032 +1
31199031199032
1198941198941=1119948119948120783120783 ⎦⎥⎥⎤
⎣⎢⎢⎡ 11988111988111989511989521198951198952prime(119948119948120784120784)1198761198761199481199481207841207841198951198952(1205911205912)1198761198761199481199481207841207841198951198952prime
dagger (1205911205912)3119903119903
1198951198952prime=31199031199032 +1
31199031199032
1198951198952=1119948119948120784120784
+ 11988111988111989411989421198941198942primelowast (119948119948120784120784)1198761198761199481199481207841207841198941198942prime(1205911205912)1198761198761199481199481207841207841198941198942
dagger (1205911205912)3119903119903
1198941198942prime=31199031199032 +1
31199031199032
1198941198942=1119948119948120784120784 ⎦⎥⎥⎤119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888 = 1198921198921 + 1198921198922 + 1198921198923 + 1198921198924
where
1198921198921 = 18ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0
sum sum 11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989511989521198951198952prime(119948119948120784120784)1le1198951198952le
31199031199032 lt1198951198952
primele311990311990311994811994812078412078411989511989521198951198952prime
1le1198951198951le31199031199032 lt1198951198951
primele311990311990311994811994812078312078311989511989511198951198951prime
times lang1198791198791205911205911198761198761199481199481207831207831198951198951(1205911205911)1198761198761199481199481207831207831198951198951primedagger (1205911205911)1198761198761199481199481207841207841198951198952(1205911205912)1198761198761199481199481207841207841198951198952prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(522)
1198921198922 = 18ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0
sum sum 11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942primelowast (119948119948120784120784)
1le1198941198942le31199031199032 lt1198941198942
primele311990311990311994811994812078412078411989411989421198941198942prime
1le1198951198951le31199031199032 lt1198951198951
primele311990311990311994811994812078312078311989511989511198951198951prime
times lang1198791198791205911205911198761198761199481199481207831207831198951198951(1205911205911)1198761198761199481199481207831207831198951198951primedagger (1205911205911)1198761198761199481199481207841207841198941198942prime (1205911205912)1198761198761199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(523)
1198921198923 = 18ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0
sum sum 11988111988111989411989411198941198941primelowast (119948119948120783120783)11988111988111989511989521198951198952prime(119948119948120784120784)
1le1198951198952le31199031199032 lt1198951198952
primele311990311990311994811994812078412078411989511989521198951198952prime
1le1198941198941le31199031199032 lt1198941198941
primele311990311990311994811994812078312078311989411989411198941198941prime
times lang1198791198791205911205911198761198761199481199481207831207831198941198941prime (1205911205911)1198761198761199481199481207831207831198941198941dagger (1205911205911)1198761198761199481199481207841207841198951198952(1205911205912)1198761198761199481199481207841207841198951198952prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(524)
22
1198921198924 = 18ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0
sum sum 11988111988111989411989411198941198941primelowast (119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)1le1198941198942le
31199031199032 lt1198941198942
primele311990311990311994811994812078412078411989411989421198941198942prime
1le1198941198941le31199031199032 lt1198941198941
primele311990311990311994811994812078312078311989411989411198941198941prime
times lang1198791198791205911205911198761198761199481199481207831207831198941198941prime (1205911205911)1198761198761199481199481207831207831198941198941dagger (1205911205911)1198761198761199481199481207841207841198941198942prime (1205911205912)1198761198761199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(525)
In what follows we will expand the term 1198921198922 The expansion of all other terms follows the same lines
and eventually results in the same expression Expanding the product 11987611987611994811994812078312078311989511989511198761198761199481199481207831207831198951198951primedagger 1198761198761199481199481207841207841198941198942prime1198761198761199481199481207841207841198941198942
dagger we can
rewrite Eq (523) as follows
1198921198922 = 132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)119881119881
11989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032 lt1198941198942
primele311990311990311994811994812078412078411989411989421198941198942prime
1le1198951198951le31199031199032 lt1198951198951
primele311990311990311994811994812078312078311989511989511198951198951prime
times 119868119868 (526)
Where
119868119868 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911) + 119886119886minus1199481199481207831207831198951198951dagger (1205911205911) 119886119886minus1199481199481207831207831198951198951prime(1205911205911) + 1198861198861199481199481207831207831198951198951prime
dagger (1205911205911) 1198861198861199481199481207841207841198941198942prime (1205911205912) + 119886119886minus1199481199481207841207841198941198942primedagger (1205911205912) 119886119886minus1199481199481207841207841198941198942(1205911205912) + 1198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0c
= lang119879119879120591120591 1198861198861199481199481207831207831198951198951119886119886minus1199481199481207831207831198951198951prime + 11988611988611994811994812078312078311989511989511198861198861199481199481207831207831198951198951primedagger + 119886119886minus1199481199481207831207831198951198951
dagger 119886119886minus1199481199481207831207831198951198951prime + 119886119886minus1199481199481207831207831198951198951dagger 1198861198861199481199481207831207831198951198951prime
dagger 12059112059111198861198861199481199481207841207841198941198942prime119886119886minus1199481199481207841207841198941198942 + 1198861198861199481199481207841207841198941198942prime1198861198861199481199481207841207841198941198942
dagger + 119886119886minus1199481199481207841207841198941198942primedagger 119886119886minus1199481199481207841207841198941198942
+ 119886119886minus1199481199481207841207841198941198942primedagger 1198861198861199481199481207841207841198941198942
dagger 1205911205912119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0c
= lang119879119879120591120591 11988611988611994811994812078312078311989511989511198861198861199481199481207831207831198951198951primedagger + 119886119886minus1199481199481207831207831198951198951
dagger 119886119886minus1199481199481207831207831198951198951prime12059112059111198861198861199481199481207841207841198941198942prime1198861198861199481199481207841207841198941198942
dagger + 119886119886minus1199481199481207841207841198941198942primedagger 119886119886minus11994811994812078412078411989411989421205911205912
119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
Where the symbol [hellip ]1205911205911 signifies that the operators in the brackets are given in the interaction
picture The last equality in above equation comes from the fact that the contractions of the product
of the operators are equal to zero when the number of creation and annihilation operators are not the
same (see Wickrsquos theorem in Ref [2]) Using Wickrsquos theorem [2] we are able to calculate the above
equation term by term as follows
⎩⎪⎨
⎪⎧ 1198681198681 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951prime
dagger (1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)1198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888
1198681198682 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
1198681198683 = lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)1198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
1198681198684 = lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(527)
We shall now calculate them term by term
23
1198681198681 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951primedagger (1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)1198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942
dagger (1205911205912)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
+ lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951prime
dagger (1205911205911)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0
+ lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)1198861198861199481199481207841207841198941198942prime(1205911205912)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0119888119888
= 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199481199481207841207841198951198952
0 (1205911205912 1205911205912)12057512057511989411989421198941198942prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582
+ 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 1198661198661199481199481207831207831198951198951
0 (1205911205911 1205911205911)12057512057511989511989511198951198951prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582
+ 1198661198661199481199481207831207831198951198951prime0 (1205911205912 1205911205911)1205751205751199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951prime ∙ 119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582
= 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582
+ 1198661198661199481199481207831207831198951198951prime0 (1205911205912 1205911205911)1205751205751199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951prime ∙ 119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582
The last equality results from the fact that 1198951198951 and 1198951198951prime are indices of different ranges (belonging to
different subsystems) and the same holds for 1198941198942 and 1198941198942prime Similarity
1198681198684 = lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0
= 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 119866119866minus11994811994812078412078411989411989420 (1205911205912 1205911205912)12057512057511989411989421198941198942prime
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 119866119866minus1199481199481207831207831198951198951prime 0 (1205911205911 1205911205912)1205751205751199481199481207831207831199481199481207841207841205751205751198951198951prime1198941198942prime
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205911 1205911205911)12057512057511989511989511198951198951prime
+ 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205912 1205911205911)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942
= 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 119866119866minus1199481199481207831207831198951198951prime 0 (1205911205911 1205911205912)1205751205751199481199481207831207831199481199481207841207841205751205751198951198951prime1198941198942prime
+ 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205912 1205911205911)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942
Where again the last equality results from the fact that 1198951198951 and 1198951198951prime are indices of different ranges
(belonging to different subsystems) and the same holds for 1198941198942 and 1198941198942prime Similarly we obtain 1198681198682 =
1198681198683 = 0 for the same reason as shown below
24
1198681198682 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951prime
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0
+ lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)1198861198861199481199481207831207831198951198951(1205911205911)rang0
+ lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0
= 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582 ∙ 119866119866minus11994811994812078412078411989411989420 (1205911205912 1205911205912)12057512057511989411989421198941198942prime
+ 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)120575120575minus11994811994812078412078411994811994812078312078312057512057511989511989511198941198942prime ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205911)12057512057511989511989511198951198951prime
+ 1198661198661199481199481207831207831198951198951prime0 (1205911205912 1205911205911)120575120575minus11994811994812078412078411994811994812078312078312057512057511989411989421198951198951prime ∙ 119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 = 0
1198681198683 = lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)1198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0
+ lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0
+ lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0
= 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 11986611986611994811994812078412078411989411989420 (1205911205912 1205911205912)12057512057511989411989421198941198942prime
+ 11986611986611994811994812078412078411989411989420 (1205911205911 1205911205912)120575120575minus1199481199481207831207831199481199481207841207841205751205751198951198951prime1198941198942 ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205911 1205911205911)12057512057511989511989511198951198951prime
+ 1198661198661199481199481207841207841198941198942prime0 (1205911205912 1205911205911)120575120575minus1199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951 ∙ 119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 = 0
The two non-vanishing terms (1198681198681 and 1198681198684) produce the following contributions to 1198921198922 of Eq (526)
11989211989221 = 132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime1198681198681 =
132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime1198661198661199481199481207831207831198951198951
0 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙
1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582 + 1
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime1198661198661199481199481207831207831198951198951prime
0 (1205911205912 1205911205911)1205751205751199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951prime ∙
119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 =
⎩⎪⎨
⎪⎧
120575120575119954119954119954119954prime
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast (119954119954prime)
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (1205911205911 1205911205912) ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
120575120575119954119954119954119954prime
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032+1 1198661198661199541199541198951198951prime
0 (1205911205912 1205911205911) ∙ 119866119866119954119954prime120582120582prime0 (1205911205911 0)119866119866119954119954120582120582
0 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032
0 else
and
25
11989211989224 = 132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime1198681198684 =
132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙
119866119866minus1199481199481207831207831198951198951prime 0 (1205911205911 1205911205912)1205751205751199481199481207831207831199481199481207841207841205751205751198951198951prime1198941198942prime + 1
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙
1198661198661199541199541205821205820 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 119866119866minus1199481199481207831207831198951198951
0 (1205911205912 1205911205911)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 =
⎩⎪⎨
⎪⎧120575120575119954119954119954119954prime
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032+1 1198661198661199541199541198951198951prime
0 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime0 (1205911205912 0)119866119866119954119954120582120582
0 (120591120591 1205911205911) 120582120582 120582120582prime isin 1 31199031199032
120575120575119954119954119954119954prime
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast (119954119954prime)
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
0 else
Here we used the following properties 120596120596minus119954119954120582120582 = 120596120596119954119954120582120582 and 1198811198811205821205821198951198951prime(minus119954119954) = 1198811198811205821205821198951198951primelowast (119954119954) [see Eqs (112) and
(314)]
The equivalence of 11989211989221 and 11989211989224 can be seen by changing the integration variables 1205911205912 harr 1205911205911
Substituting 11989211989221 and 11989211989224 into Eq (526) 1198921198922 is given by
1198921198922 =
⎩⎪⎨
⎪⎧
120575120575119954119954119954119954prime
16 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast (119954119954prime)
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime
0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
120575120575119954119954119954119954prime
16 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032+1 1198661198661199541199541198951198951prime
0 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime0 (1205911205911 0)119866119866119954119954120582120582
0 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032
0 else
(528)
By changing the indices and integration variables itrsquos straightforward to show that the other three
terms (119892119892111989211989231198921198924 ) are identical to 1198921198922 thus the final expression of 119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) [Eq (516)) can be
calculated as
119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) = 41198921198922 =
⎩⎪⎨
⎪⎧120575120575119954119954119954119954prime
4 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032
120575120575119954119954119954119954prime
4 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast 119954119954prime
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime
0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 31199031199032
+ 13119903119903
0 else
(529)
To calculate 119866119866119954119954120582120582119954119954prime120582120582prime(2) in frequency space we expand 119866119866119954119954120582120582119954119954prime120582120582prime
(2) (120591120591) in a Fourier series
119866119866119954119954120582120582119954119954prime120582120582prime
(2) (120591120591) = 1120573120573ℏsum 119890119890minus119894119894120596120596119899119899120591120591119866119866119954119954120582120582119954119954prime120582120582prime
(2) (119894119894120596120596119899119899)119899119899
119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = int d120591120591119890119890119894119894120596120596119899119899120591120591119866119866119954119954120582120582119954119954prime120582120582prime
(2) (120591120591)120573120573ℏ0
(530)
Using Eq (421) we get for 120582120582 120582120582prime isin 1 31199031199032
26
119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) =
120575120575119954119954119954119954prime4
1198891198891205911205911120573120573ℏ
0 1198891198891205911205912120573120573ℏ
0
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1198661198661199541199541198951198951prime0 (1205911205912 1205911205911)119866119866119954119954120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912)
=120575120575119954119954119954119954prime
4
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1198891198891205911205911120573120573ℏ
0 1198891198891205911205912120573120573ℏ
0
1120573120573ℏ
119890119890minus1198941198941205961205961198991198992(1205911205912minus1205911205911)1198661198661199541199541198951198951prime0 1198941198941205961205961198991198992
1198991198992
∙1120573120573ℏ
119890119890minus11989411989412059612059611989911989911205911205911119866119866119954119954120582120582prime0 1198941198941205961205961198991198991
1198991198991
∙1120573120573ℏ
119890119890minus119894119894120596120596119899119899(120591120591minus1205911205912)1198661198661199541199541205821205820 (119894119894120596120596119899119899)
119899119899
=120575120575119954119954119954119954prime
4
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1120573120573ℏ
119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954120582120582prime
0 11989411989412059612059611989911989911198661198661199541199541198951198951prime0 1198941198941205961205961198991198992
11989911989911989911989911198991198992⎩⎪⎨
⎪⎧ 1120573120573ℏ
1198891198891205911205911120573120573ℏ
0119890119890minus1198941198941205961205961198991198991minus12059612059611989911989921205911205911
times1120573120573ℏ
1198891198891205911205912120573120573ℏ
0119890119890minus1198941198941205961205961198991198992minus1205961205961198991198991205911205912
⎭⎪⎬
⎪⎫
=120575120575119954119954119954119954prime
4
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1120573120573ℏ
119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)
11989911989911989911989911198991198992
119866119866119954119954120582120582prime0 11989411989412059612059611989911989911198661198661199541199541198951198951prime
0 1198941198941205961205961198991198992120575120575119899119899111989911989921205751205751198991198991198991198992
=120575120575119954119954119954119954prime120573120573ℏ
119890119890minus119894119894120596120596119899119899120591120591
119899119899
1198661198661199541199541205821205820 (119894119894120596120596119899119899)
14
119881119881120582120582prime1198951198951prime(119954119954
prime)1198811198811205821205821198951198951primelowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899)119866119866119954119954120582120582prime
0 (119894119894120596120596119899119899)
Comparing above expression with Eq (530) the Fourier transform of 119866119866119954119954120582120582119954119954120582120582prime(2) (120591120591) reads as
119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) 120575120575119954119954119954119954prime
4sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 119866119866119954119954120582120582prime
0 (119894119894120596120596119899119899) (531)
where we have introduced the notation
Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) equiv
120575120575119954119954119954119954prime
4sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) (532)
and 119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954120582120582prime(2) (119894119894120596120596119899119899) ∙ 119866119866119954119954120582120582prime
0 (119894119894120596120596119899119899) Similarly for 120582120582 120582120582prime isin 31199031199032
+ 13119903119903 we
have
Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) =
120575120575119954119954119954119954prime
4sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast 119954119954prime
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (119894119894120596120596119899119899) (533)
and all together we have for Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899)
Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) =
⎩⎪⎪⎨
⎪⎪⎧120575120575119954119954119954119954prime
4sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 1 3119903119903
2
120575120575119954119954119954119954prime
4sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast 119954119954prime
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
0 else
(534)
27
533 Dysonrsquos equation Substituting Eqs (521) and (534) into Eq (516) and performing a Fourier transform we have
119866119866119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) = 119866119866119954119954120582120582119954119954prime120582120582prime0 (119894119894120596120596119899119899) + 119866119866119954119954120582120582119954119954prime120582120582prime
(1) (119894119894120596120596119899119899) + 119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) + ⋯
= 1198661198661199541199541205821205820 (119894119894120596120596119899119899)120575120575119954119954119954119954prime120575120575120582120582120582120582prime + 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) ∙ 119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899) + 1198661198661199541199541205821205820 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime
(2) (119894119894120596120596119899119899)
∙ 119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899) + ⋯ = 119866119866119954119954120582120582
0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime120575120575120582120582120582120582prime + 1198661198661199541199541205821205820 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) ∙ 119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899)
or
119866119866119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) = 1198661198661199541199541205821205820 (119894119894120596120596119899119899)120575120575119954119954119954119954prime120575120575120582120582120582120582prime + 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) ∙ 119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899) (535)
Eq (535) is the so-called Dysonrsquos equation (see Ref [2]) where
Σ119954119954120582120582119954119954120582120582prime(119894119894120596120596119899119899) = Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) + Σ119954119954120582120582119954119954prime120582120582prime
(2) (119894119894120596120596119899119899) + ⋯
is called self-energy and Σ119954119954120582120582119954119954120582120582prime(119899119899) (119894119894120596120596119899119899)119899119899 = 12⋯ is the self-energy of nth order approximation Up
to the second order approximation the self-energy is written as
Σ119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) =
⎩⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎧ minus
120575120575119954119954119954119954prime
2
119881119881120582120582120582120582primelowast (119954119954)
120596120596119954119954120582120582120596120596119954119954120582120582prime 1 le 120582120582 le 3119903119903
2lt 120582120582prime le 3119903119903
minus120575120575119954119954119954119954prime
2
119881119881120582120582prime120582120582(119954119954)
120596120596119954119954120582120582120596120596119954119954120582120582prime 1 le 120582120582prime le 3119903119903
2lt 120582120582 le 3119903119903
120575120575119954119954119954119954prime
4sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 1 3119903119903
2
120575120575119954119954119954119954prime
4sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast 119954119954prime
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
(536)
54 Fermirsquos golden Rule
To get the Fermirsquos golden Rule we need to calculate the retarded Greenrsquos function which can be
calculated from 119866119866119954119954120582120582119954119954120582120582prime(119894119894120596120596119899119899) by analytic continuation to the real axis via 119894119894120596120596119899119899 rarr 120596120596 + 119894119894119894119894 with an
infinitesimal positive 119894119894 [12]
119866119866119954119954120582120582119954119954120582120582prime119903119903 (120596120596) equiv 119866119866119954119954120582120582119954119954120582120582prime(119894119894120596120596119899119899 rarr 120596120596 + 119894119894119894119894) 119894119894 rarr 0 (537)
Similarity the retarded self-energy is defined as
Σ119954119954120582120582119954119954120582120582prime119903119903 (120596120596) equiv Σ119954119954120582120582119954119954120582120582prime(119894119894120596120596119899119899 rarr 120596120596 + 119894119894119894119894) 119894119894 rarr 0 (538)
The transition rate of phonons of branch 120582120582 at 119954119954 Γ119954119954120582120582(120596120596) is related to the retarded self energy
Σ119954119954120582120582119903119903 (120596120596) via [13]
Γ119954119954120582120582(120596120596) = minus2ImΣ119954119954120582120582119954119954120582120582119903119903 (120596120596) (539)
Using the expression for 1198661198661199541199541205821205820 (119894119894120596120596119899119899) [Eq (421)] and focused on the second order approximation [13]
we have
28
Σ119954119954120582120582119954119954120582120582119903119903 (119894119894120596120596119899119899) =
⎩⎪⎨
⎪⎧14sum
1198811198811205821205821198951198951prime(119954119954)
2
1205961205961199541199541198951198951prime120596120596119954119954120582120582
31199031199031198951198951prime=
31199031199032 +1
1
120596120596minus1205961205961199541199541198951198951prime +119894119894119894119894
120582120582 isin 1 31199031199032
14sum 1198811198811198951198951120582120582(119954119954)2
1205961205961199541199541198951198951120596120596119954119954120582120582
311990311990321198951198951=1
1120596120596minus1205961205961199541199541198951198951+119894119894119894119894
120582120582 isin 31199031199032
+ 13119903119903
(540)
Using the relation
Im 1
120596120596minus1205961205961199541199541198951198951prime +119894119894119894119894
= minus120587120587120575120575120596120596 minus 1205961205961199541199541198951198951prime (541)
The transition rate reads as
Γ119954119954120582120582(120596120596) =
⎩⎪⎨
⎪⎧1205871205872sum
1198811198811205821205821198951198951prime(119954119954)
2
1205961205961199541199541198951198951prime120596120596119954119954120582120582
31199031199031198951198951prime=
31199031199032 +1
120575120575120596120596 minus 1205961205961199541199541198951198951prime 120582120582 isin 1 31199031199032
1205871205872sum 1198811198811198951198951120582120582(119954119954)2
1205961205961199541199541198951198951120596120596119954119954120582120582
311990311990321198951198951=1
120575120575120596120596 minus 1205961205961199541199541198951198951 120582120582 isin 31199031199032
+ 13119903119903
(542)
Or equivalently
Γ119954119954120582120582(119864119864 = ℏ120596120596) =
⎩⎪⎨
⎪⎧120587120587ℏ3
2sum
1198811198811205821205821198951198951prime(119954119954)
2
1198641198641199541199541198951198951prime119864119864119954119954120582120582
31199031199031198951198951prime=
31199031199032 +1
120575120575119864119864 minus 1198641198641199541199541198951198951prime 120582120582 isin 1 31199031199032
120587120587ℏ3
2sum 1198811198811198951198951120582120582(119954119954)2
1198641198641199541199541198951198951119864119864119954119954120582120582
311990311990321198951198951=1
120575120575119864119864 minus 1198641198641199541199541198951198951 120582120582 isin 31199031199032
+ 13119903119903
(543)
Γ119954119954120582120582(119864119864) represents the probability per unit time of a transition with energy 119864119864 from phonon branch 120582120582
at wave number 119954119954 in subsystem I (II) to a set of phonon branches in subsystem II (I) with the same
wave number Here state 119954119954120582120582 in one subsystem is coupled to state 1199541199541198951198951prime in the other subsystem via
1198811198811205821205821198951198951prime(119954119954) Since the two expressions in Eq (543) are completely equivalent just representing
transitions of opposite directions we consider only the first one in what follows Assuming that
subsystem II sufficiently large such that its density of states is nearly continuous the sum over its
states in the above expression can be replaced by an integral over the energy sum (⋯ )119895119895 rarr
int(⋯ )120588120588119864119864119954119954d119864119864119954119954 giving
Γ119954119954120582120582(119864119864) = 120587120587ℏ3
2 intd119864119864119954119954prime120588120588119864119864119954119954prime 119881119881120582120582119895119895prime(119954119954)
2119864119864119954119954119895119895prime=119864119864119954119954prime
119864119864119954119954prime119864119864119954119954120582120582120575120575119864119864 minus 119864119864119954119954prime = 120587120587ℏ3
2120588120588(119864119864)
119881119881120582120582119895119895prime(119954119954)2119864119864119954119954119895119895prime=119864119864
119864119864119864119864119954119954120582120582 (544)
Eq (544) represents the transition rate of the phonons with energy 119864119864 from branch 120582120582 and wave
number 119954119954 in subsystem I to the continuous manifold of states in subsystem II The total transition
rate between the two subsystems is then given by the sum of transition rates from all states in
subsystem I weighted by their phonon populations to the manifold of states in subsystem II
Assuming that the two subsystems are weakly coupled such that the width of state 119954119954120582120582 in subsystem
I is small and the probability to leave it at an energy other than 119864119864119954119954120582120582 is negligible we can now write
29
total transition rate as
Γtot = 1119885119885119890119890minus120573120573119864119864119954119954120582120582Γ119954119954120582120582119864119864119954119954120582120582
119954119954120582120582
=120587120587ℏ3
2
119890119890minus120573120573119864119864119954119954120582120582
119885119885
120588120588119864119864119954119954120582120582 119881119881120582120582119895119895prime(119954119954)2119864119864119954119954119895119895prime=119864119864119954119954120582120582
1198641198641199541199541205821205822119954119954120582120582
=120587120587ℏ3
2
119890119890minus120573120573119864119864119954119954120582120582
119885119885
120588120588119864119864119954119954120582120582 119881119881120582120582120582120582+31199031199032(119954119954)
2
1198641198641199541199541205821205822119954119954120582120582
Finally we get
Γtot = Γ119954119954120582120582(119864119864) =120587120587ℏ3
2sum 119890119890minus120573120573119864119864119954119954120582120582
119885119885
120588120588119864119864119954119954120582120582119881119881120582120582120582120582+31199031199032(119954119954)
2
1198641198641199541199541205821205822119954119954120582120582 (545)
where 119885119885 = sum 119890119890minus120573120573119864119864119954119954120582120582119954119954120582120582 is the partition function Here we choose the index of phonon branches 120582120582
such that the phonon branch with lower energy has a smaller index ie 1198641198641199541199541205821205821 lt 1198641198641199541199541205821205822 for index 1205821205821 lt
1205821205822 Note that phonon branches 120582120582 and 1198951198951prime belong to the subsystem I (with index from 1 to 31199031199032) and
subsystem II (with index from 1 to 1 + 31199031199032 to 3119903119903) respectively we choose 119895119895prime = 120582120582 + 31199031199032
to make
sure that 119864119864119954119954119895119895prime = 119864119864119954119954120582120582 since phonon energy of two uncoupled system is identical Eq (545) is the
Fermirsquos golden rule for inter-phonon coupling
References
[1] Y Xu J-S Wang W Duan B-L Gu and B Li Nonequilibrium Greens function method for phonon-phonon interactions and ballistic-diffusive thermal transport Phys Rev B 78 224303 (2008) [2] A L Fetter and J D Walecka Quantum theory of many-particle systems (Courier Corporation 2012) [3] G E Stedman Fermis Golden RulemdashAn Exercise in Quantum Field Theory Am J Phys 39 205 (1971)
4
observations can be made (i) the ITR is weakly dependent on the stack thickness indicating that the
thickness dependence arises from the conductivity of the optimally-stacked interfacing slabs
Specifically in the thickness range considered the heat conductivity grows with slab thickness due to
reduction of phonon-phonon interactions and increased contribution of long wave-length phonons
below the mean-free path [28-30] (ii) the ITR strongly depends on the twist angle demonstrating a
~10-fold (4-fold) increase when the twist angle at the graphene (h-BN) interface is varied from 0deg
to 15deg This clearly indicates that the twist angle can be utilized to control the cross-plane thermal
conductivity of hexagonal two-dimensional (2D) materials and to effectively thermally isolate the top
layers from the underlying substrate
Figure 2 Twist-angle dependence of the cross-plane thermal conductivity of the entire stack (a b)
and the interfacial thermal resistance (c d) of the twisted contact formed between the optimally-
stacked slabs of graphite (a c) and bulk h-BN (b d) respectively Red circles and black triangles
correspond to the results obtained using 8 and 16 layer models respectively
The strong dependence of the cross-plane thermal conductivity of graphene and h-BN on the stacking
fault twist angle is related to the degree of coupling between the phonon modes of the two contacting
layers at the twisted interface Note that the term ldquocouplingrdquo used herein is not related to the standard
notion of phonon-phonon couplings due to anharmonic effects Instead we regard to the off-diagonal
terms of the Hessian when represented in the basis of the harmonic phonon modes of the isolated
5
layers To demonstrate this we write the dynamical matrix (the mass-reduced Fourier transform of
the force constant matrix) in block form as follows
120625120625(119954119954) = 12062512062511(119954119954) 12062512062512(119954119954)12062512062521(119954119954) 12062512062522(119954119954) (1)
where 12062512062511(119954119954) and 12062512062522(119954119954) are the block matrices relating to the first and second layer and 12062512062512(119954119954)
and 12062512062521(119954119954) = 12062512062512dagger (119954119954) all evaluated at wave-vector 119954119954 The interlayer phonon-phonon couplings are
obtained by diagonalizing separately 12062512062511(119954119954) and 12062512062522(119954119954) such that 12062512062511(119954119954) =
1199321199321dagger(119954119954)12062512062511(119954119954)1199321199321(119954119954) and 12062512062522(119954119954) = 1199321199322
dagger(119954119954)12062512062522(119954119954)1199321199322(119954119954) are diagonal matrices containing the
frequencies (120596120596119894119894) of the phonon modes of the two layers and 1199321199321(119954119954) and 1199321199322(119954119954) are unitary
matrices of the corresponding eigenvectors We now construct a global block diagonal transformation
matrix of the form
119932119932(119954119954) = 1199321199321(119954119954) 120782120782120782120782 1199321199322(119954119954) (2)
and transform the full dynamical matrix as follows
119932119932dagger(119954119954)120625120625(119954119954)119932119932(119954119954) = 12062512062511(119954119954) 12062512062512(119954119954)12062512062521(119954119954) 12062512062522(119954119954)
(3)
where 12062512062512(119954119954) = 1199321199321dagger(119954119954)12062512062512(119954119954)1199321199322(119954119954) and 12062512062521(119954119954) = 12062512062512
dagger (119954119954) are the interlayer phonon-phonon
coupling blocks Naturally when the two layers are infinitely separated these coupling blocks vanish
and the diagonal blocks converge to those of the isolated layers
The overall coupling between the two layers can be obtained from the individual phonon-phonon
coupling matrix elements via Fermirsquos golden rule [31] which reads as (see Sec 4 of SI for a detailed
derivation)
Γtot = 120587120587ℏ3
2sum 119890119890minus120573120573119864119864119954119954120582120582
119885119885
120588120588119864119864119954119954120582120582119881119881120582120582120582120582+31199031199032(119954119954)
2
1198641198641199541199541205821205822119954119954120582120582 (4)
where 119885119885 = sum 119890119890minus120573120573119864119864119954119954120582120582119954119954120582120582 is the partition function 119864119864119954119954120582120582 is the energy of phonons at branch 120582120582 with
wave number 119954119954 120588120588119864119864119954119954120582120582 is the density-of-states (DOS) at 119864119864119954119954120582120582 and 119881119881120582120582120582120582+31199031199032(119954119954)
2 is the coupling
matrix element between branches of phonons of similar energy in the two layers whose number of
atoms in one unit cell is 119903119903
Using Eq (4) we can rationalize the misfit angle dependence of the heat flux across the twisted
interface from the calculated inter-phonon coupling To that end we performed room temperature
(300K) simulations (technical details can be found in Sec 3 of SI) for tBLG with different misfit
angles using the Greenrsquos function molecular dynamics (GFMD) developed by Kong et al [32] as
6
implemented in LAMMPS [33] The simulations allow us to evaluate the dynamical matrix from
which the phonon-phonon couplings can be extracted (see details in Sec 3 of SI) and the overall heat
transfer rate calculated Figure 3 shows the resulting heat transfer rate (normalized to is value for the
aligned contact (120579120579 = 0) ) as a function of the misfit angle compared to the interfacial thermal
conductivity defined as the inverse of the ITR presented in Figure 2(c) ITC equiv 1ITR The
remarkable agreement between the calculated interfacial thermal conductivity and Fermirsquos golden
rule results indicate that the dependence of the interlayer phonon-phonon couplings on the misfit
angle is responsible for the strong angle dependence of the interfacial conductivity Notably the sharp
heat conductivity drop at misfit angles in the range of 0deg-5deg as well as the small conductivity for larger
misfit angles are well captured by Fermirsquos golden rule
Figure 3 Comparison between Fermirsquos golden rule results (open blue squares) for the interfacial heat-
transfer rate of a tBLG and the calculated interfacial thermal conductivity at various misfit angles
ITC simulation results are presented for both 8 layers (open red circles) and 16 layers (open black
triangles) showing similar behavior For comparison purposes all data sets are normalized to their
value obtained for the aligned contact
To correlate our results with experimentally measured thermal conductivities that are often obtained
for thick samples we repeated our calculations for increasing stack thicknesses at fixed misfit angles
Figure 4 presents results for the calculated heat conductivity of (a) graphite and (b) h-BN stacks either
aligned (open red circles) or twisted by 120579120579 = 3016deg (open black diamond symbols) as a function of
number of layers in the stack As discussed above for both systems the misoriented stack exhibits
lower heat conductivity compared to the aligned system however its thickness dependence is
considerably stronger This can be attributed to the significantly higher interface resistance of the
twisted interface that when plugged in Eq (S2) of the SI for the overall conductivity induces stronger
7
thickness dependence
Comparing our calculated heat conductivities for the aligned contact (open red circles) to available
experimental data for ~35 nm thick graphite slabs [34] (dashed green line) we find that at the thickest
model system considered of 104 graphene layers (~34 nm thick) the calculated value of 085plusmn005
W m sdot Kfrasl is in remarkable agreement with the measured value of ~07 W m sdot Kfrasl Furthermore
experimental values for bulk graphite [5] indicate that the thermal conductivity continues to grow up
to ~68 W m sdot Kfrasl (black dash-dotted line) which is consistent with the general trend of the calculated
heat conductivity that does not saturate for the thickest model system considered These results
strongly enforce the validity of our force-field and model systems to model the heat conductivity of
twisted layered materials interfaces Available experimental results for the heat conductivity of bulk
h-BN are marked by the dashed-dotted black and dashed-green lines in Figure 4(b) In line with our
findings for the graphitic interface our calculated finite slab heat conductivities for the aligned
interface (open red circles) continue to grow with the number of layers and are consistently below the
bulk value
Figure 4 Thickness dependence of the thermal conductivity 120581120581CP of aligned (open red circles) and
twisted by 3016deg (open black diamond symbols) graphite (a) and h-BN (b) stacks Blue squares
represent results obtained using the isotropic Lennard-Jones potential for the aligned contacts The
green dashed and black dash-dotted lines represent experimental results measured for graphite (a)
(Refs [534]) and bulk h-BN (b) (Refs [1415]) Note that both axis scales are logarithmic Error bar
estimation procedure is discussed in Sec 1 of the SI
Another important factor that may affect the interlayer thermal transport properties of 2D material
stacks is the average temperature of the system which was taken to be ~300K in all abovementioned
simulations To evaluate the sensitivity of our results towards this parameter we repeated the heat
conductivity and interfacial resistance calculations of optimally stacked graphite and h-BN stacks for
an average temperature of 400 K The results presented in see Sec 4 of the SI indicate that 120581120581CP and
8
ITR weakly depend on the average temperature over the entire thickness range considered This
conclusion is consistent with recent experimental findings [11] We may attribute this to the fact that
the thickness of the graphite and h-BN stack model considered is much smaller than their phonon
mean-free path (~200 nm for graphite [1011133536] and ~100 nm for bulk h-BN [15]) such that
the phonon transport is dominated by phonon-boundary scattering which weakly depends on
temperature in the range considered Therefore temperature dependent Umklapp processes have only
marginal contribution to our results [11]
Finally we note that previous calculations of the heat conductivity of twisted graphitic interfaces
relied on Lennard-Jones (LJ) potentials describing the interlayer interactions [8-1013] To
demonstrate the importance of using registry-dependent interlayer potentials we have repeated our
calculations of the heat conductivity of graphitic slabs with the REBO intralayer potential augmented
by LJ interlayer interactions [37] (120576120576 = 284 meV120590120590 = 34 Å ) We find that the calculated heat
conductivities obtained using the LJ interlayer potential are consistently higher than those obtained
by our ILP and that the difference between them grows with the model system thickness Notably the
heat conductivity obtained using the LJ potential for a graphitic slab of thickness ~34 nm is 154
W m sdot Kfrasl overestimating the experimental value by more than a factor of 2
The excellent agreement of our ILP calculations with experimental data of nanoscale graphitic stacks
therefore demonstrates the reliability of our predictions for the strong interfacial misfit angle
dependence of cross-layer thermal conductivity in graphite and h-BN The observed sharp
conductivity decrease of twisted graphitic interfaces at misfit angles lt 5deg opens the way to control
the thermal evacuation rate and thermal isolation of active layers in graphene-based electronic and
mechanical devices The revealed underlying mechanism suggests that design rules can be obtained
by carefully tailoring the phonon-phonon couplings across the twisted interface While the misfit
angle dependence of h-BN is found to be weaker than that of graphite the overall thermal
conductivity of the former is found to be higher This may be utilized to achieve higher conductivity
and controllability in twisted heterogeneous junctions of layered materials
ASSOCIATED CONTENT
Supporting Information
The supporting Information section includes a description of the methodology thermal conductivity
of twisted bilayer graphene theory for calculating the phonon coupling of twisted bilayer graphene
9
derivation of Fermirsquos golden rule and temperature dependence of the cross-plane thermal
conductivity
AUTHOR INFORMATION
Corresponding Author
E-mail urbakhtauextauacil
Notes
The authors declare no competing financial interest
ACKNOWLEDGMENTS
The authors would like to thank Prof Abraham Nitzan Dr Guy Cohen and Dr Yiming Pan for
helpful discussions WO acknowledges the financial support from a fellowship program for
outstanding postdoctoral researchers from China and India in Israeli Universities and the support from
the National Natural Science Foundation of China (Nos 11890673 and 11890674) HQ
acknowledges the financial support from the National Natural Science Foundation of China (No
11890674) MU acknowledges the financial support of the Israel Science Foundation Grant
No 114118 and the ISF-NSFC joint grant 319119 OH is grateful for the generous financial
support of the Israel Science Foundation under grant no 158617 and the Naomi Foundation for
generous financial support via the 2017 Kadar Award This work is supported in part by COST Action
MP1303
10
References
[1] A A Balandin Thermal properties of graphene and nanostructured carbon materials Nat Mater 10 569 (2011) [2] S Ghosh I Calizo D Teweldebrhan E P Pokatilov D L Nika A A Balandin W Bao F Miao and C N Lau Extremely high thermal conductivity of graphene Prospects for thermal management applications in nanoelectronic circuits Appl Phys Lett 92 151911 (2008) [3] A A Balandin S Ghosh W Bao I Calizo D Teweldebrhan F Miao and C N Lau Superior thermal conductivity of single-layer graphene Nano Lett 8 902 (2008) [4] J H Seol et al Two-Dimensional Phonon Transport in Supported Graphene Science 328 213 (2010) [5] R Taylor The thermal conductivity of pyrolytic graphite J Philosophical Magazine 13 157 (1966) [6] S Ghosh W Bao D L Nika S Subrina E P Pokatilov C N Lau and A A Balandin Dimensional crossover of thermal transport in few-layer graphene Nat Mater 9 555 (2010) [7] Z Wang R Xie C T Bui D Liu X Ni B Li and J T L Thong Thermal Transport in Suspended and Supported Few-Layer Graphene Nano Lett 11 113 (2011) [8] Z Wei Z Ni K Bi M Chen and Y Chen Interfacial thermal resistance in multilayer graphene structures Phys Lett A 375 1195 (2011) [9] Y Ni Y Chalopin and S Volz Significant thickness dependence of the thermal resistance between few-layer graphenes Appl Phys Lett 103 061906 (2013) [10] J Chen J H Walther and P Koumoutsakos Strain Engineering of Kapitza Resistance in Few-Layer Graphene Nano Lett 14 819 (2014) [11] Q Fu J Yang Y Chen D Li and D Xu Experimental evidence of very long intrinsic phonon mean free path along the c-axis of graphite Appl Phys Lett 106 031905 (2015) [12] G Fugallo A Cepellotti L Paulatto M Lazzeri N Marzari and F Mauri Thermal Conductivity of Graphene and Graphite Collective Excitations and Mean Free Paths Nano Lett 14 6109 (2014) [13] Z Wei J Yang W Chen K Bi D Li and Y Chen Phonon mean free path of graphite along the c-axis Appl Phys Lett 104 081903 (2014) [14] E K Sichel R E Miller M S Abrahams and C J Buiocchi Heat capacity and thermal conductivity of hexagonal pyrolytic boron nitride Phys Rev B 13 4607 (1976) [15] P Jiang X Qian R Yang and L Lindsay Anisotropic thermal transport in bulk hexagonal boron nitride Phys Rev Mater 2 064005 (2018) [16] M-H Wang Y-E Xie and Y-P Chen Thermal transport in twisted few-layer graphene Chinese Physics B 26 116503 (2017) [17] X Nie L Zhao S Deng Y Zhang and Z Du How interlayer twist angles affect in-plane and cross-plane thermal conduction of multilayer graphene A non-equilibrium molecular dynamics study Int J Heat Mass Tran 137 161 (2019) [18] A N Kolmogorov and V H Crespi Registry-dependent interlayer potential for graphitic systems Phys Rev B 71 235415 (2005) [19] M Reguzzoni A Fasolino E Molinari and M C Righi Potential energy surface for graphene on graphene Ab initio derivation analytical description and microscopic interpretation Phys Rev B 86 (2012) [20] M M van Wijk A Schuring M I Katsnelson and A Fasolino Moire Patterns as a Probe of Interplanar Interactions for Graphene on h-BN Phys Rev Lett 113 135504 (2014) [21] D Mandelli W Ouyang M Urbakh and O Hod The Princess and the Nanoscale Pea Long-Range Penetration of Surface Distortions into Layered Materials Stacks ACS Nano 13 7603 (2019) [22] E Koren E Loumlrtscher C Rawlings A W Knoll and U Duerig Adhesion and friction in mesoscopic graphite contacts Science 348 679 (2015) [23] E Koren I Leven E Loumlrtscher A Knoll O Hod and U Duerig Coherent commensurate electronic states at the interface between misoriented graphene layers Nat Nanotechnol 11 752 (2016)
11
[24] W Ouyang I Azuri D Mandelli A Tkatchenko L Kronik M Urbakh and O Hod Mechanical and Tribological Properties of Layered Materials under High Pressure Assessing the Importance of Many-Body Dispersion Effects J Chem Theory Comput 16 666 (2020) [25] W Ouyang D Mandelli M Urbakh and O Hod Nanoserpents Graphene Nanoribbon Motion on Two-Dimensional Hexagonal Materials Nano Lett 18 6009 (2018) [26] I Leven T Maaravi I Azuri L Kronik and O Hod Interlayer Potential for Grapheneh-BN Heterostructures J Chem Theory Comput 12 2896 (2016) [27] T Maaravi I Leven I Azuri L Kronik and O Hod Interlayer Potential for Homogeneous Graphene and Hexagonal Boron Nitride Systems Reparametrization for Many-Body Dispersion Effects J Phys Chem C 121 22826 (2017) [28] L H Liang and B Li Size-dependent thermal conductivity of nanoscale semiconducting systems Phys Rev B 73 153303 (2006) [29] P K Schelling S R Phillpot and P Keblinski Comparison of atomic-level simulation methods for computing thermal conductivity Phys Rev B 65 144306 (2002) [30] J Shiomi Nonequilirium molecular dynamics methods for lattice heat conduction calculations Annual Review of Heat Transfer 17 (2014) [31] G E Stedman Fermis Golden RulemdashAn Exercise in Quantum Field Theory Am J Phys 39 205 (1971) [32] L T Kong G Bartels C Campantildeaacute C Denniston and M H Muumlser Implementation of Greens function molecular dynamics An extension to LAMMPS Comput Phys Commun 180 1004 (2009) [33] S Plimpton Fast Parallel Algorithms for Short-Range Molecular Dynamics J Comput Phys 117 1 (1995) [34] M Harb et al The c-axis thermal conductivity of graphite film of nanometer thickness measured by time resolved X-ray diffraction Appl Phys Lett 101 233108 (2012) [35] H Zhang X Chen Y-D Jho and A J Minnich Temperature-Dependent Mean Free Path Spectra of Thermal Phonons Along the c-Axis of Graphite Nano Lett 16 1643 (2016) [36] D P Sellan E S Landry J E Turney A J H McGaughey and C H Amon Size effects in molecular dynamics thermal conductivity predictions Phys Rev B 81 214305 (2010) [37] S J Stuart A B Tutein and J A Harrison A reactive potential for hydrocarbons with intermolecular interactions J Chem Phys 112 6472 (2000)
S1
Supporting information for ldquoControllable Thermal Conductivity of
Twisted Graphene and Hexagonal Boron Nitride Interfacesrdquo
Wengen Ouyang1 Huasong Qin2 Michael Urbakh1 and Oded Hod1
1Department of Physical Chemistry School of Chemistry The Raymond and Beverly
Sackler Faculty of Exact Sciences and The Sackler Center for Computational
Molecular and Materials Science Tel Aviv University Tel Aviv 6997801 Israel
2State Key Laboratory for Strength and Vibration of Mechanical Structures School of
Aerospace Xirsquoan Jiaotong University Xirsquoan 710049 China
The Supporting information includes following sections
1 Methodology
2 Thermal conductivity of twisted bilayer graphene
3 Theory for calculating the phonon coupling of twisted bilayer graphene
4 Derivation of Fermirsquos golden rule
5 Temperature dependence of cross-plane thermal conductivity
S2
1 Methodology
11 Model system
The initial interlayer distance across the layered stack was set equal to 34 Aring and 33 Aring for graphite
and bulk h-BN respectively Periodic boundary conditions were applied in all directions It should be
noted that the lattice structure is rigorously periodic only at some specific twist angles the values of
which are listed in Table S1 in section 3 below While the cross-sectional area for each misfit angle
θ is different all systems considered have a contact area exceeding 12 nm2 which was shown to
provide converged results with respect to unit-cell dimensions [1] The intralayer interactions within
each graphene and h-BN layer were modeled via the second generation REBO potential [2] and
Tersoff potential [3] respectively The interlayer interactions between the layers of graphite and bulk
h-BN were described via our dedicated interlayer potential (ILP) [4] which is implemented in the
LAMMPS [5] suite of codes [6]
Fig S1 Schematic representation of the simulation setup (a) and steady-state temperature profile (b) respectively In panel (a) two identical AB-stacked graphite slabs (gray and orange respectively) are twisted with respect to each other to create a stacking fault of misfit angle θ A thermal bias is induced by applying Langevin thermostats to the two layers marked by dashed red (Thot) and green (Tcold) rectangles The arrows indicate the direction of the vertical heat flux Since periodic boundary conditions are applied also in the vertical direction two twisted interfaces are shown across which heat flows in opposite directions The steady-state temperature profiles are illustrated in panel (b) where N is the total number of layers in the model system and RAB dAB and Rθ and dθ mark the interfacial Kapitza resistance [89] and interlayer distance for contacting graphene layers with AB-stacking and misfit angle θ respectively The red lines in panel (b) mark the temperature variation across the twisted interface where the vertical axis corresponds to the position of the various layers along the stack and the horizontal axis marks the temperature of the various layers
S3
12 Simulation Protocol
All MD simulations were performed with the LAMMPS simulation package [5] The velocity-Verlet
algorithm with a time-step of 05 fs was used to propagate the equations of motion A Noseacute-Hoover
thermostat with a time constant of 025 ps was used for constant temperature simulations To maintain
a specified hydrostatic pressure the three translational vectors of the simulation cell were adjusted
independently by a Noseacute-Hoover barostat with a time constant of 10 ps [7] To relax the box we first
equilibrated the systems in the NPT ensemble at a temperature of T = 300 K and zero pressure for
250 ps (see Fig S2) After equilibration Langevin thermostats with damping coefficients 10 ps-1
were applied to the bottom and middle layer of the graphene stack (see Fig S1) with target
temperature Thot = 375 K (hot reservoir) and Tcold = 225 K (cold reservoir) respectively Then the
system was allowed to reach steady-state over a subsequent simulation period of 750 ps (see Fig S2)
during which the dynamics of all non-thermostated layers followed the NVE ensemble For the larger
model systems the length of the NPT and Langevin stages was doubled (for the 32 and 48 layers
systems) or tripled (for the 104 layers graphitic system) to ensure convergence of the obtained steady-
state Once steady-state was obtained the last 500 ps were used to calculate the thermal conductivity
of the twisted graphite and bulk h-BN The statistical errors were estimated using ten different data
sets each calculated over a time interval of 50 ps
Fig S2 Time evolution of the temperature of thermostated layers for 16 layers twisted graphite with misfit angle (a) θ = 0deg (b) θ = 509deg (c) θ = 1518deg and (d) θ = 3016deg Note that the thermal fluctuations increase with increasing the misfit angle due to the growing interfacial thermal resistance that enhances phonon back scattering at the twisted junction
S4
13 Calculation of the interfacial thermal resistance
According to Fourierrsquos law the cross-plane thermal conductivity (120581120581CP) of a twisted graphitic interface
of misfit angle θ can be calculated as
120581120581CP = 119876119860119860Δ119879119879Δ119911119911
(S1)
where 119876 is the heat flux A is the cross-section area and Δ119879119879Δ119911119911 is the temperature gradient along
the direction of heat flux (perpendicular to the basal plane in our case) Fig S1(b) shows a schematic
temperature profile along the z-direction where the vertical axis corresponds to the position of the
various layers along the stack and the horizontal axis marks the temperature of the various layers The
actual temperature profiles extracted from the NEMD simulations for twisted graphite with different
number of layers can be found in Fig S3 For Bernal-stacked graphite (ie θ = 0deg red circles in Fig
S3) only the linear region of the temperature profile was used to calculate 120581120581CP and the points
corresponding to the layers where the thermostats were applied were omitted (marked with green
triangle in Fig S3) The 120581120581CP of the system was calculated using Eq (S1) by averaging over the two
linear regions of the temperature profiles For the twisted case (θ ne 0deg) we found a sudden
temperature decrease Δ119879119879120579120579 at the position of the twisted interface (see black squares in Fig S3) 120581120581CP
in this case was calculated using the temperature gradient calculated for the same layer range as that
for θ = 0deg To characterize the thermal properties of the twisted interface the concept of interfacial
thermal resistance (ITR) ie Kapitza resistance [89] was introduced According the definition of
the Kapitza resistance [8] 119877119877 = 119860119860Δ119879119879119876 and noticing that Δ119879119879tot = (1198731198732 minus 3)Δ119879119879AB + Δ119879119879120579120579 and
Δ119911119911 = (1198731198732 minus 3)119889119889AB + 119889119889120579120579 Eq (1) can be rewritten considering two-resistors in series as [see Fig
S1(b)]
(1198731198732 minus 3)119877119877AB + 119877119877120579120579 = [(1198731198732 minus 3)119889119889AB + 119889119889120579120579] 120581120581CPfrasl (S2)
Here 119877119877AB 119889119889AB Δ119879119879AB and 119877119877120579120579 119889119889120579120579 Δ119879119879120579120579 are the ITR interlayer distance and temperature
difference for adjacent AB-stacked and twisted graphene layers respectively For the aligned contact
(120579120579 = 0deg) the ITR can be simply calculated as 119877119877AB = 119889119889AB120581120581AB Once 119877119877AB is known 119877119877120579120579 can be
calculated from Eq (S2) We note that the sharp temperature drop at the twisted interface indicates
that 119877119877120579120579 should be much larger than 119877119877AB
S5
Fig S3 Temperature profiles for graphitic stacks consisting of (a) 8 and (b) 16 layers The red circles and black squares represent the temperature profiles for the aligned (120579120579 = 0deg) and twisted (120579120579 =3016deg) junctions respectively Green triangles represent data points that were omitted in the 120581120581CP calculation
2 Thermal conductivity of twisted bilayer graphene
As comparison we also calculated the interfacial thermal conductivity (ITC) and ITR of twisted
bilayer graphene (tBLG) with the transient MD simulation approach [15-17] since the NEMD
simulation protocol used in the main text becomes invalid in this case In this protocol the system
was first equilibrated within the NPT ensemble at T = 200 K and zero pressure for 100 ps which was
followed by a 100 ps NVT ensemble equilibration stage and a 100 ps of NVE ensemble equilibration
stage After the system reached steady-state an ultrafast heat impulse was imposed on the top layer
of the t-BLG for 50 fs to increase the temperature of the top layer from 200 K to 400 K while that of
bottom layer of tBLG remained unchanged After the external heat source was removed thermal
energy flowed from the top layer to the bottom layer due to the temperature difference and the
temperature of both layers approached 300 K when quasi-steady-state was reached During the
thermal relaxation time interval (500 ps) the temperature and energy of the system sections were
recorded The ITR could then be extracted using the following equation [15-17]
part119864119864t120597120597120597120597
= 119860119860119877119877119879119879bot(119905119905) minus 119879119879top(119905119905) (S3)
where 119864119864t is the total energy of the top graphene layer R is the ITR of the tBLG A is the interfacial
cross-section area and 119879119879bot and 119879119879top are the instantaneous temperatures measured for the bottom
and top layers of the tBLG respectively Note that in Eq S3 we assume a linear dependence of the
heat flux on the temperature difference between the layers The ITC of the tBLG is simply defined as
ITC equiv 119889119889ITR where d is the average interlayer distance
S6
The ITC and ITR of tBLG as functions of misfit angle calculated with the transient MD simulation
protocol are illustrated in Fig S4 demonstrating similar misfit-angle dependence as that for the
NEMD protocol with Langevin thermostat exercised to obtain the results presented in the main text
This further validates the reliability of the simulation protocol adopted in the main text which is more
suitable to treat thick slabs and allows to obtain a true stead-state
Fig S4 Misfit-angle dependence of (a) ITC and (b) ITR for a twisted bilayer graphene obtained using the transient MD simulation approach
3 Theory for calculating the phonon-phonon coupling of twisted bilayer graphene
31 Brillouin Zone of supercell in tBLG
For tBLG the lattice structure is rigorously periodic only at some specific misfit angles θ where the
lattice vector 1199231199231 = 11989911989911199381199381 + 11989911989921199381199382 in the bottom layer equals the vector 1199231199232 = 11989811989811199381199381 + 11989811989821199381199382 in the
top layer with certain integers m1 m2 and n1 n2 Here 1199381199381 = 119886119886(10) and 1199381199382 = 11988611988612radic32 are
the primitive lattice vectors of the bottom layer and a is the lattice constant of monolayer graphene
Thus the exact superlattice period is then given by [18]
119871119871 = |11989911989911199381199381 + 11989911989921199381199382| = 11988611988611989911989912 + 11989911989922 + 11989911989911198991198992 = |1198991198991minus1198991198992|1198861198862 sin(1205791205792) (S4)
where θ is the angle between two lattice vectors 1199231199231 and 1199231199232 In the simulations below we always
rotated the supercell such that its lattice vector is 1199231199231 = 119871119871(10) and 1199231199232 = 11987111987112radic32 In this case
the corresponding reciprocal lattice vector of the moireacute superlattice satisfies the relation 119918119918119894119894 ∙ 119923119923119895119895 =
2120587120587120575120575119894119894119895119895 such that
1199181199181 = 4120587120587radic3119871119871
radic32
minus12 1199181199182 = 4120587120587
radic3119871119871(01) (S5)
S7
Both the lattice vectors and the corresponding reciprocal lattice vectors of the superlattice of the tBLG
are presented in Fig S5 Table S1 reports the parameters used to construct rhombus periodic
supercells of different misfit angles that can be duplicated to construct a rectangular periodic supercell
Table S1 The parameters used to construct periodic supercells of various misfit angles
120579120579 (deg) 119860119860 (nm2) 1198991198991 1198991198992 1198981198981 1198981198982
0 60383688 24 0 24 0
0696407 709613169 48 47 47 48
1121311 273718420 30 29 29 30
2000628 85648391 17 16 16 17
3006558 152322047 23 21 21 23
4048894 189013524 26 23 23 26
5085849 53255058 14 12 12 14
7926470 49376245 14 11 11 14
9998709 86277388 19 14 14 19
15178179 31554670 15 4 4 15
19932013 42876612 15 8 8 15
25039660 13942761 9 4 4 9
30158276 18974735 11 4 4 11
32204228 21805221 12 4 4 12
32 Special points for Brillouin Zone integration
The calculation of the sum over wave vector q in Eq (4) in the main text can be transformed to an
integral using the relation sum (⋯ )119954119954 = 1119881119881119887119887int (⋯ )BZ d119954119954 where 119881119881119887119887 = (2120587120587)3119881119881 is the volume of the
Brillouin Zone (BZ) and V is volume of the real-space unit-cell The calculation of integral is usually
inefficient since it requires calculating the value of the function over a large set of k points in the first
BZ To calculate such integrations more efficiently simple k-point meshes can be replaced by a
carefully selected set of special points in the BZ 119954119954119894119894 [19-22] over which the function is evaluated
The integral can then be estimated via
119868119868 = 1119881119881119887119887int 119891119891(119954119954)BZ d119954119954 asymp 1
119873119873sum 119908119908119894119894119891119891(119954119954119894119894)119894119894 (S6)
S8
where 119881119881119887119887 is the volume of the BZ 119908119908119894119894 is the weight of the ith data point and N normalizes the
weighting factors to unity 119873119873 = sum 119908119908119894119894119894119894 The set of selected 119954119954119894119894 forms a grid in the irreducible
Brillouin zone (IBZ) as is illustrated by the red points in Fig S5(b) The coordinates of these points
for a hexagonal lattice are presented in Eq (S7)
Fig S5 (a) Twisted bilayer graphene of misfit angle θ = 509deg L1 and L2 are the superlattice vectors (b) The corresponding first Brillouin Zone of (a) G1 and G2 are the reciprocal lattice vectors of the superlattice The triangle ΔΓMK represents the irreducible Brillouin zone Red circles mark the position of the special points used to evaluate the integral over the first Brillouin Zone
⎩⎪⎨
⎪⎧ 1199541199541 = 1
9 1205971205973 1199541199542 = 2
9 1205971205973 1199541199543 = 1
18 512059712059718
1199541199544 = 518
512059712059718 1199541199545 = 1
9 21205971205979 1199541199546 = 2
9 21205971205979
1199541199547 = 118
312059712059718 1199541199548 = 1
9 1205971205979 1199541199549 = 1
18 12059712059718
(S7)
Here 119905119905 = radic3 and the unit of the coordinates is 2120587120587119871119871 The weighting factors 119908119908119894119894 are [19]
119908119908124689 = 112
119908119908357 = 16 (S8)
Using Eqs (S6-S8) Eq (4) in the main text can be evaluated as follows
Γtot = 120587120587ℏ3
2sum 119908119908119896119896 sum
119890119890minus120573120573119864119864119954119954119948119948120582120582
119885119885
120588120588119864119864119954119954119948119948120582120582119881119881120582120582120582120582+31199031199032(119954119954119948119948)
2
1198641198641199541199541199481199481205821205822120582120582
9119896119896=1 (S9)
This equation was used to calculate the transition rate presented in Fig 3 in the main text
4 Derivation of Fermirsquos golden rule
The derivation of Fermirsquos golden rule is provided in a separate file (Fermirsquos Golden Rule for
phononspdf) due to its extent
S9
5 Temperature dependence of interfacial thermal conductivity
In the main text the target temperatures of the Langevin thermostats for the bottom and middle layers
of graphene and h-BN were set to 225 K and 375 K respectively After reaching the steady-state the
average temperature of the system was found to be ~300 K To check the effect of average temperature
on our results we calculated the cross-plane thermal conductivity (120581120581CP ) and the corresponding
interfacial thermal resistance (ITR) at a different temperature gradient (325 K ndash 475 K) resulting the
average steady-state temperature of ~400K The protocol described in Section 1 above was used to
perform these calculations as well Both ILP and Lennard-Jones (LJ) potential were tested for
graphite whereas for the bulk h-BN simulations only the ILP was used The results for graphite and
bulk h-BN are illustrated in Fig S6 and Fig S7 respectively For the ILP we find that the overall
values of 120581120581CP (ITR) decrease (increase) slightly with increasing average temperature which is
consistent with a recent experiment [10] The LJ potential calculations as well exhibit very week
dependence on average temperature within the range studied Altogether the layer dependences of
both quantities remain mostly insensitive to the average temperature The reason is that the thickness
of the graphite and h-BN model systems was chosen to be much smaller than their phonon mean free
path (~200 nm for graphite [110-13] and ~100 nm for bulk h-BN [14]) such that phonon transport is
dominated by phonon-boundary scattering and the Umklapp process only makes marginal
contribution [10]
S10
Fig S6 Layer dependence of 120581120581CP (a c) and ITR (b d) for Bernal-stacked graphite at average steady-state temperatures of 300 K (red circles) and 400 K (black squares) The left and right columns correspond to the 120581120581CP and ITR calculated with ILP and LJ potential respectively
Fig S7 Layer dependence of 120581120581CP (a) and ITR (b) for AArsquo-stacked h-BN at average temperatures of 300 K (red circles) and 400 K (blue squares)
S11
References [1] Z Wei J Yang W Chen K Bi D Li and Y Chen Phonon mean free path of graphite along the c-axis Appl Phys Lett 104 081903 (2014) [2] D W Brenner O A Shenderova J A Harrison S J Stuart B Ni and S B Sinnott A second-generation reactive empirical bond order (REBO) potential energy expression for hydrocarbons J Phys Condens Matter 14 783 (2002) [3] A Kınacı J B Haskins C Sevik and T Ccedilağın Thermal conductivity of BN-C nanostructures Phys Rev B 86 (2012) [4] W Ouyang I Azuri D Mandelli A Tkatchenko L Kronik M Urbakh and O Hod Mechanical and Tribological Properties of Layered Materials under High Pressure Assessing the Importance of Many-Body Dispersion Effects J Chem Theory Comput 16 666 (2020) [5] S Plimpton Fast Parallel Algorithms for Short-Range Molecular Dynamics J Comput Phys 117 1 (1995) [6] W Ouyang D Mandelli M Urbakh and O Hod Nanoserpents Graphene Nanoribbon Motion on Two-Dimensional Hexagonal Materials Nano Lett 18 6009 (2018) [7] W Shinoda M Shiga and M Mikami Rapid estimation of elastic constants by molecular dynamics simulation under constant stress Phys Rev B 69 134103 (2004) [8] G L Pollack Kapitza Resistance Rev Mod Phys 41 48 (1969) [9] H-S Yang G R Bai L J Thompson and J A Eastman Interfacial thermal resistance in nanocrystalline yttria-stabilized zirconia Acta Mater 50 2309 (2002) [10] Q Fu J Yang Y Chen D Li and D Xu Experimental evidence of very long intrinsic phonon mean free path along the c-axis of graphite Appl Phys Lett 106 031905 (2015) [11] J Chen J H Walther and P Koumoutsakos Strain Engineering of Kapitza Resistance in Few-Layer Graphene Nano Lett 14 819 (2014) [12] H Zhang X Chen Y-D Jho and A J Minnich Temperature-Dependent Mean Free Path Spectra of Thermal Phonons Along the c-Axis of Graphite Nano Lett 16 1643 (2016) [13] D P Sellan E S Landry J E Turney A J H McGaughey and C H Amon Size effects in molecular dynamics thermal conductivity predictions Phys Rev B 81 214305 (2010) [14] P Jiang X Qian R Yang and L Lindsay Anisotropic thermal transport in bulk hexagonal boron nitride Phys Rev Mater 2 064005 (2018) [15] J Zhang Y Hong and Y Yue Thermal transport across graphene and single layer hexagonal boron nitride J Appl Phys 117 134307 (2015) [16] B Liu F Meng C D Reddy J A Baimova N Srikanth S V Dmitriev and K Zhou Thermal transport in a graphenendashMoS2 bilayer heterostructure a molecular dynamics study RSC Advances 5 29193 (2015) [17] Z Ding Q-X Pei J-W Jiang W Huang and Y-W Zhang Interfacial thermal conductance in grapheneMoS2 heterostructures Carbon 96 888 (2016) [18] N N T Nam and M Koshino Lattice relaxation and energy band modulation in twisted bilayer graphene Phys Rev B 96 075311 (2017)
S12
[19] R Ramiacutearez and M C Boumlhm Simple geometric generation of special points in brillouin-zone integrations Two-dimensional bravais lattices Int J Quantum Chem 30 391 (1986) [20]S L Cunningham Special points in the two-dimensional Brillouin zone Phys Rev B 10 4988 (1974) [21] H J Monkhorst and J D Pack Special points for Brillouin-zone integrations Phys Rev B 13 5188 (1976) [22] D J Chadi Special points for Brillouin-zone integrations Phys Rev B 16 1746 (1977)
1
Fermirsquos golden rule for phonons
1 Basic theory for phonons 11 Basic notations
Let us consider a 3D crystal with a total of 119873119873 = 119873119873111987311987321198731198733 unit cells and periodic boundary conditions
To be specific let 119938119938119894119894 119894119894 = 123 be the lattice vectors that define the unit cell We index unit cells
with n = (n1n2n3) where each ni = 12 ∙∙∙ Ni and their locations are 119929119929119899119899 = sum 1198991198991198941198941199381199381198941198943119894119894=1 Assume that
there are r atoms in each unit cell which are indexed with 119904119904 = 1⋯ 119903119903 The mass and the equilibrium
distance of the sth atom are notated as 119872119872s and 119929119929s0 respectively Then the location of the sth atom in
the nth unit cell at time t can be expressed as
119955119955119899119899119899119899(119905119905) = 119929119929119899119899 + 119929119929s0 + 119958119958119899119899119899119899(119905119905) (11)
where 119958119958119899119899119899119899(119905119905) is its displacement from its equilibrium position
The Lagrangian for this classical problem can be written as
ℒ = sum sum 119872119872119904119904|119955119899119899119904119904|2(119905119905)2
119903119903119899119899=1
119873119873119899119899=1 minus 119881119881 (12)
where the second term is the sum of interactions between all pairs of atoms
Under the harmonic approximation ie expanding the total potential energy 119881119881 around the
equilibrium positions The Lagrangian can be simplified as
ℒ = sum sum 119872119872119904119904|119958119899119899119904119904|2(119905119905)2
119903119903119899119899=1
119873119873119899119899=1 minus 1
2sum sum sum 120601120601120572120572120572120572prime
119899119899119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime120572120572120572120572prime119899119899119899119899prime119899119899119899119899prime (13)
where 119906119906119899119899119899119899120572120572 120572120572 = 123 are the Cartesian coordinates of the displacement 119958119958119899119899119899119899(119905119905) and
120601120601120572120572120572120572prime 119899119899119899119899prime119904119904 119904119904prime = 1205971205972119881119881
120597120597119903119903119899119899119904119904119899119899119903119903119899119899prime119904119904prime119899119899primeeq
= 120601120601120572120572prime120572120572 119899119899prime119899119899119904119904prime 119904119904 = 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime (14)
Note that the first order term vanishes because we are expanding around the equilibrium positions
120601120601120572120572120572120572prime 119899119899119899119899prime119904119904 119904119904prime represents the component of the force acted on the sth atom in the nth unit cell along 120572120572
direction when the atom 119904119904prime in the unit cell 119899119899prime moves a unit displacement along 120572120572prime direction The
symmetries of 120601120601120572120572120572120572prime 119899119899 119899119899prime119904119904 119904119904prime appearing in Eq (14) arise from the intechangability of the second
derivative and the translational invariance of the interactions
12 Dynamical matrix
The equation of motion of the sth atom in the nth unit cell can be derived using the EulerndashLagrange
equation as follows
2
119872119872119899119899119906119899119899119899119899120572120572 = minussum 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119899119899prime119899119899prime120572120572prime 119906119906119899119899prime119899119899prime120572120572prime (15)
If we displace all atoms equally ie shifting 119906119906119899119899prime119899119899prime120572120572prime to 119906119906119899119899prime119899119899prime120572120572prime + 120575120575 the total force on the sth atom in
the nth unit cell does not change From the above equation we have
sum 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119899119899prime120572120572prime = 0 (16)
We are looking for normal modes (because any general solution can be written as a linear combination
of them) these are solutions where all atoms oscillate with the same frequency Moreover because
of the lattice structure we expect solutions to reflect this periodicity So we guess solutions of the
form
119906119906119899119899119899119899120572120572(119905119905) = 1119872119872119904119904
119890119890119899119899120572120572119890119890119894119894(119954119954∙119929119929119899119899minus120596120596119905119905) (17)
where 119890119890119899119899120572120572 are real-space solutions that will be determined later and 119954119954 is the wave-vector in
reciprocal space Substituting Eq (17) into Eq (15) we can get
1205961205962119890119890119899119899120572120572(119954119954) = sum 119863119863119899119899120572120572119899119899prime120572120572prime(119954119954)119890119890119899119899prime120572120572prime(119954119954)119899119899prime120572120572prime (18)
where
119863119863119899119899120572120572119899119899prime120572120572prime(119954119954) = sum 1
119872119872119904119904119872119872119904119904prime120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119890119890
minus119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime119899119899prime = sum 1
119872119872119904119904119872119872119904119904prime120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime 119890119890
minus119894119894119954119954∙119929119929119897119897119897119897 (19)
is called dynamical matrix (dimension 3119903119903 times 3119903119903) Note that we have defined the relative cell distance
vector 119929119929119897119897 equiv 119929119929119899119899 minus 119929119929119899119899prime and number index 119897119897 equiv 119899119899 minus 119899119899prime where the infinite sum over 119899119899prime can be
replaced by the sum over 119897119897 for any value of the index 119899119899 Note that dynamical matrix is Hermitian
symmetric (ie 119863119863119899119899120572120572119899119899prime120572120572prime(119954119954)
lowast= 119863119863119899119899prime120572120572prime
119899119899120572120572 (minus119954119954)) because ϕ is symmetric so all the eigenvalues of Eq (18)
(120596120596120582120582(119954119954) 120582120582 = 12⋯ 3119903119903) are real for each 119954119954 in the Brillouin zone (BZ) which is determined by
det1205961205961205821205822(119954119954)120575120575119899119899120572120572120575120575119899119899prime120572120572prime minus 119863119863119899119899120572120572119899119899prime120572120572prime(119954119954) = 0 (110)
Taking the conjugate of this equation we have
0 = det 1205961205961205821205822(119954119954)lowast120575120575119899119899120572120572120575120575119899119899prime120572120572prime minus 119863119863119899119899120572120572119899119899
prime120572120572prime(119954119954)lowast = det1205961205961205821205822(119954119954)120575120575119899119899120572120572120575120575119899119899prime120572120572prime minus 119863119863119899119899120572120572119899119899
prime120572120572prime(minus119954119954) (111)
while replacing 119954119954 by minus119954119954 in Eq (110) we have det1205961205961205821205822(minus119954119954)120575120575119899119899120572120572120575120575119899119899prime120572120572prime minus 119863119863119899119899120572120572119899119899prime120572120572prime(minus119954119954) = 0
Itrsquos clear that 1205961205961205821205822(119954119954) and 1205961205961205821205822(minus119954119954) obey the same equation thus we have
120596120596120582120582(minus119954119954) = 120596120596120582120582(119954119954) (112)
The corresponding eigenvectors are orthonormal
sum 119890119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582
prime119899119899120572120572 = 120575120575120582120582120582120582prime (113)
The complex conjugate of Eq (18) gives
1205961205962119890119890119899119899120572120572lowast (119954119954) = sum 119863119863119899119899120572120572119899119899prime120572120572prime(119954119954)
lowast119890119890119899119899prime120572120572primelowast (119954119954)119899119899prime120572120572prime = sum 119863119863119899119899prime120572120572prime
119899119899120572120572 (minus119954119954)119890119890119899119899prime120572120572primelowast (119954119954)119899119899prime120572120572prime (114)
3
While replacing 119954119954 by minus119954119954 in Eq (18) we get the following equation
1205961205962119890119890119899119899120572120572(minus119954119954) = sum 119863119863119899119899120572120572119899119899prime120572120572prime(minus119954119954)119890119890119899119899prime120572120572prime(minus119954119954)119899119899prime120572120572prime (115)
From Eq (114) and Eq (115) we see that eigenvectors 119890119890119899119899120572120572120582120582 (minus119954119954)lowast and 119890119890119899119899120572120572120582120582 (119954119954) obey the same
eigenvalue equation Since the eigenvectors are normalized we get the following property
119890119890119899119899120572120572120582120582 (minus119954119954)lowast
= 119890119890119899119899120572120572120582120582 (119954119954) (116)
2 Second quantization The general solution is a linear combination of all these normal modes thus we have
119906119906119899119899119899119899120572120572(119905119905) = sum 119862119862120582120582(119954119954)119873119873119872119872119904119904
119890119890119899119899120572120572120582120582 (119954119954)119890119890minus119894119894120596120596119954119954120582120582119905119905119890119890119894119894119954119954∙119929119929119899119899119954119954120582120582 = sum 119876119876120582120582(119954119954119905119905)119873119873119872119872119904119904
119890119890119899119899120572120572120582120582 (119954119954)119890119890119894119894119954119954∙119929119929119899119899119954119954120582120582 (21)
where the we define the normal coordinates as 119876119876120582120582(119954119954 119905119905) equiv 119862119862120582120582(119954119954)119890119890minus119894119894120596120596120582120582(119954119954)119905119905 in the eigenvectors
representation To ensure that the displacements 119906119906119899119899119899119899120572120572(119905119905) are real (namely 119906119906119899119899119899119899120572120572lowast (119905119905) = 119906119906119899119899119899119899120572120572(119905119905)) the
following relation on 119876119876120582120582(119954119954 119905119905) and 119890119890119899119899120572120572120582120582 is enforced
119876119876120582120582(119954119954 119905119905)119890119890119899119899120572120572120582120582 (119954119954)lowast
= 119876119876120582120582(minus119954119954 119905119905)119890119890119899119899120572120572120582120582 (minus119954119954) (22)
where we used the fact that the sum over 119954119954 runs symmetrically over both negative and positive
values Using Eq (116) we have
[119876119876120582120582(119954119954 119905119905)]lowast = 119876119876120582120582(minus119954119954 119905119905) (23)
21 Kinetic energy term
Using Eq (21) the kinetic energy of the system can be expressed as
119879119879 = 12sum 1198721198721198991198991199061198991198991198991198991205721205722 (119905119905)119899119899119899119899120572120572 = 1
2sum 119872119872119904119904
119873119873119872119872119904119904sum sum 119876120582120582(119954119954 119905119905)119876120582120582prime(119954119954prime 119905119905)119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582
prime(119954119954prime) sum 119890119890119894119894119954119954+119954119954prime∙119929119929119899119899119899119899119954119954prime120582120582prime119954119954120582120582119899119899120572120572 =
12sum sum 119876120582120582(119954119954 119905119905)119876120582120582prime(minus119954119954 119905119905)119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582
prime(minus119954119954)119954119954119954119954prime120582120582prime =119899119899120572120572
12sum 119876120582120582(119954119954 119905119905)119876120582120582prime
lowast (119954119954 119905119905) sum 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582prime(119954119954)
lowast119899119899120572120572 =119954119954120582120582120582120582prime
12sum 119876120582120582(119954119954 119905119905)119876120582120582lowast(119954119954 119905119905)119954119954120582120582 =
12sum 119876120582120582(119954119954 119905119905)119876120582120582(minus119954119954 119905119905)119954119954120582120582
ie
119879119879 = 12sum 119876120582120582(119954119954 119905119905)119876120582120582(minus119954119954 119905119905)119954119954120582120582 (24)
To derive Eq (24) we used Eqs (113) and (116) as well as the following equations
1119873119873sum 119890119890119894119894119954119954+119954119954prime∙119929119929119899119899119899119899 = 120575120575119954119954+119954119954prime120782120782 (25)
4
22 Potential energy term
Similarity the potential energy can be rewritten in terms of the normal coordinates as follows
119880119880 =12120601120601119899119899119899119899120572120572119899119899prime119899119899prime120572120572prime119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
120572120572120572120572prime119899119899119899119899prime119899119899119899119899prime=
12120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119899119899+119954119954prime∙119929119929119899119899prime
119873119873119872119872119899119899119872119872119899119899prime
[119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(119954119954prime 119905119905)] 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime120582120582prime (119954119954prime)
119954119954prime120582120582prime119954119954120582120582119899119899119899119899prime120572120572120572120572prime119899119899119899119899prime
=12120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894(119954119954+119954119954prime)∙119929119929119899119899119890119890minus119894119894119954119954prime∙119929119929119897119897
119873119873119872119872119899119899119872119872119899119899prime
[119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(119954119954prime 119905119905)] 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime120582120582prime (119954119954prime)
119954119954prime120582120582prime119954119954120582120582119899119899119897119897120572120572120572120572prime119899119899119899119899prime
=12120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890minus119894119894119954119954prime∙119929119929119897119897
119872119872119899119899119872119872119899119899prime[119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(119954119954prime 119905119905)] 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (119954119954prime)1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899
119899119899=120575120575119902119902+119902119902prime=0
119954119954prime120582120582prime119954119954120582120582119897119897120572120572120572120572prime119899119899119899119899prime
=12120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894119954119954119929119929119897119897
119872119872119899119899119872119872119899119899prime [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)] 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (minus119954119954)119954119954120582120582120582120582prime119897119897120572120572120572120572prime119899119899119899119899prime
=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]119890119890119899119899120572120572120582120582 (119954119954)
119899119899120572120572
119863119863119899119899120572120572119899119899prime120572120572prime(minus119954119954)119890119890119899119899prime120572120572prime
120582120582prime (minus119954119954)119899119899prime120572120572prime119954119954120582120582120582120582prime
=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]119890119890119899119899120572120572120582120582 (119954119954)
119899119899120572120572
1205961205961205821205822(minus119954119954)119890119890119899119899120572120572120582120582prime (minus119954119954)
119954119954120582120582120582120582prime
=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]1205961205961205821205822(119954119954)120575120575120582120582120582120582prime =119954119954120582120582120582120582prime
121205961205961205821205822(119954119954)119876119876120582120582(119954119954 119905119905)119876119876120582120582(minus119954119954 119905119905)
119954119954120582120582
ie
119880119880 = 12sum 1205961205961205821205822(119954119954)119876119876120582120582(119954119954 119905119905)119876119876120582120582(minus119954119954 119905119905)119954119954120582120582 (26)
where in above deriviation the orthogonality of its eigenvectores and the symmetry of its eigenvalues
are used Thus the Lagrangian reads as
ℒ = 119879119879 minus 119881119881 = 12sum 119876120582120582(minus119954119954 119905119905)119876120582120582(119954119954 119905119905) minus 120596120596120582120582
2(119954119954)119876119876120582120582(minus119954119954 119905119905)119876119876120582120582(119954119954 119905119905)119954119954120582120582 (27)
Using the relation 119875119875120582120582(119954119954 119905119905) = 120597120597ℒ part119876120582120582(119954119954 119905119905) the Hamiltonian of the system then can be written as
119867119867 = 119879119879 + 119881119881 = 12sum [119875119875120582120582(minus119954119954 119905119905)119875119875120582120582(119954119954 119905119905) + 120596120596120582120582
2(119954119954)119876119876120582120582(minus119954119954 119905119905)119876119876120582120582(119954119954 119905119905)]119954119954120582120582 (28)
Now we quantize H by asking the momenta and coordinates to be operators
119867119867 = 12sum 119875119875minus119954119954120582120582119875119875119954119954120582120582 + 120596120596119954119954120582120582
2 119876119876minus119954119954120582120582119876119876119954119954120582120582119954119954120582120582 (29)
here 119875119875119954119954120582120582 and 119876119876119954119954120582120582 obey the following commutation relations
119876119876119954119954120582120582119875119875119954119954prime120582120582prime = 119894119894ℏ120575120575119954119954119954119954prime120575120575120582120582120582120582prime
119876119876119954119954120582120582119876119876119954119954prime120582120582prime = 0 119875119875119954119954120582120582119875119875119954119954prime120582120582prime = 0 (210)
Similar to the case of ordinary quantum harmonic oscillators it is convenient to define ladder
operators for each mode as follows
119876119876119954119954120582120582 = ℏ
2120596120596119954119954120582120582119887119887119954119954120582120582 + 119887119887minus119954119954120582120582
dagger 119875119875119954119954120582120582 = 119894119894ℏ120596120596119954119954120582120582
2119887119887119954119954120582120582
dagger minus 119887119887minus119954119954120582120582 (211)
where 119887119887119954119954120582120582dagger and 119887119887119954119954120582120582 are Bosonic creation and annihilation operators for phonons with momentum
119954119954 branch index 120582120582 and frequency 120596120596119954119954120582120582 which obeys the Bosonic commutation relation
5
119887119887119954119954120582120582 119887119887119954119954prime120582120582prime
dagger = 120575120575119954119954119954119954prime120575120575120582120582120582120582prime
119887119887119954119954120582120582 119887119887119954119954prime120582120582prime = 0 119887119887119954119954120582120582dagger 119887119887119954119954prime120582120582prime
dagger = 0 (212)
Substituting Eqs (211) into (29) and using the properties Eq (210) and (212) we have
119867119867 =12119875119875minus119954119954120582120582119875119875119954119954120582120582 +120596120596119954119954120582120582
2 119876119876minus119954119954120582120582119876119876119954119954120582120582119954119954120582120582
=12minus
ℏ120596120596119954119954120582120582
2119887119887minus119954119954120582120582
dagger minus 119887119887119954119954120582120582119887119887119954119954120582120582dagger minus 119887119887minus119954119954120582120582+ 120596120596119954119954120582120582
2 ℏ2120596120596119954119954120582120582
119887119887minus119954119954120582120582 + 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 + 119887119887minus119954119954120582120582
dagger 119954119954120582120582
=ℏ4120596120596119954119954120582120582minus119887119887minus119954119954120582120582
dagger 119887119887119954119954120582120582dagger minus 119887119887minus119954119954120582120582
dagger 119887119887minus119954119954120582120582 minus 119887119887119954119954120582120582119887119887119954119954120582120582dagger + 119887119887119954119954120582120582119887119887minus119954119954120582120582
119954119954120582120582
+ 119887119887minus119954119954120582120582119887119887119954119954120582120582 + 119887119887minus119954119954120582120582119887119887minus119954119954120582120582dagger + 119887119887119954119954120582120582
dagger 119887119887119954119954120582120582 + 119887119887119954119954120582120582dagger 119887119887minus119954119954120582120582
dagger
=ℏ4120596120596119954119954120582120582119887119887minus119954119954120582120582
dagger 119887119887minus119954119954120582120582 + 119887119887minus119954119954120582120582119887119887minus119954119954120582120582dagger + 119887119887119954119954120582120582119887119887119954119954120582120582
dagger + 119887119887119954119954120582120582dagger 119887119887119954119954120582120582
119954119954120582120582
=ℏ41205961205961199541199541205821205822119887119887minus119954119954120582120582
dagger 119887119887minus119954119954120582120582 + 1+ 2119887119887119954119954120582120582dagger 119887119887119954119954120582120582 + 1
119954119954120582120582
= ℏ120596120596119954119954120582120582 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 +
12
119954119954120582120582
Therefore we finally get the quantized representation of non-interacting phonons
119867119867 = sum ℏ120596120596119954119954120582120582 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 + 1
2119954119954120582120582 (213)
The operator of atom displacements (Eq 114) is expressed in terms of the phonon operators by
119906119906119899119899119899119899120572120572 = sum ℏ
2119873119873119872119872119904119904120596120596119954119954120582120582119890119890119899119899120572120572120582120582 119890119890119894119894119954119954∙119929119929119899119899119887119887119954119954120582120582 + 119887119887minus119954119954120582120582
dagger 119954119954120582120582 (214)
These equations will be used below
3 Inter-phonon coupling within harmonic approximation 31 Hamiltonian with inter-phonon coupling
In this section we consider systems that consist of two (or more) covalently bonded units that are
weakly coupled between them Unlike previous studies that considered phonon-phonon couplings
resulting from anharmonicity effects [1] here all phonon mode considered are Harmonic and the
couplings arise from the division of the entire system into subunits The Hamiltonian of the whole
system can be written as
119867119867 = 1198671198671 + 1198671198672 + 11986711986712 (31)
To derive the expressions of 1198671198671 1198671198672 and 11986711986712 in Eq (31) we consider the Hamiltonian of the
whole system written as the function of the atomic displacements
119867119867 = 119879119879 + 119881119881 = sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)2119899119899119899119899120572120572 + 1
2sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime120572120572120572120572prime119899119899119899119899prime119899119899119899119899prime (32)
6
Letrsquos assume that subsystem I and II contain atoms with indeices ranging from 119904119904 = 12⋯ 1199031199032 and
119904119904 = 1199031199032
+ 1 1199031199032
+ 2⋯ 119903119903 respectively Then the kinetic energy term in Eq (32) can be rewritten as
119879119879 = sum sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)2
1199031199032119899119899=1 + sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)
2119903119903119899119899=1199031199032+1
119899119899120572120572 (33)
While the potential energy term is
119881119881 =12 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
1199031199032
119899119899119899119899prime=1
+ 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
119903119903
119899119899119899119899prime=1199031199032+1
+ 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime119899119899119899119899prime
+ 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
1199031199032
119899119899prime=1
119903119903
119899119899=1199031199032+1
= 11988111988111 + 11988111988122 + 11988111988112 + 11988111988121
Or equivalently
⎩⎪⎪⎨
⎪⎪⎧ 11988111988111 = 1
2sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
1199031199032119899119899119899119899prime=1120572120572120572120572prime119899119899119899119899prime
11988111988122 = 12sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
119903119903119899119899119899119899prime=1199031199032+1120572120572120572120572prime119899119899119899119899prime
11988111988112 = 12sum sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
119903119903119899119899prime=1199031199032+1
1199031199032119899119899=1120572120572120572120572prime119899119899119899119899prime
11988111988121 = 12sum sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
1199031199032119899119899prime=1
119903119903119899119899=1199031199032+1120572120572120572120572prime119899119899119899119899prime
(34)
32 Second quantization
We may now quantize this Hamiltonian and write it in the basis of the eigenstates of the coupled
subunits In second quantization the atomic displacement operators of the two subunits are given in
the following form
119906119906119899119899119899119899120572120572 =
⎩⎨
⎧ sum ℏ
2119873119873119872119872119904119904ω119954119954120582120582119890119890119899119899120572120572120582120582 119890119890119894119894119954119954∙119929119929119899119899119886119886119954119954120582120582 + 119886119886minus119954119954120582120582
dagger 119954119954120582120582 119904119904 isin 1 1199031199032
sum ℏ
2119873119873119872119872119904119904120596120596119954119954prime120582120582prime119890119899119899120572120572120582120582
prime119890119890119894119894119954119954prime∙119929119929119899119899 119886119886 119954119954prime120582120582prime + 119886119886minus119954119954prime120582120582primedagger 119954119954prime120582120582prime 119904119904 isin 119903119903
2+ 1 119903119903
(35)
Here we use different notations for the creation and annihilation operators (119886119886119954119954120582120582119886119886minus119954119954120582120582dagger and 119886119886119954119954120582120582119886119886minus119954119954120582120582
dagger )
eigenvalues (ω119954119954120582120582120596120596119954119954120582120582) and eigenvectors (119890119890119899119899120572120572120582120582 119890119899119899120572120572120582120582 ) for the two subunits Note that 119890119890119899119899120572120572120582120582 and 119890119899119899120572120572120582120582 are
of the dimensions of the whole system however their non-zero elements appear only on the relevant
subunits such that they obey the following relations sum 119890119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582
prime119899119899120572120572 = 120575120575120582120582120582120582prime sum 119890119899119899120572120572120582120582
lowast119890119899119899120572120572120582120582
prime119899119899120572120572 =
120575120575120582120582120582120582prime sum 119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582
prime119899119899120572120572 = sum 119890119890119899119899120572120572120582120582
lowast119890119899119899120572120572120582120582
prime119899119899120572120572 = 0 Defining the normal coordinates of the two subunits as
119876119876119954119954120582120582 equiv ℏ
2ω119954119954120582120582119886119886119954119954120582120582 + 119886119886minus119954119954120582120582
dagger 119876119876119954119954120582120582 equiv ℏ
2ω119954119954120582120582119886119886119954119954120582120582 + 119886119886minus119954119954120582120582
dagger (36)
and the corresponding momenta operators as
7
119875119875119954119954120582120582 = 119894119894ℏω119954119954120582120582
2119886119886119954119954120582120582
dagger minus 119886119886minus119954119954120582120582 119875119875119954119954120582120582 = 119894119894ℏω1199541199541205821205822
119886119886119954119954120582120582dagger minus 119886119886minus119954119954120582120582 (37)
Eq (35) reads
119906119906119899119899119899119899120572120572 = sum 119890119890119904119904119899119899120582120582 (119954119954)
119873119873119872119872119904119904119890119890119894119894119954119954∙119929119929119899119899119876119876119954119954120582120582119954119954120582120582 119904119904 isin 1 119903119903
2
sum 119890119904119904119899119899120582120582 (119954119954)119873119873119872119872119904119904
119890119890119894119894119954119954∙119929119929119899119899119876119876119954119954120582120582119954119954120582120582 119904119904 isin 1199031199032
+ 1 119903119903 (38)
The second quantized kinetic energy operator is then written as
119879119879 = 1198791198791 + 1198791198792 = 12sum 119875119875119954119954120582120582119875119875minus119954119954120582120582119954119954120582120582 + 1
2sum 119875119875119954119954prime120582120582prime119875119875minus119954119954prime120582120582prime119954119954prime120582120582prime (39)
Correspondingly the various potential energy terms are obtained by substituting Eq (38) in Eq (34)
11988111988111 =12 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119899119899119899119899prime
119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime120582120582prime (119954119954prime)
119872119872119899119899119872119872119899119899prime
1119873119873119890119890119894119894119954119954∙119929119929119899119899+119894119894119954119954prime∙119929119929119899119899prime119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954120582120582119954119954prime120582120582prime
1199031199032
119899119899119899119899prime=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954119954119954prime120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (119954119954prime)1119873119873 119890119890
119894119894(119954119954+119954119954prime)119929119929119899119899prime
119899119899prime120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime
119872119872119899119899119872119872119899119899prime119899119899
1199031199032
119899119899119899119899prime=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954119954119954prime120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (119954119954prime) 120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime
1199031199032
119899119899119899119899prime=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime
119954119954120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (minus119954119954)120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
1199031199032
119899119899119899119899prime=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime
119954119954120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954) 119863119863119899119899120572120572119899119899
prime120572120572prime(minus119954119954)119890119890119899119899prime120572120572prime120582120582prime (minus119954119954)
120572120572prime
1199031199032
119899119899prime=1
1199031199032
119899119899=1120572120572
=12 ω119954119954120582120582prime
2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582120582120582prime
119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582prime (119954119954)
lowast
1199031199032
119899119899=1
=120572120572
12ω119954119954120582120582
2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582
ie
11988111988111 = 12sum ω119954119954120582120582
2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582 (310)
Going from the second line to the third line we used the fact that the sum over 119899119899prime runs between plusmninfin
and the summand depends only on the difference between 119899119899 and 119899119899prime hence the sum is independent
of the value of the index 119899119899 Therefore we can replace the sum over 119899119899prime by a sum over 119897119897 equiv 119899119899 minus 119899119899prime
amd define 119929119929119897119897 equiv 119929119929119899119899 minus 119929119929119899119899prime Following the same procedure we can get the corresponding
expressions for the second diagonal term
11988111988122 = 12sum 1205961205961199541199541205821205822 119876119876119954119954120582120582119876119876minus119954119954120582120582119954119954120582120582 (311)
Similarly for the off-diagonal terms we get
8
11988111988112 =12 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119899119899119899119899prime
119903119903
119899119899prime=1199031199032+1
119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime
120582120582prime (119954119954prime)
119872119872119899119899119872119872119899119899prime
1119873119873119890119890119894119894119954119954∙119929119929119899119899+119894119894119954119954prime∙119929119929119899119899prime119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954120582120582119954119954prime120582120582prime
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954119954119954prime120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime
120582120582prime (119954119954prime)1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime
119872119872119899119899119872119872119899119899prime119899119899
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954119954119954prime120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime
120582120582prime (119954119954prime) 120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime
119954119954120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime
120582120582prime (minus119954119954)120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954
119890119899119899prime120572120572prime120582120582prime (119954119954)
lowast120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119890119890119899119899120572120572120582120582 (119954119954)119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954
Where we have defined
119881119881120582120582120582120582prime(119954119954) equiv sum sum sum 119890119899119899prime120572120572prime120582120582prime (119954119954)
lowast119881119881119899119899120572120572119899119899
prime120572120572prime(119954119954)119890119890119899119899120572120572120582120582 (119954119954)119903119903119899119899prime=1199031199032+1
1199031199032119899119899=1120572120572120572120572prime (312)
where
119881119881119899119899120572120572119899119899prime120572120572prime(119954119954) equiv sum 119890119890119894119894119954119954∙119929119929119897119897
119872119872119904119904119872119872119904119904prime119897119897 120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime (313)
Itrsquos easy to show that 119881119881120582120582120582120582prime(119954119954) has the following property
119881119881120582120582120582120582primelowast (119954119954) = 119881119881120582120582120582120582prime(minus119954119954) (314)
Following the same procedure we have
9
11988111988121 =12 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119899119899119899119899prime
119903119903
119899119899=1199031199032+1
119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (119954119954prime)
119872119872119899119899119872119872119899119899prime
1119873119873119890119890119894119894119954119954∙119929119929119899119899+119894119894119954119954prime∙119929119929119899119899prime119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954120582120582119954119954prime120582120582prime
1199031199032
119899119899prime=1120572120572120572120572prime
=12 120601120601120572120572prime120572120572
119899119899prime minus 119899119899119904119904prime 119904119904
119899119899prime119899119899
119903119903
119899119899prime=1199031199032+1
119890119899119899prime120572120572prime120582120582prime (119954119954prime)119890119890119899119899120572120572120582120582 (119954119954)
119872119872119899119899prime119872119872119899119899
1119873119873 119890119890
119894119894119954119954prime∙119929119929119899119899prime+119894119894119954119954∙119929119929119899119899119876119876119954119954prime120582120582prime119876119876119954119954120582120582119954119954prime120582120582prime119954119954120582120582
1199031199032
119899119899=1120572120572prime120572120572
=12 119876119876119954119954prime120582120582prime119876119876119954119954120582120582
119954119954119954119954prime120582120582120582120582prime119890119899119899prime120572120572prime120582120582prime (119954119954prime)119890119890119899119899120572120572120582120582 (119954119954)
1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime
119872119872119899119899119872119872119899119899prime119899119899
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954prime120582120582prime119876119876119954119954120582120582
119954119954119954119954prime120582120582120582120582prime119890119899119899prime120572120572prime120582120582prime (119954119954prime)119890119890119899119899120572120572120582120582 (119954119954) 120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954prime120582120582prime119876119876minus119954119954prime120582120582119890119899119899prime120572120572prime
120582120582prime (119954119954prime)119890119890119899119899120572120572120582120582 (minus119954119954prime)120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890minus119894119894119954119954prime∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897119954119954prime120582120582120582120582prime
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954prime120582120582prime119876119876119954119954prime120582120582
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954prime 119890119890119899119899120572120572120582120582 (119954119954prime)
lowast120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890minus119894119894119954119954prime∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119890119899119899prime120572120572prime120582120582prime (119954119954prime)
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582prime119876119876119954119954120582120582
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954
119890119890119899119899120572120572120582120582 (119954119954)lowast120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890minus119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119890119899119899prime120572120572prime120582120582prime (119954119954)
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger dagger
3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954
119890119899119899prime120572120572prime120582120582prime (119954119954)
lowast120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119890119890119899119899120572120572120582120582 (119954119954)119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
lowast
=
⎩⎨
⎧12 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954⎭⎬
⎫dagger
= 11988111988112dagger
Define
1198671198671 = 11987911987911 + 119881119881111198671198672 = 11987911987922 + 1198811198812211986711986712 = 11988111988112 + 11988111988121
(315)
we finally get the expressions of 1198671198671 1198671198672 and 11986711986712 in Eq (31) as follows
⎩⎪⎨
⎪⎧ 1198671198671 = sum sum ℏ120596120596119954119954120582120582 119886119886119954119954120582120582
dagger 119886119886119954119954120582120582 + 1231199031199032
120582120582=1119954119954
1198671198672 = sum sum ℏ120596120596119954119954120582120582prime 119886119886119954119954120582120582primedagger 119886119886119954119954120582120582prime + 1
23119903119903
120582120582prime=1+31199031199032119954119954
11986711986712 = 12sum sum sum 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903120582120582prime=31199031199032 +1
31199031199032120582120582=1 + h c 119954119954
(316)
Here hc means the Hermitian conjugate Since the indices of 119886119886119954119954120582120582 119886119886119954119954120582120582prime and 119876119876119954119954120582120582 119876119876119954119954120582120582prime belong to
the two system sections we can define an abbreviated notation 119886119886119954119954120582120582 and 119876119876119954119954120582120582 using the index 120582120582 to
identify which subsystem they belong to In this case the Hamiltonian of the coupled system can be
10
simplified as follows
119867119867 = 1198671198670 + 119867119867119862119862 (317)
where
1198671198670 = sum sum ℏ120596120596119954119954120582120582 119886119886119954119954120582120582
dagger 119886119886119954119954120582120582 + 123119903119903
120582120582=1119954119954
119867119867119862119862 = 12sum sum sum 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903120582120582prime=31199031199032 +1
31199031199032120582120582=1 + ℎ 119888119888 119954119954
(318)
4 Greenrsquos function 41 Dynamics of the ladder operators
In order to describe the dynamics of the ladder operators appearing in the Hamiltonian of Eq (317)
we express them in the Heisenberg picture as follows
119886119886119901119901(120591120591) = 119890119890119894119894ℏ119867119867119905119905119886119886119901119901119890119890
minus119894119894ℏ119867119867119905119905 = 119890119890
119867119867120591120591ℏ 119886119886119901119901119890119890
minus119867119867120591120591ℏ (41)
where we define the imaginary time 120591120591 equiv 119894119894119905119905 and introduce the notation 119953119953 equiv (119954119954 120582120582) and 119953119953 equiv (minus119954119954 120582120582)
The corresponding equation of motion for the ladder operators is given by
ℏ part119886119886119953119953(120591120591)
120597120597120591120591= 119867119867119886119886119953119953(120591120591) (42)
For the uncoupled system (119867119867119862119862 = 0) this gives
ℏpart119886119886119953119953(120591120591)120597120597120591120591
= 1198671198670119886119886119953119953(120591120591) = 1198671198670 1198901198901198671198670120591120591ℏ 119886119886119953119953119890119890
minus1198671198670120591120591ℏ = 1198671198670119890119890
1198671198670120591120591ℏ 119886119886119953119953119890119890
minus1198671198670120591120591ℏ minus 119890119890
1198671198670120591120591ℏ 119886119886119953119953119890119890
minus1198671198670120591120591ℏ 1198671198670
= 1198901198901198671198670120591120591ℏ 1198671198670119886119886119953119953 minus 1198861198861199531199531198671198670119890119890
minus1198671198670120591120591ℏ = 119890119890
1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890
minus1198671198670120591120591ℏ
ie
ℏ part119886119886119953119953(120591120591)
120597120597120591120591= 119890119890
1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890
minus1198671198670120591120591ℏ (43)
The commutator on the right-hand-side of Eq (43) can be evaluated as follows
1198671198670119886119886119953119953prime = sum ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953 + 1
2119953119953 119886119886119953119953prime = sum ℏ120596120596119953119953119886119886119953119953
dagger119886119886119953119953119886119886119953119953prime119953119953 = sum ℏ120596120596119953119953119886119886119953119953dagger119886119886119953119953119886119886119953119953prime minus 119886119886119953119953prime119886119886119953119953
dagger119886119886119953119953119953119953 =
sum ℏ120596120596119953119953119886119886119953119953dagger119886119886119953119953119886119886119953119953prime minus 120575120575119953119953prime119953119953 + 119886119886119953119953
dagger119886119886119953119953prime119886119886119953119953119953119953 = minusℏ120596120596119953119953prime119886119886119953119953prime + sum ℏ120596120596119953119953119886119886119953119953dagger119886119886119953119953119886119886119953119953prime minus 119886119886119953119953
dagger119886119886119953119953119886119886119953119953prime119953119953 = minusℏ120596120596119953119953prime119886119886119953119953prime
ie
1198671198670119886119886119953119953prime = minusℏ120596120596119953119953prime119886119886119953119953prime (44)
Therefore we have
ℏ part119886119886119953119953(120591120591)
120597120597120591120591= 119890119890
1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890
minus1198671198670120591120591ℏ = minusℏ120596120596119953119953119890119890
1198671198670120591120591ℏ 119886119886119953119953119890119890
minus1198671198670120591120591ℏ = minusℏ120596120596119953119953119886119886119953119953(120591120591) (45)
11
The solution of Eq (45) is 119886119886119953119953(120591120591) = 119886119886119953119953(0)119890119890minus120596120596119953119953120591120591 = 119886119886119953119953119890119890minus120596120596119953119953120591120591 For 119886119886119953119953dagger(120591120591) we have
1198671198670119886119886119953119953primedagger (0) = 1198671198670119886119886119953119953prime
dagger = ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953 +
12
119953119953
119886119886119953119953primedagger = ℏ120596120596119953119953119886119886119953119953
dagger119886119886119953119953119886119886119953119953primedagger
119953119953
= ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953119886119886119953119953prime
dagger minus 119886119886119953119953primedagger 119886119886119953119953
dagger119886119886119953119953119953119953
= ℏ120596120596119953119953 119886119886119953119953dagger 120575120575119953119953119953119953prime + 119886119886119953119953prime
dagger 119886119886119953119953 minus 119886119886119953119953primedagger 119886119886119953119953
dagger119886119886119953119953119953119953
= ℏ120596120596119953119953prime119886119886119953119953primedagger + ℏ120596120596119953119953 119886119886119953119953
dagger119886119886119953119953primedagger 119886119886119953119953 minus 119886119886119953119953prime
dagger 119886119886119953119953dagger119886119886119953119953
119953119953
= ℏ120596120596119953119953prime119886119886119953119953primedagger
In summary we have
119886119886119953119953(120591120591) = 119886119886119953119953(0)119890119890minus120596120596119953119953120591120591 = 119886119886119953119953119890119890minus120596120596119953119953120591120591
119886119886119953119953dagger(120591120591) = 119886119886119953119953
dagger(0)119890119890120596120596119953119953120591120591 = 119886119886119953119953dagger119890119890120596120596119953119953120591120591 (46)
42 Greenrsquos function
To describe thermal transport properties between different system sections we use the formalism of
thermal (or imaginary time) phonon Greenrsquos function [2] To this end we define the thermal Greenrsquos
function for phonons as
119866119866119953119953119953119953prime(120591120591 120591120591prime) = minuslang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime)rang = minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime) (47)
where 119879119879120591120591 is the time ordering operator and 120588120588119867119867 is the statistical operator for the grand canonical
ensemble (note that the chemical potential for phonons is zero)
120588120588119867119867 = 119890119890minus120573120573119867119867119885119885119867119867 (48)
where the partition function is given by 119885119885119867119867 equiv Tr119890119890minus120573120573119867119867 and 120573120573 = 1119896119896119861119861119879119879 with 119896119896119861119861 being
Boltzmannrsquos constant and 119879119879 the temperature For time independent Hamiltonians 119866119866119953119953119953119953prime(120591120591 120591120591prime)
depends only on 120591120591 minus 120591120591prime ie
119866119866119953119953119953119953prime(120591120591 120591120591prime) = 119866119866119953119953119953119953prime(120591120591 minus 120591120591prime 0) (49)
To show this we shall first assume that 120591120591 gt 120591120591prime such that
119866119866119953119953119953119953prime(120591120591 120591120591prime) = minuslang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime)rang = minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime) = minusTr 120588120588119867119867119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime)
= minusTr 120588120588119867119867119890119890119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591ℏ 119890119890
119867119867120591120591primeℏ 119886119886119953119953prime
dagger 119890119890minus119867119867120591120591primeℏ = minusTr 120588120588119867119867119890119890
minus119867119867120591120591primeℏ 119890119890
119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591minus120591120591prime
ℏ 119886119886119953119953primedagger
= minusTr 120588120588119867119867119890119890119867119867120591120591minus120591120591prime
ℏ 119886119886119953119953119890119890minus119867119867120591120591minus120591120591prime
ℏ 119886119886119953119953primedagger = minusTr 120588120588119867119867119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953prime
dagger (0)
= minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953primedagger (0) = minus lang119879119879120591120591119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953prime
dagger (0)rang = 119866119866119953119953119953119953prime(120591120591 minus 120591120591prime 0)
Where we have used the commutativity of 120588120588119867119867 and 119890119890plusmn119867119867120591120591prime
ℏ and the invariance of the trace operation
12
towards cyclic permutations Similarly for 120591120591 lt 120591120591prime we have
119866119866119953119953119953119953prime(120591120591 120591120591prime) = minuslang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime)rang = minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime) = minusTr 120588120588119867119867119886119886119953119953primedagger (120591120591prime)119886119886119953119953(120591120591)
= minusTr 120588120588119867119867119890119890119867119867120591120591primeℏ 119886119886119953119953prime
dagger 119890119890minus119867119867120591120591primeℏ 119890119890
119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591ℏ = minusTr 120588120588119867119867119886119886119953119953prime
dagger 119890119890119867119867120591120591minus120591120591prime
ℏ 119886119886119953119953119890119890minus119867119867120591120591ℏ 119890119890
119867119867120591120591primeℏ
= minusTr 120588120588119867119867119886119886119953119953primedagger 119890119890
119867119867120591120591minus120591120591primeℏ 119886119886119953119953119890119890
minus119867119867120591120591minus120591120591prime
ℏ = minusTr 120588120588119867119867119886119886119953119953primedagger (0)119886119886119953119953(120591120591 minus 120591120591prime)
= minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953primedagger (0)119886119886119953119953(120591120591 minus 120591120591prime) = minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953prime
dagger (0)
= minuslang119879119879120591120591119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953primedagger (0)rang = 119866119866119953119953119953119953prime(120591120591 minus 120591120591prime 0)
For simplicity we will introduce the notation 119866119866119953119953119953119953prime(120591120591 0) equiv 119866119866119953119953119953119953prime(120591120591) We further note that when using
imaginary time 119866119866119953119953119953119953prime(120591120591) is a periodic functions in the domain [minus120573120573ℏ120573120573ℏ] with a period of 120573120573ℏ (see
Page 236 of Ref [2])
119866119866119953119953119953119953prime(120591120591 + 120573120573ℏ 0) = 119866119866119953119953119953119953prime(120591120591 0) 120591120591 lt 0119866119866119953119953119953119953prime(120591120591 minus 120573120573ℏ 0) = 119866119866119953119953119953119953prime(120591120591 0) 120591120591 gt 0 (410)
To show this we shall again assume first that minus120573120573ℏ lt 120591120591 lt 0 to write that is
119866119866119953119953119953119953prime(120591120591 + 120573120573ℏ 0) = minus lang119879119879120591120591119886119886119953119953(120591120591 + 120573120573ℏ)119886119886119953119953primedagger (0)rang = minuslang119886119886119953119953(120591120591 + 120573120573ℏ)119886119886119953119953prime
dagger (0)rang
= minusTr 120588120588119867119867119890119890119867119867(120591120591+120573120573ℏ)
ℏ 119886119886119953119953119890119890minus119867119867(120591120591+120573120573ℏ)
ℏ 119886119886119953119953prime = minusTr 120588120588119867119867119890119890120573120573119867119867119890119890
119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591ℏ 119890119890minus120573120573119867119867119886119886119953119953prime
= minusTr120588120588119867119867119890119890120573120573119867119867119886119886119953119953(120591120591)119890119890minus120573120573119867119867119886119886119953119953prime(0) = minus
1119885119885119867119867
Tr119890119890minus120573120573119867119867119890119890120573120573119867119867119886119886119953119953(120591120591)119890119890minus120573120573119867119867119886119886119953119953prime(0)
= minus1119885119885119867119867
Tr119886119886119953119953(120591120591)119890119890minus120573120573119867119867119886119886119953119953prime(0) = minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119886119886119953119953prime(0)119886119886119953119953(120591120591)
= minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime(0) = minusTr120588120588119867119867119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime = 119866119866119953119953119953119953prime(120591120591 0)
Similarly for 120573120573ℏ gt 120591120591 gt 0 we have
119866119866119953119953119953119953prime(120591120591 minus 120573120573ℏ 0) = minus lang119879119879120591120591119886119886119953119953(120591120591 minus 120573120573ℏ)119886119886119953119953primedagger (0)rang = minuslang119886119886119953119953prime
dagger (0)119886119886119953119953(120591120591 minus 120573120573ℏ)rang
= minusTr 120588120588119867119867119886119886119953119953prime119890119890119867119867(120591120591minus120573120573ℏ)
ℏ 119886119886119953119953119890119890minus119867119867(120591120591minus120573120573ℏ)
ℏ = minusTr 120588120588119867119867119886119886119953119953prime119890119890minus120573120573119867119867119890119890
119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591ℏ 119890119890120573120573119867119867
= minusTr120588120588119867119867119886119886119953119953prime(0)119890119890minus120573120573119867119867119886119886119953119953(120591120591)119890119890120573120573119867119867 = minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119886119886119953119953prime(0)119890119890minus120573120573119867119867119886119886119953119953(120591120591)119890119890120573120573119867119867
= minus1119885119885119867119867
Tr119886119886119953119953prime(0)119890119890minus120573120573119867119867119886119886119953119953(120591120591) = minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119886119886119953119953(120591120591)119886119886119953119953prime(0)
= minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime(0) = minusTr120588120588119867119867119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime = 119866119866119953119953119953119953prime(120591120591 0)
13
Therefore 119866119866119953119953119953119953prime(120591120591) can be expanded as a Fourier series in the domain [0120573120573ℏ] as follows
119866119866119953119953119953119953prime(120591120591) = 1120573120573ℏsum 119890119890minus119894119894120596120596119899119899120591120591119866119866119953119953119953119953prime(119894119894120596120596119899119899)infinminusinfin (411)
where 120596120596119899119899 = 2120587120587119899119899120573120573ℏ
and the associated Fourier coefficient is given by
119866119866119953119953119953119953prime(119894119894120596120596119899119899) = int d120591120591119890119890119894119894120596120596119899119899120591120591119866119866119953119953119953119953prime(120591120591)120573120573ℏ0 120596120596119899119899 = 2119899119899120587120587
120573120573ℏ (412)
Having proven the translational time invariance and the periodicity of the Greenrsquos functios we can
now calculate it for the uncoupled system
119866119866119953119953119953119953prime0 (120591120591 120591120591prime) = minuslang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime)rang0 == minus lang119886119886119953119953(0)119890119890minus120596120596119953119953120591120591119886119886119953119953prime
dagger (0)119890119890120596120596119953119953prime120591120591primerang 120591120591 minus 120591120591prime gt 0
minus lang119886119886119953119953primedagger (0)119890119890120596120596119953119953prime120591120591
prime119886119886119953119953(0)119890119890minus120596120596119953119953120591120591rang 120591120591 minus 120591120591prime lt 0
= minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591
primelang119886119886119953119953(0)119886119886119953119953prime
dagger (0)rang 120591120591 minus 120591120591prime gt 0
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang 120591120591 minus 120591120591prime lt 0
= minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591
primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0) + 119886119886119953119953(0)119886119886119953119953primedagger (0)rang 120591120591 minus 120591120591prime gt 0
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang 120591120591 minus 120591120591prime lt 0
= minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591
primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang + 120575120575119953119953119953119953prime 120591120591 minus 120591120591prime gt 0
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang 120591120591 minus 120591120591prime lt 0
ie
119866119866119953119953119953119953prime0 (120591120591 120591120591prime) =
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang + 120575120575119953119953119953119953prime 120591120591 minus 120591120591prime gt 0
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang 120591120591 minus 120591120591prime lt 0 (413)
where we have used Eq (46) for 119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime) and Commutation relation 119886119886119953119953(0)119886119886119953119953prime
dagger (0) = 120575120575119953119953119953119953prime
To calculate lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang we first prove following equation
119890119890119860119860119861119861119890119890minus119860119860 = sum 1119899119899119860(119899119899)119861119861infin
119899119899=0 (414)
where 119860(119896119896)119861119861 equiv 119860 119860(119896119896minus1)119861119861 To prove this identity we define the operator 119891119891(119905119905) = 119890119890119905119905A119861119861119890119890minus119905119905119860119860
and taylor expand it around 119905119905 = 0
119891119891(119905119905) = 119891119891(0) + 119905119905119891119891prime(0) + 1199051199052
2119891119891primeprime(0) + ⋯ = sum 119905119905119899119899
119899119899119889119889119899119899119891119889119889119905119905119899119899
119905119905=0
infin119899119899=0 (415)
The corresponding derivatives are given by
⎩⎪⎨
⎪⎧ 119891119891prime(119905119905) = 119890119890119905119905119860119860119860119861119861119890119890minus119905119905119860119860 minus 119890119890119905119905119860119860119861119861119860119890119890minus119905119905119860119860 = 119890119890119905119905119860119860119860119861119861119890119890minus119905119905119860119860
119891119891primeprime(119905119905) = 119890119890119905119905119860119860119860119860119861119861 minus 119860119861119861119860119890119890minus119905119905119860119860 = 119890119890119905119905119860119860119860(2)119861119861119890119890minus119905119905119860119860
⋮119891119891(119899119899)(119905119905) = 119890119890119905119905119860119860119860(119899119899)119861119861119890119890minus119905119905119860119860
(416)
14
Substituting Eq (416) into Eq (415) we have
119890119890119905119905A119861119861119890119890minus119905119905119860119860 = sum 119905119905119899119899
119899119899119860(119899119899)119861119861infin
119899119899=0 (417)
Itrsquos clear that Eq (413) is a spectial case of Eq (416) with 119905119905 = 1 With this we can proceed as
follows
lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang =
11198851198851198671198670
Tr119890119890minus1205731205731198671198670119886119886119953119953primedagger (0)119886119886119953119953(0) =
11198851198851198671198670
Tr119890119890minus1205731205731198671198670119886119886119953119953primedagger (0)1198901198901205731205731198671198670119890119890minus1205731205731198671198670119886119886119953119953(0)
=11198851198851198671198670
Tr (minus120573120573)119899119899
1198991198991198671198670
(119899119899)119886119886119953119953primedagger (0)
infin
119899119899=0
119890119890minus1205731205731198671198670119886119886119953119953(0)
Note that 1198671198670119886119886119953119953primedagger (0) = ℏ120596120596119953119953prime119886119886119953119953prime
dagger (0) we have
1198671198670(119899119899)119886119886119953119953prime
dagger (0) = ℏ120596120596119953119953prime119899119899119886119886119953119953primedagger (0) (418)
such that
lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang =
11198851198851198671198670
Tr (minus120573120573)119899119899
1198991198991198671198670
(119899119899)119886119886119953119953primedagger (0)
infin
119899119899=0
119890119890minus1205731205731198671198670119886119886119953119953(0)
=11198851198851198671198670
Tr minus120573120573ℏ120596120596119953119953prime
119899119899
119899119899119886119886119953119953primedagger (0)
infin
119899119899=0
119890119890minus1205731205731198671198670119886119886119953119953(0)
= 119890119890minus120573120573ℏ120596120596119953119953prime11198851198851198671198670
Tr119890119890minus1205731205731198671198670119886119886119953119953(0)119886119886119953119953primedagger (0) = 119890119890minus120573120573ℏ120596120596119953119953primelang119886119886119953119953(0)119886119886119953119953prime
dagger (0)rang
= 119890119890minus120573120573ℏ120596120596119953119953primelang119886119886119953119953primedagger (0)119886119886119953119953(0)rang + 120575120575119953119953119953119953prime
therefore we obtain lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang 119890119890120573120573ℏ120596120596119953119953prime minus 1 = 120575120575119953119953119953119953prime For 119953119953prime ne 119953119953 and general 119890119890120573120573ℏ120596120596119953119953prime we have
lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang = 0 For 119953119953prime = 119953119953 we obtain lang119886119886119953119953
dagger(0)119886119886119953119953(0)rang = 119890119890120573120573ℏ120596120596119953119953 minus 1minus1
Hence we have
lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang = 120575120575119953119953prime119953119953
1119890119890120573120573ℏ120596120596119953119953minus1
equiv 120575120575119953119953prime119953119953119899119899119861119861120596120596119953119953 (419)
where 119899119899119861119861120596120596119953119953 is the Bose-Einstein distribution for phonons Substituting Eq (418) in Eq (413)
we obtain
119866119866119953119953119953119953prime0 (120591120591 120591120591prime) =
minus120575120575119953119953119953119953prime1 + 119899119899119861119861120596120596119953119953119890119890minus120596120596119953119953120591120591minus120591120591prime 120591120591 minus 120591120591prime gt 0
minus120575120575119953119953119953119953prime119899119899119861119861120596120596119953119953119890119890minus120596120596119953119953120591120591minus120591120591prime 120591120591 minus 120591120591prime lt 0 (420)
Note that only those Greenrsquos functions of the form 119866119866119953119953119953119953prime0 (120591120591 120591120591prime) = 1198661198661199531199530(120591120591 minus 120591120591prime)120575120575119953119953119953119953prime are non-zero due
to the orthogonality of normal modes Looking at the non-vanishing terms and setting 120591120591prime = 0
without loss of generality we can calculate the Fourier transform of 1198661198661199531199530(120591120591) from Eq (412)
1198661198661199531199530(119894119894120596120596119899119899) = int d1205911205911198901198901198941198941205961205961198991198991205911205911198661198661199531199530(120591120591)120573120573ℏ0 = minusint d120591120591119890119890119894119894120596120596119899119899120591120591120573120573ℏ
0 1 + 119899119899119861119861120596120596119953119953119890119890minus120596120596119953119953120591120591 = minus1+119899119899119861119861120596120596119953119953119894119894120596120596119899119899minus120596120596119953119953
119890119890119894119894120596120596119899119899minus120596120596119953119953120573120573ℏ minus
15
1 = minus1+ 1
119890119890120573120573ℏ120596120596119953119953minus1
119894119894120596120596119899119899minus120596120596119953119953119890119890119894119894
2120587120587119899119899120573120573ℏ 120573120573ℏ119890119890minus120573120573ℏ120596120596119953119953 minus 1 = minus 119890119890120573120573ℏ120596120596119953119953
119894119894120596120596119899119899minus120596120596119953119953
119890119890minus120573120573ℏ120596120596119953119953minus1
119890119890120573120573ℏ120596120596119953119953minus1= minus 1
119894119894120596120596119899119899minus120596120596119953119953
1minus119890119890120573120573ℏ120596120596119953119953
119890119890120573120573ℏ120596120596119953119953minus1= 1
119894119894120596120596119899119899minus120596120596119953119953
ie
1198661198661199531199530(119894119894120596120596119899119899) == 1119894119894120596120596119899119899minus120596120596119953119953
(421)
5 Fermirsquos golden rule 51 Interaction picture
Next we can proceed with calculating the Greenrsquos function of the coupled system To this end we
define the coupling Hamiltonian operator term in the interaction picture as
119867119867119862119862119868119868 (120591120591) = 1198901198901198671198670120591120591ℏ 119867119867119862119862119890119890
minus1198671198670120591120591ℏ (51)
The time evolution of 119867119867119862119862(120591120591) is then given by
ℏ part119867119867119862119862119868119868 (120591120591)120597120597120591120591
= 1198671198670119867119867119862119862119868119868 (120591120591) (52)
Note that the Greenrsquos function defined above was given in the Heisenberg picture To proceed we
need to transform it to the interaction picture
119866119866119953119953119953119953prime(120591120591 0) = minus lang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953primedagger (0)rang = minusTr 120588120588119867119867119879119879120591120591 119890119890
119867119867120591120591ℏ 119886119886119953119953(0)119890119890minus
119867119867120591120591ℏ 119886119886119953119953prime
dagger (0)
= minusTr 120588120588119867119867119879119879120591120591 119890119890119867119867120591120591ℏ 119890119890minus
1198671198670120591120591ℏ 119890119890
1198671198670120591120591ℏ 119886119886119953119953(0)119890119890minus
1198671198670120591120591ℏ 119890119890
1198671198670120591120591ℏ 119890119890minus
119867119867120591120591ℏ 119886119886119953119953prime
dagger (0)
= minusTr 120588120588119867119867119879119879120591120591 119880119880(0 120591120591)119886119886119953119953119868119868 (120591120591)119880119880(120591120591 0)119886119886119953119953primedagger (0)
where
119886119886119953119953119868119868 (120591120591) equiv 1198901198901198671198670120591120591ℏ 119886119886119953119953(0)119890119890minus
1198671198670120591120591ℏ (53)
is the operator in interaction picture and the operator 119880119880 is defined by
119880119880(1205911205911 1205911205912) equiv 11989011989011986711986701205911205911ℏ 119890119890minus
119867119867 (1205911205911minus1205911205912)ℏ 119890119890minus
11986711986701205911205912ℏ (54)
Note that while 119880119880 is not unitary it satisfies the following group property
119880119880(1205911205911 1205911205912)119880119880(1205911205912 1205911205913) = 119880119880(1205911205911 1205911205913) (55)
and the boundary condition 119880119880(1205911205911 1205911205911) = 1 In addition the 120591120591 derivative of 119880119880 is simply
ℏpart119880119880(120591120591 120591120591prime)
120597120597120591120591= 1198671198670119890119890
1198671198670120591120591ℏ 119890119890minus
119867119867120591120591minus120591120591primeℏ minus 119890119890
1198671198670120591120591ℏ 119867119867119890119890minus
119867119867120591120591minus120591120591primeℏ 119890119890minus
1198671198670120591120591primeℏ
= 1198901198901198671198670120591120591ℏ 1198671198670 minus 119867119867119890119890minus
1198671198670120591120591ℏ 119890119890
1198671198670120591120591ℏ 119890119890minus
119867119867120591120591minus120591120591primeℏ 119890119890minus
1198671198670120591120591primeℏ = 119890119890
1198671198670120591120591ℏ minus119867119867119862119862119890119890
minus1198671198670120591120591ℏ 119880119880(120591120591 120591120591prime)
= minus119867119867119862119862119868119868 (120591120591)119880119880(120591120591 120591120591prime)
16
ie
ℏ part119880119880120591120591120591120591prime120597120597120591120591
= minus119867119867119862119862119868119868 (120591120591)119880119880(120591120591 120591120591prime) (56)
The solution of Eq (56) is (see page 235 of Ref [2])
119880119880(120591120591 120591120591prime) = summinus1ℏ
119899119899
119899119899 int 1198891198891205911205911120591120591120591120591prime ⋯int 119889119889120591120591119899119899
120591120591120591120591prime 119879119879120591120591119867119867119862119862119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119899119899)infin
119899119899=0 (57)
The exact thermal Greens function now may be rewritten in the interaction picture as
119866119866119953119953119953119953prime(120591120591 0) = minus1119885119885119867119867
Tr 119890119890minus120573120573119867119867119879119879120591120591119880119880(0 120591120591)119886119886119953119953119868119868 (120591120591)119880119880(120591120591 0)119886119886119953119953primedagger (0)
= minusTr 119890119890minus120573120573119867119867119879119879120591120591 119880119880(0 120591120591)119886119886119953119953119868119868 (120591120591)119880119880(120591120591 0)119886119886119953119953prime
dagger (0)
Tr119890119890minus120573120573119867119867
= minusTr 119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)119879119879120591120591 119886119886119953119953119868119868 (120591120591)119880119880(0 120591120591)119880119880(120591120591 0)119886119886119953119953prime
dagger (0)
Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)
= minusTr 119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)119879119879120591120591 119886119886119953119953119868119868 (120591120591)119880119880(00)119886119886119953119953prime
dagger (0)
Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)=
= minusTr 119890119890minus1205731205731198671198670119879119879120591120591 119880119880(120573120573ℏ 0)119886119886119953119953119868119868 (120591120591)119886119886119953119953prime
dagger (0)
Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)
Where we used the fact that we are free to change the order of the operators within the time ordering
operation (see pages 241-242 of Ref [2])
52 Wickrsquos theorem
Thus the Greenrsquos function can be expanded as [2]
119866119866119953119953119953119953prime(120591120591 0) = minussum 1
119898119898minus1ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)119886119886119953119953119868119868 (120591120591)119886119886
119953119953primedagger (0)rang0infin
119898119898=0
sum 1119898119898minus
1ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)rang0infin
119898119898=0 (58)
where lang⋯ rang0 represents the ensemble average with respect to the non-interacting basis
Tr119890119890minus1205731205731198671198670(⋯ ) Or explicitly
119866119866119953119953119953119953prime(120591120591 0) = minus119866119866119953119953119953119953prime0 (120591120591)minus1ℏint 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953primedagger (0)rang0+
12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0+⋯
1minus1ℏint 1198891198891205911205911120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)rang0+12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0+⋯
(59)
For consiceness we introduce the following notation
119863119863119898119898 equiv 1119898119898minus 1
ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)rang0 (510)
To simplify the calculation in Eq (58) we adopt Wickrsquos theorem (see pages 237-241 of Ref
[2])which can be expressed as follows
lang119879119879120591120591119860119861119861119862119863119863 ⋯119884119884119885rang0 = lang119879119879120591120591119860119861119861rang0lang119879119879120591120591119862119863119863rang0⋯ lang119879119879120591120591119884119884119885rang0 + lang119879119879120591120591119860119862rang0lang119879119879120591120591119861119861119863119863rang0⋯ lang119879119879120591120591119883119883119885rang0 + ⋯ (511)
17
Here 119860119861119861 hellip 119884119884 119885 represent 119886119886119953119953(120591120591) or 119886119886119953119953dagger(120591120591) In Eq (511) each term corresponds to a particular
pairing of the operators 119860119861119861119862119863119863 ⋯119884119884119885 and all possible pairings are taken into account Here the only
non-vanishing propagators will have the form [see Eq (420)]
lang119879119879120591120591 119886119886119953119953primedagger (120591120591prime)119886119886119953119953(120591120591)rang0 = minuslang119879119879120591120591 119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime)rang0 = 119866119866119953119953119953119953prime0 (120591120591 120591120591prime) = 1198661198661199531199530(120591120591 minus 120591120591prime)120575120575119953119953119953119953prime (512)
Therefore the second term in the numerator of Eq (59) can be calculated as
1198681198682 = minus1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0
= minus1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 lang119879119879120591120591119886119886119953119953prime
dagger (0)119886119886119953119953(120591120591)rang0 minus1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
= 119866119866119953119953119953119953prime0 (120591120591 0) minus
1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 minus
1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
The first term in above equation is called disconnected part since the pairing is performed separately
on 119867119867119862119862(1205911205911) and 119886119886119953119953(120591120591)119886119886119953119953primedagger (0) All other terms have pairs that mix creation and annihilation operators
of the Hamiltonian with 119886119886119953119953(120591120591) or 119886119886119953119953primedagger (0) and are said to have connected party For simplicity we
include all these terms in the notation lang119879119879120591120591119867119867119862119862(1205911205911)119886119886119953119953(120591120591)119886119886119953119953primedagger (0)rang0119888119888 Furthermore we define 119866119866119953119953119953119953prime
(1) (120591120591) equiv
minus1ℏ int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862(1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 Then we have
1198681198683 = minus1ℏint 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 = 119866119866119953119953119953119953prime0 (120591120591 0)1198631198631 + 119866119866119953119953119953119953prime
(1) (120591120591) (513)
Using this method the third term in the numerator of Eq (59) can be calculated as
1198681198683 = 12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 = 12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0 lang119879119879120591120591119886119886119953119953prime
dagger (0)119886119886119953119953(120591120591)rang0 +
12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 + 12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 +
12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 = 119866119866119953119953119953119953prime0 (120591120591 0) 1
2ℏ2 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0 +
12ℏ2 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 + 12ℏ2 int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 +
12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
Here the last term is denoted as 119866119866119953119953119953119953prime(2) (120591120591)
Using Eq (513) the second and third terms on the right hand above equation can be simplified as
follows
18
12ℏ2
1198891198891205911205912120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 +1
2ℏ2 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
=12 minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0
minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
+12 minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0
minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
=12 minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0
minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
+12 minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0
minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
= minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0
minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 = 1198631198631119866119866119953119953119953119953prime(1) (120591120591)
Where we performed the following integration variables interchange 1205911205911 ⟷ 1205911205912 Thus we have
1198681198683 = 119866119866119953119953119953119953prime(2) (120591120591) + 119866119866119953119953119953119953prime
(1) (120591120591)1198631198631 + 119866119866119953119953119953119953prime0 (120591120591 0)1198631198632 (514)
Similarity we can calculate the fourth term of Eq (59) as
1198681198684 = minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 =
minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0 lang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 + 3 times
minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 + 3 times
minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 +
minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 =
119866119866119953119953119953119953prime0 (120591120591 0) minus1
6ℏ3 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0+
12ℏ2 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0
minus1ℏ int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888+
12ℏ2 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 minus1ℏ int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0+ 119866119866119953119953119953119953prime
(3) (120591120591) = 119866119866119953119953119953119953prime(3) (120591120591) +
119866119866119953119953119953119953prime(2) (120591120591)1198631198631 + 119866119866119953119953119953119953prime
(1) (120591120591)1198631198632 + 119866119866119953119953119953119953prime0 (120591120591 0)1198631198633
Higher order terms can be treated in the same manner (see pages 95-96 of Ref [2]) Then when
substituting 1198681198682 1198681198683 1198681198684 and all higher order terms into Eq (59) we can simplify the numerator as
follows
119866119866119953119953119953119953prime0 (120591120591) minus
1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0
+1
2ℏ2 1198891198891205911205911120573120573ℏ
0 1198891198891205911205912120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 + ⋯
= 119866119866119953119953119953119953prime0 (120591120591) + 119866119866119953119953119953119953prime
(1) (120591120591) + 119866119866119953119953119953119953prime(2) (120591120591) + ⋯ (1 + 1198631198631 + 1198631198632 + ⋯ )
Noting that the expression (1 + 1198631198631 + 1198631198632 + ⋯ ) is exactly canceled with the denominator the
Greenrsquos function can be simplified as
19
119866119866119953119953119953119953prime(120591120591) = 119866119866119953119953119953119953prime0 (120591120591) + 119866119866119953119953119953119953prime
(1) (120591120591) + 119866119866119953119953119953119953prime(2) (120591120591) + ⋯ (515)
where
119866119866119953119953119953119953prime
(1) (120591120591) = minus1ℏ int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888
119866119866119953119953119953119953prime(2) (120591120591) = 1
2ℏ2 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888 (516)
53 Calculation of Greenrsquos function
531 First order appriximation In Eq (516) we go back to the full notation 119886119886119954119954120582120582 to distinguish phonons of different branches
then 119866119866119953119953119953119953prime(1) (120591120591) is calculated as
119866119866119953119953119953119953prime(1) (120591120591) = minus
1ℏ
12 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591 119881119881119895119895119895119895prime(119948119948)119876119876119948119948119895119895(1205911205911)119876119876119948119948119895119895prime
dagger (1205911205911)3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ 119881119881119895119895119895119895primelowast (119948119948)119876119876119948119948119895119895prime(1205911205911)119876119876119948119948119895119895
dagger (1205911205911)3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= minus12ℏ
1198891198891205911205911120573120573ℏ
0lang119879119879120591120591
ℏ119881119881119895119895119895119895prime(119948119948)2120596120596119948119948119895119895120596120596119948119948119895119895prime
119886119886119948119948119895119895(1205911205911) + 119886119886minus119948119948119895119895dagger (1205911205911) 119886119886minus119948119948119895119895prime(1205911205911) + 119886119886119948119948119895119895prime
dagger (1205911205911)3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ ℏ119881119881119895119895119895119895prime
lowast (119948119948)
2120596120596119948119948119895119895120596120596119948119948119895119895prime119886119886119948119948119895119895prime(1205911205911) + 119886119886minus119948119948119895119895prime
dagger (1205911205911) 119886119886minus119948119948119895119895(1205911205911) + 119886119886119948119948119895119895dagger (1205911205911)
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
According to Wickrsquos theorem [2] the terms that contain 119886119886119948119948119895119895119886119886minus119948119948119895119895prime and 119886119886minus119948119948119895119895dagger 119886119886119948119948119895119895prime
dagger are equal to zero
thus the first term of above equation are calculated as
11989211989211 = minus14
119881119881119895119895119895119895prime(119948119948)
120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119886119886minus119948119948119895119895
dagger (1205911205911)119886119886minus119948119948119895119895prime(1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ lang119879119879120591120591119886119886119948119948119895119895(1205911205911)119886119886119948119948119895119895primedagger (1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888
= minus14
119881119881119895119895119895119895prime(119948119948)
120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119886119886minus119948119948119895119895
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus119948119948119895119895prime(1205911205911)rang0
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ lang119879119879120591120591119886119886119948119948119895119895primedagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)119886119886119948119948119895119895(1205911205911)rang0
= minus14
119881119881119895119895119895119895prime(119948119948)
120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
01198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954minus119948119948120575120575120582120582119895119895119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119954119954primeminus119948119948120575120575120582120582prime119895119895prime3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ 1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119948119948120575120575120582120582119895119895prime119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119954119954prime119948119948120575120575120582120582prime119895119895
20
Simplification of above equation gives
11989211989211 =
⎩⎪⎨
⎪⎧minus119881119881120582120582120582120582prime(minus119954119954)
4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954120582120582
0 (120591120591 1205911205911)119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903
2lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582(119954119954)
4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954120582120582
0 (120591120591 1205911205911)119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582prime le 3119903119903
2lt 120582120582 le 3119903119903
0 else
(517)
Similarity the second term of 119866119866119953119953119953119953prime(1) (120591120591) is calculated as
11989211989212 = minus14
minus119881119881119895119895119895119895primelowast (119948119948)
4120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119886119886119948119948119895119895prime(1205911205911)119886119886119948119948119895119895
dagger (1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ lang119879119879120591120591119886119886minus119948119948119895119895primedagger (1205911205911)119886119886minus119948119948119895119895(1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888
= minus14
minus119881119881119895119895119895119895primelowast (119948119948)
4120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)119886119886119948119948119895119895prime(1205911205911)rang0 lang119879119879120591120591119886119886119948119948119895119895dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus119948119948119895119895(1205911205911)rang0 lang119879119879120591120591119886119886minus119948119948119895119895prime
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
ie
11989211989212 =
⎩⎪⎨
⎪⎧
minus119881119881120582120582120582120582primelowast (119954119954)
4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954prime120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903
2lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582lowast (minus119954119954)
4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954prime120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119954119954prime 1 le 120582120582prime le 3119903119903
2lt 120582120582 le 3119903119903
0 else
(518)
Note that since 119881119881120582120582120582120582primelowast (119954119954) = 119881119881120582120582120582120582prime(minus119954119954) [Eq (314)] the first order approximation of Greenrsquos function
can be written as
119866119866119954119954120582120582119954119954prime120582120582prime(1) (120591120591) =
⎩⎪⎨
⎪⎧minus119881119881
120582120582120582120582prime(minus119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ
0 1198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582 le3119903119903
2lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582
(119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ
0 1198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582prime le3119903119903
2lt 120582120582 le 3119903119903
0 else
(519)
To derive the Greenrsquos function in frequency domain we use the expression for Fourier series then
we have
1198891198891205911205911120573120573ℏ
01198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0) = 1198891198891205911205911120573120573ℏ
0
1120573120573ℏ
119890119890minus119894119894120596120596119899119899(120591120591minus1205911205911)1198661198661199541199541205821205820 (119894119894120596120596119899119899)
119899119899
1120573120573ℏ
119890119890minus119894119894120596120596119899119899prime1205911205911119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899prime)
119899119899prime
=1
(120573120573ℏ)2119890119890minus119894119894120596120596119899119899120591120591 1198891198891205911205911120573120573ℏ
0119890119890119894119894120596120596119899119899minus120596120596119899119899prime1205911205911 119866119866119954119954120582120582
0 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899prime)
119899119899119899119899prime=
=1
(120573120573ℏ)2119890119890minus119894119894120596120596119899119899120591120591[120573120573ℏ120575120575119899119899119899119899prime]1198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899prime)119899119899119899119899prime
=1120573120573ℏ
119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899)119899119899
According to the definition of Fourier series [see Eq (412)] we get the Fourier coefficient of
21
119866119866119954119954120582120582119954119954prime120582120582prime(1) (120591120591)
119866119866119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) =
⎩⎪⎨
⎪⎧minus119881119881120582120582120582120582prime(minus119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582prime1198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime 1 le 120582120582 le 31199031199032
lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582(119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582prime1198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime 1 le 120582120582prime le 31199031199032
lt 120582120582 le 3119903119903
0 else
(520)
where 119866119866119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) = 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) times Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) times 119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899) and Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) is defined as
Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) =
⎩⎪⎨
⎪⎧minus119881119881120582120582120582120582prime(minus119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582prime120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903
2lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582(119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582prime120575120575119954119954119954119954prime 1 le 120582120582prime le 3119903119903
2lt 120582120582 le 3119903119903
0 else
(521)
532 second order approximation The second order approximation of the Greenrsquos function in Eq (516) 119866119866119954119954120582120582119954119954prime120582120582prime
(2) (120591120591) is calculated as
119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) =
18ℏ2 1198891198891205911205911
120573120573ℏ
0 1198891198891205911205912120573120573ℏ
0lang119879119879120591120591
⎣⎢⎢⎡ 11988111988111989511989511198951198951prime(119948119948120783120783)1198761198761199481199481207831207831198951198951(1205911205911)1198761198761199481199481207831207831198951198951prime
dagger (1205911205911)3119903119903
1198951198951prime=31199031199032 +1
31199031199032
1198951198951=1119948119948120783120783
+ 11988111988111989411989411198941198941primelowast (119948119948120783120783)1198761198761199481199481207831207831198941198941prime(1205911205911)1198761198761199481199481207831207831198941198941
dagger (1205911205911)3119903119903
1198941198941prime=31199031199032 +1
31199031199032
1198941198941=1119948119948120783120783 ⎦⎥⎥⎤
⎣⎢⎢⎡ 11988111988111989511989521198951198952prime(119948119948120784120784)1198761198761199481199481207841207841198951198952(1205911205912)1198761198761199481199481207841207841198951198952prime
dagger (1205911205912)3119903119903
1198951198952prime=31199031199032 +1
31199031199032
1198951198952=1119948119948120784120784
+ 11988111988111989411989421198941198942primelowast (119948119948120784120784)1198761198761199481199481207841207841198941198942prime(1205911205912)1198761198761199481199481207841207841198941198942
dagger (1205911205912)3119903119903
1198941198942prime=31199031199032 +1
31199031199032
1198941198942=1119948119948120784120784 ⎦⎥⎥⎤119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888 = 1198921198921 + 1198921198922 + 1198921198923 + 1198921198924
where
1198921198921 = 18ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0
sum sum 11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989511989521198951198952prime(119948119948120784120784)1le1198951198952le
31199031199032 lt1198951198952
primele311990311990311994811994812078412078411989511989521198951198952prime
1le1198951198951le31199031199032 lt1198951198951
primele311990311990311994811994812078312078311989511989511198951198951prime
times lang1198791198791205911205911198761198761199481199481207831207831198951198951(1205911205911)1198761198761199481199481207831207831198951198951primedagger (1205911205911)1198761198761199481199481207841207841198951198952(1205911205912)1198761198761199481199481207841207841198951198952prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(522)
1198921198922 = 18ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0
sum sum 11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942primelowast (119948119948120784120784)
1le1198941198942le31199031199032 lt1198941198942
primele311990311990311994811994812078412078411989411989421198941198942prime
1le1198951198951le31199031199032 lt1198951198951
primele311990311990311994811994812078312078311989511989511198951198951prime
times lang1198791198791205911205911198761198761199481199481207831207831198951198951(1205911205911)1198761198761199481199481207831207831198951198951primedagger (1205911205911)1198761198761199481199481207841207841198941198942prime (1205911205912)1198761198761199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(523)
1198921198923 = 18ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0
sum sum 11988111988111989411989411198941198941primelowast (119948119948120783120783)11988111988111989511989521198951198952prime(119948119948120784120784)
1le1198951198952le31199031199032 lt1198951198952
primele311990311990311994811994812078412078411989511989521198951198952prime
1le1198941198941le31199031199032 lt1198941198941
primele311990311990311994811994812078312078311989411989411198941198941prime
times lang1198791198791205911205911198761198761199481199481207831207831198941198941prime (1205911205911)1198761198761199481199481207831207831198941198941dagger (1205911205911)1198761198761199481199481207841207841198951198952(1205911205912)1198761198761199481199481207841207841198951198952prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(524)
22
1198921198924 = 18ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0
sum sum 11988111988111989411989411198941198941primelowast (119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)1le1198941198942le
31199031199032 lt1198941198942
primele311990311990311994811994812078412078411989411989421198941198942prime
1le1198941198941le31199031199032 lt1198941198941
primele311990311990311994811994812078312078311989411989411198941198941prime
times lang1198791198791205911205911198761198761199481199481207831207831198941198941prime (1205911205911)1198761198761199481199481207831207831198941198941dagger (1205911205911)1198761198761199481199481207841207841198941198942prime (1205911205912)1198761198761199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(525)
In what follows we will expand the term 1198921198922 The expansion of all other terms follows the same lines
and eventually results in the same expression Expanding the product 11987611987611994811994812078312078311989511989511198761198761199481199481207831207831198951198951primedagger 1198761198761199481199481207841207841198941198942prime1198761198761199481199481207841207841198941198942
dagger we can
rewrite Eq (523) as follows
1198921198922 = 132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)119881119881
11989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032 lt1198941198942
primele311990311990311994811994812078412078411989411989421198941198942prime
1le1198951198951le31199031199032 lt1198951198951
primele311990311990311994811994812078312078311989511989511198951198951prime
times 119868119868 (526)
Where
119868119868 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911) + 119886119886minus1199481199481207831207831198951198951dagger (1205911205911) 119886119886minus1199481199481207831207831198951198951prime(1205911205911) + 1198861198861199481199481207831207831198951198951prime
dagger (1205911205911) 1198861198861199481199481207841207841198941198942prime (1205911205912) + 119886119886minus1199481199481207841207841198941198942primedagger (1205911205912) 119886119886minus1199481199481207841207841198941198942(1205911205912) + 1198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0c
= lang119879119879120591120591 1198861198861199481199481207831207831198951198951119886119886minus1199481199481207831207831198951198951prime + 11988611988611994811994812078312078311989511989511198861198861199481199481207831207831198951198951primedagger + 119886119886minus1199481199481207831207831198951198951
dagger 119886119886minus1199481199481207831207831198951198951prime + 119886119886minus1199481199481207831207831198951198951dagger 1198861198861199481199481207831207831198951198951prime
dagger 12059112059111198861198861199481199481207841207841198941198942prime119886119886minus1199481199481207841207841198941198942 + 1198861198861199481199481207841207841198941198942prime1198861198861199481199481207841207841198941198942
dagger + 119886119886minus1199481199481207841207841198941198942primedagger 119886119886minus1199481199481207841207841198941198942
+ 119886119886minus1199481199481207841207841198941198942primedagger 1198861198861199481199481207841207841198941198942
dagger 1205911205912119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0c
= lang119879119879120591120591 11988611988611994811994812078312078311989511989511198861198861199481199481207831207831198951198951primedagger + 119886119886minus1199481199481207831207831198951198951
dagger 119886119886minus1199481199481207831207831198951198951prime12059112059111198861198861199481199481207841207841198941198942prime1198861198861199481199481207841207841198941198942
dagger + 119886119886minus1199481199481207841207841198941198942primedagger 119886119886minus11994811994812078412078411989411989421205911205912
119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
Where the symbol [hellip ]1205911205911 signifies that the operators in the brackets are given in the interaction
picture The last equality in above equation comes from the fact that the contractions of the product
of the operators are equal to zero when the number of creation and annihilation operators are not the
same (see Wickrsquos theorem in Ref [2]) Using Wickrsquos theorem [2] we are able to calculate the above
equation term by term as follows
⎩⎪⎨
⎪⎧ 1198681198681 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951prime
dagger (1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)1198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888
1198681198682 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
1198681198683 = lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)1198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
1198681198684 = lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(527)
We shall now calculate them term by term
23
1198681198681 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951primedagger (1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)1198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942
dagger (1205911205912)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
+ lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951prime
dagger (1205911205911)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0
+ lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)1198861198861199481199481207841207841198941198942prime(1205911205912)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0119888119888
= 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199481199481207841207841198951198952
0 (1205911205912 1205911205912)12057512057511989411989421198941198942prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582
+ 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 1198661198661199481199481207831207831198951198951
0 (1205911205911 1205911205911)12057512057511989511989511198951198951prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582
+ 1198661198661199481199481207831207831198951198951prime0 (1205911205912 1205911205911)1205751205751199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951prime ∙ 119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582
= 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582
+ 1198661198661199481199481207831207831198951198951prime0 (1205911205912 1205911205911)1205751205751199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951prime ∙ 119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582
The last equality results from the fact that 1198951198951 and 1198951198951prime are indices of different ranges (belonging to
different subsystems) and the same holds for 1198941198942 and 1198941198942prime Similarity
1198681198684 = lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0
= 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 119866119866minus11994811994812078412078411989411989420 (1205911205912 1205911205912)12057512057511989411989421198941198942prime
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 119866119866minus1199481199481207831207831198951198951prime 0 (1205911205911 1205911205912)1205751205751199481199481207831207831199481199481207841207841205751205751198951198951prime1198941198942prime
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205911 1205911205911)12057512057511989511989511198951198951prime
+ 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205912 1205911205911)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942
= 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 119866119866minus1199481199481207831207831198951198951prime 0 (1205911205911 1205911205912)1205751205751199481199481207831207831199481199481207841207841205751205751198951198951prime1198941198942prime
+ 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205912 1205911205911)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942
Where again the last equality results from the fact that 1198951198951 and 1198951198951prime are indices of different ranges
(belonging to different subsystems) and the same holds for 1198941198942 and 1198941198942prime Similarly we obtain 1198681198682 =
1198681198683 = 0 for the same reason as shown below
24
1198681198682 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951prime
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0
+ lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)1198861198861199481199481207831207831198951198951(1205911205911)rang0
+ lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0
= 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582 ∙ 119866119866minus11994811994812078412078411989411989420 (1205911205912 1205911205912)12057512057511989411989421198941198942prime
+ 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)120575120575minus11994811994812078412078411994811994812078312078312057512057511989511989511198941198942prime ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205911)12057512057511989511989511198951198951prime
+ 1198661198661199481199481207831207831198951198951prime0 (1205911205912 1205911205911)120575120575minus11994811994812078412078411994811994812078312078312057512057511989411989421198951198951prime ∙ 119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 = 0
1198681198683 = lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)1198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0
+ lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0
+ lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0
= 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 11986611986611994811994812078412078411989411989420 (1205911205912 1205911205912)12057512057511989411989421198941198942prime
+ 11986611986611994811994812078412078411989411989420 (1205911205911 1205911205912)120575120575minus1199481199481207831207831199481199481207841207841205751205751198951198951prime1198941198942 ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205911 1205911205911)12057512057511989511989511198951198951prime
+ 1198661198661199481199481207841207841198941198942prime0 (1205911205912 1205911205911)120575120575minus1199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951 ∙ 119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 = 0
The two non-vanishing terms (1198681198681 and 1198681198684) produce the following contributions to 1198921198922 of Eq (526)
11989211989221 = 132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime1198681198681 =
132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime1198661198661199481199481207831207831198951198951
0 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙
1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582 + 1
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime1198661198661199481199481207831207831198951198951prime
0 (1205911205912 1205911205911)1205751205751199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951prime ∙
119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 =
⎩⎪⎨
⎪⎧
120575120575119954119954119954119954prime
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast (119954119954prime)
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (1205911205911 1205911205912) ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
120575120575119954119954119954119954prime
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032+1 1198661198661199541199541198951198951prime
0 (1205911205912 1205911205911) ∙ 119866119866119954119954prime120582120582prime0 (1205911205911 0)119866119866119954119954120582120582
0 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032
0 else
and
25
11989211989224 = 132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime1198681198684 =
132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙
119866119866minus1199481199481207831207831198951198951prime 0 (1205911205911 1205911205912)1205751205751199481199481207831207831199481199481207841207841205751205751198951198951prime1198941198942prime + 1
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙
1198661198661199541199541205821205820 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 119866119866minus1199481199481207831207831198951198951
0 (1205911205912 1205911205911)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 =
⎩⎪⎨
⎪⎧120575120575119954119954119954119954prime
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032+1 1198661198661199541199541198951198951prime
0 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime0 (1205911205912 0)119866119866119954119954120582120582
0 (120591120591 1205911205911) 120582120582 120582120582prime isin 1 31199031199032
120575120575119954119954119954119954prime
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast (119954119954prime)
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
0 else
Here we used the following properties 120596120596minus119954119954120582120582 = 120596120596119954119954120582120582 and 1198811198811205821205821198951198951prime(minus119954119954) = 1198811198811205821205821198951198951primelowast (119954119954) [see Eqs (112) and
(314)]
The equivalence of 11989211989221 and 11989211989224 can be seen by changing the integration variables 1205911205912 harr 1205911205911
Substituting 11989211989221 and 11989211989224 into Eq (526) 1198921198922 is given by
1198921198922 =
⎩⎪⎨
⎪⎧
120575120575119954119954119954119954prime
16 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast (119954119954prime)
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime
0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
120575120575119954119954119954119954prime
16 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032+1 1198661198661199541199541198951198951prime
0 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime0 (1205911205911 0)119866119866119954119954120582120582
0 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032
0 else
(528)
By changing the indices and integration variables itrsquos straightforward to show that the other three
terms (119892119892111989211989231198921198924 ) are identical to 1198921198922 thus the final expression of 119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) [Eq (516)) can be
calculated as
119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) = 41198921198922 =
⎩⎪⎨
⎪⎧120575120575119954119954119954119954prime
4 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032
120575120575119954119954119954119954prime
4 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast 119954119954prime
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime
0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 31199031199032
+ 13119903119903
0 else
(529)
To calculate 119866119866119954119954120582120582119954119954prime120582120582prime(2) in frequency space we expand 119866119866119954119954120582120582119954119954prime120582120582prime
(2) (120591120591) in a Fourier series
119866119866119954119954120582120582119954119954prime120582120582prime
(2) (120591120591) = 1120573120573ℏsum 119890119890minus119894119894120596120596119899119899120591120591119866119866119954119954120582120582119954119954prime120582120582prime
(2) (119894119894120596120596119899119899)119899119899
119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = int d120591120591119890119890119894119894120596120596119899119899120591120591119866119866119954119954120582120582119954119954prime120582120582prime
(2) (120591120591)120573120573ℏ0
(530)
Using Eq (421) we get for 120582120582 120582120582prime isin 1 31199031199032
26
119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) =
120575120575119954119954119954119954prime4
1198891198891205911205911120573120573ℏ
0 1198891198891205911205912120573120573ℏ
0
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1198661198661199541199541198951198951prime0 (1205911205912 1205911205911)119866119866119954119954120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912)
=120575120575119954119954119954119954prime
4
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1198891198891205911205911120573120573ℏ
0 1198891198891205911205912120573120573ℏ
0
1120573120573ℏ
119890119890minus1198941198941205961205961198991198992(1205911205912minus1205911205911)1198661198661199541199541198951198951prime0 1198941198941205961205961198991198992
1198991198992
∙1120573120573ℏ
119890119890minus11989411989412059612059611989911989911205911205911119866119866119954119954120582120582prime0 1198941198941205961205961198991198991
1198991198991
∙1120573120573ℏ
119890119890minus119894119894120596120596119899119899(120591120591minus1205911205912)1198661198661199541199541205821205820 (119894119894120596120596119899119899)
119899119899
=120575120575119954119954119954119954prime
4
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1120573120573ℏ
119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954120582120582prime
0 11989411989412059612059611989911989911198661198661199541199541198951198951prime0 1198941198941205961205961198991198992
11989911989911989911989911198991198992⎩⎪⎨
⎪⎧ 1120573120573ℏ
1198891198891205911205911120573120573ℏ
0119890119890minus1198941198941205961205961198991198991minus12059612059611989911989921205911205911
times1120573120573ℏ
1198891198891205911205912120573120573ℏ
0119890119890minus1198941198941205961205961198991198992minus1205961205961198991198991205911205912
⎭⎪⎬
⎪⎫
=120575120575119954119954119954119954prime
4
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1120573120573ℏ
119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)
11989911989911989911989911198991198992
119866119866119954119954120582120582prime0 11989411989412059612059611989911989911198661198661199541199541198951198951prime
0 1198941198941205961205961198991198992120575120575119899119899111989911989921205751205751198991198991198991198992
=120575120575119954119954119954119954prime120573120573ℏ
119890119890minus119894119894120596120596119899119899120591120591
119899119899
1198661198661199541199541205821205820 (119894119894120596120596119899119899)
14
119881119881120582120582prime1198951198951prime(119954119954
prime)1198811198811205821205821198951198951primelowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899)119866119866119954119954120582120582prime
0 (119894119894120596120596119899119899)
Comparing above expression with Eq (530) the Fourier transform of 119866119866119954119954120582120582119954119954120582120582prime(2) (120591120591) reads as
119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) 120575120575119954119954119954119954prime
4sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 119866119866119954119954120582120582prime
0 (119894119894120596120596119899119899) (531)
where we have introduced the notation
Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) equiv
120575120575119954119954119954119954prime
4sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) (532)
and 119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954120582120582prime(2) (119894119894120596120596119899119899) ∙ 119866119866119954119954120582120582prime
0 (119894119894120596120596119899119899) Similarly for 120582120582 120582120582prime isin 31199031199032
+ 13119903119903 we
have
Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) =
120575120575119954119954119954119954prime
4sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast 119954119954prime
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (119894119894120596120596119899119899) (533)
and all together we have for Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899)
Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) =
⎩⎪⎪⎨
⎪⎪⎧120575120575119954119954119954119954prime
4sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 1 3119903119903
2
120575120575119954119954119954119954prime
4sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast 119954119954prime
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
0 else
(534)
27
533 Dysonrsquos equation Substituting Eqs (521) and (534) into Eq (516) and performing a Fourier transform we have
119866119866119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) = 119866119866119954119954120582120582119954119954prime120582120582prime0 (119894119894120596120596119899119899) + 119866119866119954119954120582120582119954119954prime120582120582prime
(1) (119894119894120596120596119899119899) + 119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) + ⋯
= 1198661198661199541199541205821205820 (119894119894120596120596119899119899)120575120575119954119954119954119954prime120575120575120582120582120582120582prime + 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) ∙ 119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899) + 1198661198661199541199541205821205820 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime
(2) (119894119894120596120596119899119899)
∙ 119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899) + ⋯ = 119866119866119954119954120582120582
0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime120575120575120582120582120582120582prime + 1198661198661199541199541205821205820 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) ∙ 119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899)
or
119866119866119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) = 1198661198661199541199541205821205820 (119894119894120596120596119899119899)120575120575119954119954119954119954prime120575120575120582120582120582120582prime + 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) ∙ 119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899) (535)
Eq (535) is the so-called Dysonrsquos equation (see Ref [2]) where
Σ119954119954120582120582119954119954120582120582prime(119894119894120596120596119899119899) = Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) + Σ119954119954120582120582119954119954prime120582120582prime
(2) (119894119894120596120596119899119899) + ⋯
is called self-energy and Σ119954119954120582120582119954119954120582120582prime(119899119899) (119894119894120596120596119899119899)119899119899 = 12⋯ is the self-energy of nth order approximation Up
to the second order approximation the self-energy is written as
Σ119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) =
⎩⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎧ minus
120575120575119954119954119954119954prime
2
119881119881120582120582120582120582primelowast (119954119954)
120596120596119954119954120582120582120596120596119954119954120582120582prime 1 le 120582120582 le 3119903119903
2lt 120582120582prime le 3119903119903
minus120575120575119954119954119954119954prime
2
119881119881120582120582prime120582120582(119954119954)
120596120596119954119954120582120582120596120596119954119954120582120582prime 1 le 120582120582prime le 3119903119903
2lt 120582120582 le 3119903119903
120575120575119954119954119954119954prime
4sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 1 3119903119903
2
120575120575119954119954119954119954prime
4sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast 119954119954prime
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
(536)
54 Fermirsquos golden Rule
To get the Fermirsquos golden Rule we need to calculate the retarded Greenrsquos function which can be
calculated from 119866119866119954119954120582120582119954119954120582120582prime(119894119894120596120596119899119899) by analytic continuation to the real axis via 119894119894120596120596119899119899 rarr 120596120596 + 119894119894119894119894 with an
infinitesimal positive 119894119894 [12]
119866119866119954119954120582120582119954119954120582120582prime119903119903 (120596120596) equiv 119866119866119954119954120582120582119954119954120582120582prime(119894119894120596120596119899119899 rarr 120596120596 + 119894119894119894119894) 119894119894 rarr 0 (537)
Similarity the retarded self-energy is defined as
Σ119954119954120582120582119954119954120582120582prime119903119903 (120596120596) equiv Σ119954119954120582120582119954119954120582120582prime(119894119894120596120596119899119899 rarr 120596120596 + 119894119894119894119894) 119894119894 rarr 0 (538)
The transition rate of phonons of branch 120582120582 at 119954119954 Γ119954119954120582120582(120596120596) is related to the retarded self energy
Σ119954119954120582120582119903119903 (120596120596) via [13]
Γ119954119954120582120582(120596120596) = minus2ImΣ119954119954120582120582119954119954120582120582119903119903 (120596120596) (539)
Using the expression for 1198661198661199541199541205821205820 (119894119894120596120596119899119899) [Eq (421)] and focused on the second order approximation [13]
we have
28
Σ119954119954120582120582119954119954120582120582119903119903 (119894119894120596120596119899119899) =
⎩⎪⎨
⎪⎧14sum
1198811198811205821205821198951198951prime(119954119954)
2
1205961205961199541199541198951198951prime120596120596119954119954120582120582
31199031199031198951198951prime=
31199031199032 +1
1
120596120596minus1205961205961199541199541198951198951prime +119894119894119894119894
120582120582 isin 1 31199031199032
14sum 1198811198811198951198951120582120582(119954119954)2
1205961205961199541199541198951198951120596120596119954119954120582120582
311990311990321198951198951=1
1120596120596minus1205961205961199541199541198951198951+119894119894119894119894
120582120582 isin 31199031199032
+ 13119903119903
(540)
Using the relation
Im 1
120596120596minus1205961205961199541199541198951198951prime +119894119894119894119894
= minus120587120587120575120575120596120596 minus 1205961205961199541199541198951198951prime (541)
The transition rate reads as
Γ119954119954120582120582(120596120596) =
⎩⎪⎨
⎪⎧1205871205872sum
1198811198811205821205821198951198951prime(119954119954)
2
1205961205961199541199541198951198951prime120596120596119954119954120582120582
31199031199031198951198951prime=
31199031199032 +1
120575120575120596120596 minus 1205961205961199541199541198951198951prime 120582120582 isin 1 31199031199032
1205871205872sum 1198811198811198951198951120582120582(119954119954)2
1205961205961199541199541198951198951120596120596119954119954120582120582
311990311990321198951198951=1
120575120575120596120596 minus 1205961205961199541199541198951198951 120582120582 isin 31199031199032
+ 13119903119903
(542)
Or equivalently
Γ119954119954120582120582(119864119864 = ℏ120596120596) =
⎩⎪⎨
⎪⎧120587120587ℏ3
2sum
1198811198811205821205821198951198951prime(119954119954)
2
1198641198641199541199541198951198951prime119864119864119954119954120582120582
31199031199031198951198951prime=
31199031199032 +1
120575120575119864119864 minus 1198641198641199541199541198951198951prime 120582120582 isin 1 31199031199032
120587120587ℏ3
2sum 1198811198811198951198951120582120582(119954119954)2
1198641198641199541199541198951198951119864119864119954119954120582120582
311990311990321198951198951=1
120575120575119864119864 minus 1198641198641199541199541198951198951 120582120582 isin 31199031199032
+ 13119903119903
(543)
Γ119954119954120582120582(119864119864) represents the probability per unit time of a transition with energy 119864119864 from phonon branch 120582120582
at wave number 119954119954 in subsystem I (II) to a set of phonon branches in subsystem II (I) with the same
wave number Here state 119954119954120582120582 in one subsystem is coupled to state 1199541199541198951198951prime in the other subsystem via
1198811198811205821205821198951198951prime(119954119954) Since the two expressions in Eq (543) are completely equivalent just representing
transitions of opposite directions we consider only the first one in what follows Assuming that
subsystem II sufficiently large such that its density of states is nearly continuous the sum over its
states in the above expression can be replaced by an integral over the energy sum (⋯ )119895119895 rarr
int(⋯ )120588120588119864119864119954119954d119864119864119954119954 giving
Γ119954119954120582120582(119864119864) = 120587120587ℏ3
2 intd119864119864119954119954prime120588120588119864119864119954119954prime 119881119881120582120582119895119895prime(119954119954)
2119864119864119954119954119895119895prime=119864119864119954119954prime
119864119864119954119954prime119864119864119954119954120582120582120575120575119864119864 minus 119864119864119954119954prime = 120587120587ℏ3
2120588120588(119864119864)
119881119881120582120582119895119895prime(119954119954)2119864119864119954119954119895119895prime=119864119864
119864119864119864119864119954119954120582120582 (544)
Eq (544) represents the transition rate of the phonons with energy 119864119864 from branch 120582120582 and wave
number 119954119954 in subsystem I to the continuous manifold of states in subsystem II The total transition
rate between the two subsystems is then given by the sum of transition rates from all states in
subsystem I weighted by their phonon populations to the manifold of states in subsystem II
Assuming that the two subsystems are weakly coupled such that the width of state 119954119954120582120582 in subsystem
I is small and the probability to leave it at an energy other than 119864119864119954119954120582120582 is negligible we can now write
29
total transition rate as
Γtot = 1119885119885119890119890minus120573120573119864119864119954119954120582120582Γ119954119954120582120582119864119864119954119954120582120582
119954119954120582120582
=120587120587ℏ3
2
119890119890minus120573120573119864119864119954119954120582120582
119885119885
120588120588119864119864119954119954120582120582 119881119881120582120582119895119895prime(119954119954)2119864119864119954119954119895119895prime=119864119864119954119954120582120582
1198641198641199541199541205821205822119954119954120582120582
=120587120587ℏ3
2
119890119890minus120573120573119864119864119954119954120582120582
119885119885
120588120588119864119864119954119954120582120582 119881119881120582120582120582120582+31199031199032(119954119954)
2
1198641198641199541199541205821205822119954119954120582120582
Finally we get
Γtot = Γ119954119954120582120582(119864119864) =120587120587ℏ3
2sum 119890119890minus120573120573119864119864119954119954120582120582
119885119885
120588120588119864119864119954119954120582120582119881119881120582120582120582120582+31199031199032(119954119954)
2
1198641198641199541199541205821205822119954119954120582120582 (545)
where 119885119885 = sum 119890119890minus120573120573119864119864119954119954120582120582119954119954120582120582 is the partition function Here we choose the index of phonon branches 120582120582
such that the phonon branch with lower energy has a smaller index ie 1198641198641199541199541205821205821 lt 1198641198641199541199541205821205822 for index 1205821205821 lt
1205821205822 Note that phonon branches 120582120582 and 1198951198951prime belong to the subsystem I (with index from 1 to 31199031199032) and
subsystem II (with index from 1 to 1 + 31199031199032 to 3119903119903) respectively we choose 119895119895prime = 120582120582 + 31199031199032
to make
sure that 119864119864119954119954119895119895prime = 119864119864119954119954120582120582 since phonon energy of two uncoupled system is identical Eq (545) is the
Fermirsquos golden rule for inter-phonon coupling
References
[1] Y Xu J-S Wang W Duan B-L Gu and B Li Nonequilibrium Greens function method for phonon-phonon interactions and ballistic-diffusive thermal transport Phys Rev B 78 224303 (2008) [2] A L Fetter and J D Walecka Quantum theory of many-particle systems (Courier Corporation 2012) [3] G E Stedman Fermis Golden RulemdashAn Exercise in Quantum Field Theory Am J Phys 39 205 (1971)
5
layers To demonstrate this we write the dynamical matrix (the mass-reduced Fourier transform of
the force constant matrix) in block form as follows
120625120625(119954119954) = 12062512062511(119954119954) 12062512062512(119954119954)12062512062521(119954119954) 12062512062522(119954119954) (1)
where 12062512062511(119954119954) and 12062512062522(119954119954) are the block matrices relating to the first and second layer and 12062512062512(119954119954)
and 12062512062521(119954119954) = 12062512062512dagger (119954119954) all evaluated at wave-vector 119954119954 The interlayer phonon-phonon couplings are
obtained by diagonalizing separately 12062512062511(119954119954) and 12062512062522(119954119954) such that 12062512062511(119954119954) =
1199321199321dagger(119954119954)12062512062511(119954119954)1199321199321(119954119954) and 12062512062522(119954119954) = 1199321199322
dagger(119954119954)12062512062522(119954119954)1199321199322(119954119954) are diagonal matrices containing the
frequencies (120596120596119894119894) of the phonon modes of the two layers and 1199321199321(119954119954) and 1199321199322(119954119954) are unitary
matrices of the corresponding eigenvectors We now construct a global block diagonal transformation
matrix of the form
119932119932(119954119954) = 1199321199321(119954119954) 120782120782120782120782 1199321199322(119954119954) (2)
and transform the full dynamical matrix as follows
119932119932dagger(119954119954)120625120625(119954119954)119932119932(119954119954) = 12062512062511(119954119954) 12062512062512(119954119954)12062512062521(119954119954) 12062512062522(119954119954)
(3)
where 12062512062512(119954119954) = 1199321199321dagger(119954119954)12062512062512(119954119954)1199321199322(119954119954) and 12062512062521(119954119954) = 12062512062512
dagger (119954119954) are the interlayer phonon-phonon
coupling blocks Naturally when the two layers are infinitely separated these coupling blocks vanish
and the diagonal blocks converge to those of the isolated layers
The overall coupling between the two layers can be obtained from the individual phonon-phonon
coupling matrix elements via Fermirsquos golden rule [31] which reads as (see Sec 4 of SI for a detailed
derivation)
Γtot = 120587120587ℏ3
2sum 119890119890minus120573120573119864119864119954119954120582120582
119885119885
120588120588119864119864119954119954120582120582119881119881120582120582120582120582+31199031199032(119954119954)
2
1198641198641199541199541205821205822119954119954120582120582 (4)
where 119885119885 = sum 119890119890minus120573120573119864119864119954119954120582120582119954119954120582120582 is the partition function 119864119864119954119954120582120582 is the energy of phonons at branch 120582120582 with
wave number 119954119954 120588120588119864119864119954119954120582120582 is the density-of-states (DOS) at 119864119864119954119954120582120582 and 119881119881120582120582120582120582+31199031199032(119954119954)
2 is the coupling
matrix element between branches of phonons of similar energy in the two layers whose number of
atoms in one unit cell is 119903119903
Using Eq (4) we can rationalize the misfit angle dependence of the heat flux across the twisted
interface from the calculated inter-phonon coupling To that end we performed room temperature
(300K) simulations (technical details can be found in Sec 3 of SI) for tBLG with different misfit
angles using the Greenrsquos function molecular dynamics (GFMD) developed by Kong et al [32] as
6
implemented in LAMMPS [33] The simulations allow us to evaluate the dynamical matrix from
which the phonon-phonon couplings can be extracted (see details in Sec 3 of SI) and the overall heat
transfer rate calculated Figure 3 shows the resulting heat transfer rate (normalized to is value for the
aligned contact (120579120579 = 0) ) as a function of the misfit angle compared to the interfacial thermal
conductivity defined as the inverse of the ITR presented in Figure 2(c) ITC equiv 1ITR The
remarkable agreement between the calculated interfacial thermal conductivity and Fermirsquos golden
rule results indicate that the dependence of the interlayer phonon-phonon couplings on the misfit
angle is responsible for the strong angle dependence of the interfacial conductivity Notably the sharp
heat conductivity drop at misfit angles in the range of 0deg-5deg as well as the small conductivity for larger
misfit angles are well captured by Fermirsquos golden rule
Figure 3 Comparison between Fermirsquos golden rule results (open blue squares) for the interfacial heat-
transfer rate of a tBLG and the calculated interfacial thermal conductivity at various misfit angles
ITC simulation results are presented for both 8 layers (open red circles) and 16 layers (open black
triangles) showing similar behavior For comparison purposes all data sets are normalized to their
value obtained for the aligned contact
To correlate our results with experimentally measured thermal conductivities that are often obtained
for thick samples we repeated our calculations for increasing stack thicknesses at fixed misfit angles
Figure 4 presents results for the calculated heat conductivity of (a) graphite and (b) h-BN stacks either
aligned (open red circles) or twisted by 120579120579 = 3016deg (open black diamond symbols) as a function of
number of layers in the stack As discussed above for both systems the misoriented stack exhibits
lower heat conductivity compared to the aligned system however its thickness dependence is
considerably stronger This can be attributed to the significantly higher interface resistance of the
twisted interface that when plugged in Eq (S2) of the SI for the overall conductivity induces stronger
7
thickness dependence
Comparing our calculated heat conductivities for the aligned contact (open red circles) to available
experimental data for ~35 nm thick graphite slabs [34] (dashed green line) we find that at the thickest
model system considered of 104 graphene layers (~34 nm thick) the calculated value of 085plusmn005
W m sdot Kfrasl is in remarkable agreement with the measured value of ~07 W m sdot Kfrasl Furthermore
experimental values for bulk graphite [5] indicate that the thermal conductivity continues to grow up
to ~68 W m sdot Kfrasl (black dash-dotted line) which is consistent with the general trend of the calculated
heat conductivity that does not saturate for the thickest model system considered These results
strongly enforce the validity of our force-field and model systems to model the heat conductivity of
twisted layered materials interfaces Available experimental results for the heat conductivity of bulk
h-BN are marked by the dashed-dotted black and dashed-green lines in Figure 4(b) In line with our
findings for the graphitic interface our calculated finite slab heat conductivities for the aligned
interface (open red circles) continue to grow with the number of layers and are consistently below the
bulk value
Figure 4 Thickness dependence of the thermal conductivity 120581120581CP of aligned (open red circles) and
twisted by 3016deg (open black diamond symbols) graphite (a) and h-BN (b) stacks Blue squares
represent results obtained using the isotropic Lennard-Jones potential for the aligned contacts The
green dashed and black dash-dotted lines represent experimental results measured for graphite (a)
(Refs [534]) and bulk h-BN (b) (Refs [1415]) Note that both axis scales are logarithmic Error bar
estimation procedure is discussed in Sec 1 of the SI
Another important factor that may affect the interlayer thermal transport properties of 2D material
stacks is the average temperature of the system which was taken to be ~300K in all abovementioned
simulations To evaluate the sensitivity of our results towards this parameter we repeated the heat
conductivity and interfacial resistance calculations of optimally stacked graphite and h-BN stacks for
an average temperature of 400 K The results presented in see Sec 4 of the SI indicate that 120581120581CP and
8
ITR weakly depend on the average temperature over the entire thickness range considered This
conclusion is consistent with recent experimental findings [11] We may attribute this to the fact that
the thickness of the graphite and h-BN stack model considered is much smaller than their phonon
mean-free path (~200 nm for graphite [1011133536] and ~100 nm for bulk h-BN [15]) such that
the phonon transport is dominated by phonon-boundary scattering which weakly depends on
temperature in the range considered Therefore temperature dependent Umklapp processes have only
marginal contribution to our results [11]
Finally we note that previous calculations of the heat conductivity of twisted graphitic interfaces
relied on Lennard-Jones (LJ) potentials describing the interlayer interactions [8-1013] To
demonstrate the importance of using registry-dependent interlayer potentials we have repeated our
calculations of the heat conductivity of graphitic slabs with the REBO intralayer potential augmented
by LJ interlayer interactions [37] (120576120576 = 284 meV120590120590 = 34 Å ) We find that the calculated heat
conductivities obtained using the LJ interlayer potential are consistently higher than those obtained
by our ILP and that the difference between them grows with the model system thickness Notably the
heat conductivity obtained using the LJ potential for a graphitic slab of thickness ~34 nm is 154
W m sdot Kfrasl overestimating the experimental value by more than a factor of 2
The excellent agreement of our ILP calculations with experimental data of nanoscale graphitic stacks
therefore demonstrates the reliability of our predictions for the strong interfacial misfit angle
dependence of cross-layer thermal conductivity in graphite and h-BN The observed sharp
conductivity decrease of twisted graphitic interfaces at misfit angles lt 5deg opens the way to control
the thermal evacuation rate and thermal isolation of active layers in graphene-based electronic and
mechanical devices The revealed underlying mechanism suggests that design rules can be obtained
by carefully tailoring the phonon-phonon couplings across the twisted interface While the misfit
angle dependence of h-BN is found to be weaker than that of graphite the overall thermal
conductivity of the former is found to be higher This may be utilized to achieve higher conductivity
and controllability in twisted heterogeneous junctions of layered materials
ASSOCIATED CONTENT
Supporting Information
The supporting Information section includes a description of the methodology thermal conductivity
of twisted bilayer graphene theory for calculating the phonon coupling of twisted bilayer graphene
9
derivation of Fermirsquos golden rule and temperature dependence of the cross-plane thermal
conductivity
AUTHOR INFORMATION
Corresponding Author
E-mail urbakhtauextauacil
Notes
The authors declare no competing financial interest
ACKNOWLEDGMENTS
The authors would like to thank Prof Abraham Nitzan Dr Guy Cohen and Dr Yiming Pan for
helpful discussions WO acknowledges the financial support from a fellowship program for
outstanding postdoctoral researchers from China and India in Israeli Universities and the support from
the National Natural Science Foundation of China (Nos 11890673 and 11890674) HQ
acknowledges the financial support from the National Natural Science Foundation of China (No
11890674) MU acknowledges the financial support of the Israel Science Foundation Grant
No 114118 and the ISF-NSFC joint grant 319119 OH is grateful for the generous financial
support of the Israel Science Foundation under grant no 158617 and the Naomi Foundation for
generous financial support via the 2017 Kadar Award This work is supported in part by COST Action
MP1303
10
References
[1] A A Balandin Thermal properties of graphene and nanostructured carbon materials Nat Mater 10 569 (2011) [2] S Ghosh I Calizo D Teweldebrhan E P Pokatilov D L Nika A A Balandin W Bao F Miao and C N Lau Extremely high thermal conductivity of graphene Prospects for thermal management applications in nanoelectronic circuits Appl Phys Lett 92 151911 (2008) [3] A A Balandin S Ghosh W Bao I Calizo D Teweldebrhan F Miao and C N Lau Superior thermal conductivity of single-layer graphene Nano Lett 8 902 (2008) [4] J H Seol et al Two-Dimensional Phonon Transport in Supported Graphene Science 328 213 (2010) [5] R Taylor The thermal conductivity of pyrolytic graphite J Philosophical Magazine 13 157 (1966) [6] S Ghosh W Bao D L Nika S Subrina E P Pokatilov C N Lau and A A Balandin Dimensional crossover of thermal transport in few-layer graphene Nat Mater 9 555 (2010) [7] Z Wang R Xie C T Bui D Liu X Ni B Li and J T L Thong Thermal Transport in Suspended and Supported Few-Layer Graphene Nano Lett 11 113 (2011) [8] Z Wei Z Ni K Bi M Chen and Y Chen Interfacial thermal resistance in multilayer graphene structures Phys Lett A 375 1195 (2011) [9] Y Ni Y Chalopin and S Volz Significant thickness dependence of the thermal resistance between few-layer graphenes Appl Phys Lett 103 061906 (2013) [10] J Chen J H Walther and P Koumoutsakos Strain Engineering of Kapitza Resistance in Few-Layer Graphene Nano Lett 14 819 (2014) [11] Q Fu J Yang Y Chen D Li and D Xu Experimental evidence of very long intrinsic phonon mean free path along the c-axis of graphite Appl Phys Lett 106 031905 (2015) [12] G Fugallo A Cepellotti L Paulatto M Lazzeri N Marzari and F Mauri Thermal Conductivity of Graphene and Graphite Collective Excitations and Mean Free Paths Nano Lett 14 6109 (2014) [13] Z Wei J Yang W Chen K Bi D Li and Y Chen Phonon mean free path of graphite along the c-axis Appl Phys Lett 104 081903 (2014) [14] E K Sichel R E Miller M S Abrahams and C J Buiocchi Heat capacity and thermal conductivity of hexagonal pyrolytic boron nitride Phys Rev B 13 4607 (1976) [15] P Jiang X Qian R Yang and L Lindsay Anisotropic thermal transport in bulk hexagonal boron nitride Phys Rev Mater 2 064005 (2018) [16] M-H Wang Y-E Xie and Y-P Chen Thermal transport in twisted few-layer graphene Chinese Physics B 26 116503 (2017) [17] X Nie L Zhao S Deng Y Zhang and Z Du How interlayer twist angles affect in-plane and cross-plane thermal conduction of multilayer graphene A non-equilibrium molecular dynamics study Int J Heat Mass Tran 137 161 (2019) [18] A N Kolmogorov and V H Crespi Registry-dependent interlayer potential for graphitic systems Phys Rev B 71 235415 (2005) [19] M Reguzzoni A Fasolino E Molinari and M C Righi Potential energy surface for graphene on graphene Ab initio derivation analytical description and microscopic interpretation Phys Rev B 86 (2012) [20] M M van Wijk A Schuring M I Katsnelson and A Fasolino Moire Patterns as a Probe of Interplanar Interactions for Graphene on h-BN Phys Rev Lett 113 135504 (2014) [21] D Mandelli W Ouyang M Urbakh and O Hod The Princess and the Nanoscale Pea Long-Range Penetration of Surface Distortions into Layered Materials Stacks ACS Nano 13 7603 (2019) [22] E Koren E Loumlrtscher C Rawlings A W Knoll and U Duerig Adhesion and friction in mesoscopic graphite contacts Science 348 679 (2015) [23] E Koren I Leven E Loumlrtscher A Knoll O Hod and U Duerig Coherent commensurate electronic states at the interface between misoriented graphene layers Nat Nanotechnol 11 752 (2016)
11
[24] W Ouyang I Azuri D Mandelli A Tkatchenko L Kronik M Urbakh and O Hod Mechanical and Tribological Properties of Layered Materials under High Pressure Assessing the Importance of Many-Body Dispersion Effects J Chem Theory Comput 16 666 (2020) [25] W Ouyang D Mandelli M Urbakh and O Hod Nanoserpents Graphene Nanoribbon Motion on Two-Dimensional Hexagonal Materials Nano Lett 18 6009 (2018) [26] I Leven T Maaravi I Azuri L Kronik and O Hod Interlayer Potential for Grapheneh-BN Heterostructures J Chem Theory Comput 12 2896 (2016) [27] T Maaravi I Leven I Azuri L Kronik and O Hod Interlayer Potential for Homogeneous Graphene and Hexagonal Boron Nitride Systems Reparametrization for Many-Body Dispersion Effects J Phys Chem C 121 22826 (2017) [28] L H Liang and B Li Size-dependent thermal conductivity of nanoscale semiconducting systems Phys Rev B 73 153303 (2006) [29] P K Schelling S R Phillpot and P Keblinski Comparison of atomic-level simulation methods for computing thermal conductivity Phys Rev B 65 144306 (2002) [30] J Shiomi Nonequilirium molecular dynamics methods for lattice heat conduction calculations Annual Review of Heat Transfer 17 (2014) [31] G E Stedman Fermis Golden RulemdashAn Exercise in Quantum Field Theory Am J Phys 39 205 (1971) [32] L T Kong G Bartels C Campantildeaacute C Denniston and M H Muumlser Implementation of Greens function molecular dynamics An extension to LAMMPS Comput Phys Commun 180 1004 (2009) [33] S Plimpton Fast Parallel Algorithms for Short-Range Molecular Dynamics J Comput Phys 117 1 (1995) [34] M Harb et al The c-axis thermal conductivity of graphite film of nanometer thickness measured by time resolved X-ray diffraction Appl Phys Lett 101 233108 (2012) [35] H Zhang X Chen Y-D Jho and A J Minnich Temperature-Dependent Mean Free Path Spectra of Thermal Phonons Along the c-Axis of Graphite Nano Lett 16 1643 (2016) [36] D P Sellan E S Landry J E Turney A J H McGaughey and C H Amon Size effects in molecular dynamics thermal conductivity predictions Phys Rev B 81 214305 (2010) [37] S J Stuart A B Tutein and J A Harrison A reactive potential for hydrocarbons with intermolecular interactions J Chem Phys 112 6472 (2000)
S1
Supporting information for ldquoControllable Thermal Conductivity of
Twisted Graphene and Hexagonal Boron Nitride Interfacesrdquo
Wengen Ouyang1 Huasong Qin2 Michael Urbakh1 and Oded Hod1
1Department of Physical Chemistry School of Chemistry The Raymond and Beverly
Sackler Faculty of Exact Sciences and The Sackler Center for Computational
Molecular and Materials Science Tel Aviv University Tel Aviv 6997801 Israel
2State Key Laboratory for Strength and Vibration of Mechanical Structures School of
Aerospace Xirsquoan Jiaotong University Xirsquoan 710049 China
The Supporting information includes following sections
1 Methodology
2 Thermal conductivity of twisted bilayer graphene
3 Theory for calculating the phonon coupling of twisted bilayer graphene
4 Derivation of Fermirsquos golden rule
5 Temperature dependence of cross-plane thermal conductivity
S2
1 Methodology
11 Model system
The initial interlayer distance across the layered stack was set equal to 34 Aring and 33 Aring for graphite
and bulk h-BN respectively Periodic boundary conditions were applied in all directions It should be
noted that the lattice structure is rigorously periodic only at some specific twist angles the values of
which are listed in Table S1 in section 3 below While the cross-sectional area for each misfit angle
θ is different all systems considered have a contact area exceeding 12 nm2 which was shown to
provide converged results with respect to unit-cell dimensions [1] The intralayer interactions within
each graphene and h-BN layer were modeled via the second generation REBO potential [2] and
Tersoff potential [3] respectively The interlayer interactions between the layers of graphite and bulk
h-BN were described via our dedicated interlayer potential (ILP) [4] which is implemented in the
LAMMPS [5] suite of codes [6]
Fig S1 Schematic representation of the simulation setup (a) and steady-state temperature profile (b) respectively In panel (a) two identical AB-stacked graphite slabs (gray and orange respectively) are twisted with respect to each other to create a stacking fault of misfit angle θ A thermal bias is induced by applying Langevin thermostats to the two layers marked by dashed red (Thot) and green (Tcold) rectangles The arrows indicate the direction of the vertical heat flux Since periodic boundary conditions are applied also in the vertical direction two twisted interfaces are shown across which heat flows in opposite directions The steady-state temperature profiles are illustrated in panel (b) where N is the total number of layers in the model system and RAB dAB and Rθ and dθ mark the interfacial Kapitza resistance [89] and interlayer distance for contacting graphene layers with AB-stacking and misfit angle θ respectively The red lines in panel (b) mark the temperature variation across the twisted interface where the vertical axis corresponds to the position of the various layers along the stack and the horizontal axis marks the temperature of the various layers
S3
12 Simulation Protocol
All MD simulations were performed with the LAMMPS simulation package [5] The velocity-Verlet
algorithm with a time-step of 05 fs was used to propagate the equations of motion A Noseacute-Hoover
thermostat with a time constant of 025 ps was used for constant temperature simulations To maintain
a specified hydrostatic pressure the three translational vectors of the simulation cell were adjusted
independently by a Noseacute-Hoover barostat with a time constant of 10 ps [7] To relax the box we first
equilibrated the systems in the NPT ensemble at a temperature of T = 300 K and zero pressure for
250 ps (see Fig S2) After equilibration Langevin thermostats with damping coefficients 10 ps-1
were applied to the bottom and middle layer of the graphene stack (see Fig S1) with target
temperature Thot = 375 K (hot reservoir) and Tcold = 225 K (cold reservoir) respectively Then the
system was allowed to reach steady-state over a subsequent simulation period of 750 ps (see Fig S2)
during which the dynamics of all non-thermostated layers followed the NVE ensemble For the larger
model systems the length of the NPT and Langevin stages was doubled (for the 32 and 48 layers
systems) or tripled (for the 104 layers graphitic system) to ensure convergence of the obtained steady-
state Once steady-state was obtained the last 500 ps were used to calculate the thermal conductivity
of the twisted graphite and bulk h-BN The statistical errors were estimated using ten different data
sets each calculated over a time interval of 50 ps
Fig S2 Time evolution of the temperature of thermostated layers for 16 layers twisted graphite with misfit angle (a) θ = 0deg (b) θ = 509deg (c) θ = 1518deg and (d) θ = 3016deg Note that the thermal fluctuations increase with increasing the misfit angle due to the growing interfacial thermal resistance that enhances phonon back scattering at the twisted junction
S4
13 Calculation of the interfacial thermal resistance
According to Fourierrsquos law the cross-plane thermal conductivity (120581120581CP) of a twisted graphitic interface
of misfit angle θ can be calculated as
120581120581CP = 119876119860119860Δ119879119879Δ119911119911
(S1)
where 119876 is the heat flux A is the cross-section area and Δ119879119879Δ119911119911 is the temperature gradient along
the direction of heat flux (perpendicular to the basal plane in our case) Fig S1(b) shows a schematic
temperature profile along the z-direction where the vertical axis corresponds to the position of the
various layers along the stack and the horizontal axis marks the temperature of the various layers The
actual temperature profiles extracted from the NEMD simulations for twisted graphite with different
number of layers can be found in Fig S3 For Bernal-stacked graphite (ie θ = 0deg red circles in Fig
S3) only the linear region of the temperature profile was used to calculate 120581120581CP and the points
corresponding to the layers where the thermostats were applied were omitted (marked with green
triangle in Fig S3) The 120581120581CP of the system was calculated using Eq (S1) by averaging over the two
linear regions of the temperature profiles For the twisted case (θ ne 0deg) we found a sudden
temperature decrease Δ119879119879120579120579 at the position of the twisted interface (see black squares in Fig S3) 120581120581CP
in this case was calculated using the temperature gradient calculated for the same layer range as that
for θ = 0deg To characterize the thermal properties of the twisted interface the concept of interfacial
thermal resistance (ITR) ie Kapitza resistance [89] was introduced According the definition of
the Kapitza resistance [8] 119877119877 = 119860119860Δ119879119879119876 and noticing that Δ119879119879tot = (1198731198732 minus 3)Δ119879119879AB + Δ119879119879120579120579 and
Δ119911119911 = (1198731198732 minus 3)119889119889AB + 119889119889120579120579 Eq (1) can be rewritten considering two-resistors in series as [see Fig
S1(b)]
(1198731198732 minus 3)119877119877AB + 119877119877120579120579 = [(1198731198732 minus 3)119889119889AB + 119889119889120579120579] 120581120581CPfrasl (S2)
Here 119877119877AB 119889119889AB Δ119879119879AB and 119877119877120579120579 119889119889120579120579 Δ119879119879120579120579 are the ITR interlayer distance and temperature
difference for adjacent AB-stacked and twisted graphene layers respectively For the aligned contact
(120579120579 = 0deg) the ITR can be simply calculated as 119877119877AB = 119889119889AB120581120581AB Once 119877119877AB is known 119877119877120579120579 can be
calculated from Eq (S2) We note that the sharp temperature drop at the twisted interface indicates
that 119877119877120579120579 should be much larger than 119877119877AB
S5
Fig S3 Temperature profiles for graphitic stacks consisting of (a) 8 and (b) 16 layers The red circles and black squares represent the temperature profiles for the aligned (120579120579 = 0deg) and twisted (120579120579 =3016deg) junctions respectively Green triangles represent data points that were omitted in the 120581120581CP calculation
2 Thermal conductivity of twisted bilayer graphene
As comparison we also calculated the interfacial thermal conductivity (ITC) and ITR of twisted
bilayer graphene (tBLG) with the transient MD simulation approach [15-17] since the NEMD
simulation protocol used in the main text becomes invalid in this case In this protocol the system
was first equilibrated within the NPT ensemble at T = 200 K and zero pressure for 100 ps which was
followed by a 100 ps NVT ensemble equilibration stage and a 100 ps of NVE ensemble equilibration
stage After the system reached steady-state an ultrafast heat impulse was imposed on the top layer
of the t-BLG for 50 fs to increase the temperature of the top layer from 200 K to 400 K while that of
bottom layer of tBLG remained unchanged After the external heat source was removed thermal
energy flowed from the top layer to the bottom layer due to the temperature difference and the
temperature of both layers approached 300 K when quasi-steady-state was reached During the
thermal relaxation time interval (500 ps) the temperature and energy of the system sections were
recorded The ITR could then be extracted using the following equation [15-17]
part119864119864t120597120597120597120597
= 119860119860119877119877119879119879bot(119905119905) minus 119879119879top(119905119905) (S3)
where 119864119864t is the total energy of the top graphene layer R is the ITR of the tBLG A is the interfacial
cross-section area and 119879119879bot and 119879119879top are the instantaneous temperatures measured for the bottom
and top layers of the tBLG respectively Note that in Eq S3 we assume a linear dependence of the
heat flux on the temperature difference between the layers The ITC of the tBLG is simply defined as
ITC equiv 119889119889ITR where d is the average interlayer distance
S6
The ITC and ITR of tBLG as functions of misfit angle calculated with the transient MD simulation
protocol are illustrated in Fig S4 demonstrating similar misfit-angle dependence as that for the
NEMD protocol with Langevin thermostat exercised to obtain the results presented in the main text
This further validates the reliability of the simulation protocol adopted in the main text which is more
suitable to treat thick slabs and allows to obtain a true stead-state
Fig S4 Misfit-angle dependence of (a) ITC and (b) ITR for a twisted bilayer graphene obtained using the transient MD simulation approach
3 Theory for calculating the phonon-phonon coupling of twisted bilayer graphene
31 Brillouin Zone of supercell in tBLG
For tBLG the lattice structure is rigorously periodic only at some specific misfit angles θ where the
lattice vector 1199231199231 = 11989911989911199381199381 + 11989911989921199381199382 in the bottom layer equals the vector 1199231199232 = 11989811989811199381199381 + 11989811989821199381199382 in the
top layer with certain integers m1 m2 and n1 n2 Here 1199381199381 = 119886119886(10) and 1199381199382 = 11988611988612radic32 are
the primitive lattice vectors of the bottom layer and a is the lattice constant of monolayer graphene
Thus the exact superlattice period is then given by [18]
119871119871 = |11989911989911199381199381 + 11989911989921199381199382| = 11988611988611989911989912 + 11989911989922 + 11989911989911198991198992 = |1198991198991minus1198991198992|1198861198862 sin(1205791205792) (S4)
where θ is the angle between two lattice vectors 1199231199231 and 1199231199232 In the simulations below we always
rotated the supercell such that its lattice vector is 1199231199231 = 119871119871(10) and 1199231199232 = 11987111987112radic32 In this case
the corresponding reciprocal lattice vector of the moireacute superlattice satisfies the relation 119918119918119894119894 ∙ 119923119923119895119895 =
2120587120587120575120575119894119894119895119895 such that
1199181199181 = 4120587120587radic3119871119871
radic32
minus12 1199181199182 = 4120587120587
radic3119871119871(01) (S5)
S7
Both the lattice vectors and the corresponding reciprocal lattice vectors of the superlattice of the tBLG
are presented in Fig S5 Table S1 reports the parameters used to construct rhombus periodic
supercells of different misfit angles that can be duplicated to construct a rectangular periodic supercell
Table S1 The parameters used to construct periodic supercells of various misfit angles
120579120579 (deg) 119860119860 (nm2) 1198991198991 1198991198992 1198981198981 1198981198982
0 60383688 24 0 24 0
0696407 709613169 48 47 47 48
1121311 273718420 30 29 29 30
2000628 85648391 17 16 16 17
3006558 152322047 23 21 21 23
4048894 189013524 26 23 23 26
5085849 53255058 14 12 12 14
7926470 49376245 14 11 11 14
9998709 86277388 19 14 14 19
15178179 31554670 15 4 4 15
19932013 42876612 15 8 8 15
25039660 13942761 9 4 4 9
30158276 18974735 11 4 4 11
32204228 21805221 12 4 4 12
32 Special points for Brillouin Zone integration
The calculation of the sum over wave vector q in Eq (4) in the main text can be transformed to an
integral using the relation sum (⋯ )119954119954 = 1119881119881119887119887int (⋯ )BZ d119954119954 where 119881119881119887119887 = (2120587120587)3119881119881 is the volume of the
Brillouin Zone (BZ) and V is volume of the real-space unit-cell The calculation of integral is usually
inefficient since it requires calculating the value of the function over a large set of k points in the first
BZ To calculate such integrations more efficiently simple k-point meshes can be replaced by a
carefully selected set of special points in the BZ 119954119954119894119894 [19-22] over which the function is evaluated
The integral can then be estimated via
119868119868 = 1119881119881119887119887int 119891119891(119954119954)BZ d119954119954 asymp 1
119873119873sum 119908119908119894119894119891119891(119954119954119894119894)119894119894 (S6)
S8
where 119881119881119887119887 is the volume of the BZ 119908119908119894119894 is the weight of the ith data point and N normalizes the
weighting factors to unity 119873119873 = sum 119908119908119894119894119894119894 The set of selected 119954119954119894119894 forms a grid in the irreducible
Brillouin zone (IBZ) as is illustrated by the red points in Fig S5(b) The coordinates of these points
for a hexagonal lattice are presented in Eq (S7)
Fig S5 (a) Twisted bilayer graphene of misfit angle θ = 509deg L1 and L2 are the superlattice vectors (b) The corresponding first Brillouin Zone of (a) G1 and G2 are the reciprocal lattice vectors of the superlattice The triangle ΔΓMK represents the irreducible Brillouin zone Red circles mark the position of the special points used to evaluate the integral over the first Brillouin Zone
⎩⎪⎨
⎪⎧ 1199541199541 = 1
9 1205971205973 1199541199542 = 2
9 1205971205973 1199541199543 = 1
18 512059712059718
1199541199544 = 518
512059712059718 1199541199545 = 1
9 21205971205979 1199541199546 = 2
9 21205971205979
1199541199547 = 118
312059712059718 1199541199548 = 1
9 1205971205979 1199541199549 = 1
18 12059712059718
(S7)
Here 119905119905 = radic3 and the unit of the coordinates is 2120587120587119871119871 The weighting factors 119908119908119894119894 are [19]
119908119908124689 = 112
119908119908357 = 16 (S8)
Using Eqs (S6-S8) Eq (4) in the main text can be evaluated as follows
Γtot = 120587120587ℏ3
2sum 119908119908119896119896 sum
119890119890minus120573120573119864119864119954119954119948119948120582120582
119885119885
120588120588119864119864119954119954119948119948120582120582119881119881120582120582120582120582+31199031199032(119954119954119948119948)
2
1198641198641199541199541199481199481205821205822120582120582
9119896119896=1 (S9)
This equation was used to calculate the transition rate presented in Fig 3 in the main text
4 Derivation of Fermirsquos golden rule
The derivation of Fermirsquos golden rule is provided in a separate file (Fermirsquos Golden Rule for
phononspdf) due to its extent
S9
5 Temperature dependence of interfacial thermal conductivity
In the main text the target temperatures of the Langevin thermostats for the bottom and middle layers
of graphene and h-BN were set to 225 K and 375 K respectively After reaching the steady-state the
average temperature of the system was found to be ~300 K To check the effect of average temperature
on our results we calculated the cross-plane thermal conductivity (120581120581CP ) and the corresponding
interfacial thermal resistance (ITR) at a different temperature gradient (325 K ndash 475 K) resulting the
average steady-state temperature of ~400K The protocol described in Section 1 above was used to
perform these calculations as well Both ILP and Lennard-Jones (LJ) potential were tested for
graphite whereas for the bulk h-BN simulations only the ILP was used The results for graphite and
bulk h-BN are illustrated in Fig S6 and Fig S7 respectively For the ILP we find that the overall
values of 120581120581CP (ITR) decrease (increase) slightly with increasing average temperature which is
consistent with a recent experiment [10] The LJ potential calculations as well exhibit very week
dependence on average temperature within the range studied Altogether the layer dependences of
both quantities remain mostly insensitive to the average temperature The reason is that the thickness
of the graphite and h-BN model systems was chosen to be much smaller than their phonon mean free
path (~200 nm for graphite [110-13] and ~100 nm for bulk h-BN [14]) such that phonon transport is
dominated by phonon-boundary scattering and the Umklapp process only makes marginal
contribution [10]
S10
Fig S6 Layer dependence of 120581120581CP (a c) and ITR (b d) for Bernal-stacked graphite at average steady-state temperatures of 300 K (red circles) and 400 K (black squares) The left and right columns correspond to the 120581120581CP and ITR calculated with ILP and LJ potential respectively
Fig S7 Layer dependence of 120581120581CP (a) and ITR (b) for AArsquo-stacked h-BN at average temperatures of 300 K (red circles) and 400 K (blue squares)
S11
References [1] Z Wei J Yang W Chen K Bi D Li and Y Chen Phonon mean free path of graphite along the c-axis Appl Phys Lett 104 081903 (2014) [2] D W Brenner O A Shenderova J A Harrison S J Stuart B Ni and S B Sinnott A second-generation reactive empirical bond order (REBO) potential energy expression for hydrocarbons J Phys Condens Matter 14 783 (2002) [3] A Kınacı J B Haskins C Sevik and T Ccedilağın Thermal conductivity of BN-C nanostructures Phys Rev B 86 (2012) [4] W Ouyang I Azuri D Mandelli A Tkatchenko L Kronik M Urbakh and O Hod Mechanical and Tribological Properties of Layered Materials under High Pressure Assessing the Importance of Many-Body Dispersion Effects J Chem Theory Comput 16 666 (2020) [5] S Plimpton Fast Parallel Algorithms for Short-Range Molecular Dynamics J Comput Phys 117 1 (1995) [6] W Ouyang D Mandelli M Urbakh and O Hod Nanoserpents Graphene Nanoribbon Motion on Two-Dimensional Hexagonal Materials Nano Lett 18 6009 (2018) [7] W Shinoda M Shiga and M Mikami Rapid estimation of elastic constants by molecular dynamics simulation under constant stress Phys Rev B 69 134103 (2004) [8] G L Pollack Kapitza Resistance Rev Mod Phys 41 48 (1969) [9] H-S Yang G R Bai L J Thompson and J A Eastman Interfacial thermal resistance in nanocrystalline yttria-stabilized zirconia Acta Mater 50 2309 (2002) [10] Q Fu J Yang Y Chen D Li and D Xu Experimental evidence of very long intrinsic phonon mean free path along the c-axis of graphite Appl Phys Lett 106 031905 (2015) [11] J Chen J H Walther and P Koumoutsakos Strain Engineering of Kapitza Resistance in Few-Layer Graphene Nano Lett 14 819 (2014) [12] H Zhang X Chen Y-D Jho and A J Minnich Temperature-Dependent Mean Free Path Spectra of Thermal Phonons Along the c-Axis of Graphite Nano Lett 16 1643 (2016) [13] D P Sellan E S Landry J E Turney A J H McGaughey and C H Amon Size effects in molecular dynamics thermal conductivity predictions Phys Rev B 81 214305 (2010) [14] P Jiang X Qian R Yang and L Lindsay Anisotropic thermal transport in bulk hexagonal boron nitride Phys Rev Mater 2 064005 (2018) [15] J Zhang Y Hong and Y Yue Thermal transport across graphene and single layer hexagonal boron nitride J Appl Phys 117 134307 (2015) [16] B Liu F Meng C D Reddy J A Baimova N Srikanth S V Dmitriev and K Zhou Thermal transport in a graphenendashMoS2 bilayer heterostructure a molecular dynamics study RSC Advances 5 29193 (2015) [17] Z Ding Q-X Pei J-W Jiang W Huang and Y-W Zhang Interfacial thermal conductance in grapheneMoS2 heterostructures Carbon 96 888 (2016) [18] N N T Nam and M Koshino Lattice relaxation and energy band modulation in twisted bilayer graphene Phys Rev B 96 075311 (2017)
S12
[19] R Ramiacutearez and M C Boumlhm Simple geometric generation of special points in brillouin-zone integrations Two-dimensional bravais lattices Int J Quantum Chem 30 391 (1986) [20]S L Cunningham Special points in the two-dimensional Brillouin zone Phys Rev B 10 4988 (1974) [21] H J Monkhorst and J D Pack Special points for Brillouin-zone integrations Phys Rev B 13 5188 (1976) [22] D J Chadi Special points for Brillouin-zone integrations Phys Rev B 16 1746 (1977)
1
Fermirsquos golden rule for phonons
1 Basic theory for phonons 11 Basic notations
Let us consider a 3D crystal with a total of 119873119873 = 119873119873111987311987321198731198733 unit cells and periodic boundary conditions
To be specific let 119938119938119894119894 119894119894 = 123 be the lattice vectors that define the unit cell We index unit cells
with n = (n1n2n3) where each ni = 12 ∙∙∙ Ni and their locations are 119929119929119899119899 = sum 1198991198991198941198941199381199381198941198943119894119894=1 Assume that
there are r atoms in each unit cell which are indexed with 119904119904 = 1⋯ 119903119903 The mass and the equilibrium
distance of the sth atom are notated as 119872119872s and 119929119929s0 respectively Then the location of the sth atom in
the nth unit cell at time t can be expressed as
119955119955119899119899119899119899(119905119905) = 119929119929119899119899 + 119929119929s0 + 119958119958119899119899119899119899(119905119905) (11)
where 119958119958119899119899119899119899(119905119905) is its displacement from its equilibrium position
The Lagrangian for this classical problem can be written as
ℒ = sum sum 119872119872119904119904|119955119899119899119904119904|2(119905119905)2
119903119903119899119899=1
119873119873119899119899=1 minus 119881119881 (12)
where the second term is the sum of interactions between all pairs of atoms
Under the harmonic approximation ie expanding the total potential energy 119881119881 around the
equilibrium positions The Lagrangian can be simplified as
ℒ = sum sum 119872119872119904119904|119958119899119899119904119904|2(119905119905)2
119903119903119899119899=1
119873119873119899119899=1 minus 1
2sum sum sum 120601120601120572120572120572120572prime
119899119899119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime120572120572120572120572prime119899119899119899119899prime119899119899119899119899prime (13)
where 119906119906119899119899119899119899120572120572 120572120572 = 123 are the Cartesian coordinates of the displacement 119958119958119899119899119899119899(119905119905) and
120601120601120572120572120572120572prime 119899119899119899119899prime119904119904 119904119904prime = 1205971205972119881119881
120597120597119903119903119899119899119904119904119899119899119903119903119899119899prime119904119904prime119899119899primeeq
= 120601120601120572120572prime120572120572 119899119899prime119899119899119904119904prime 119904119904 = 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime (14)
Note that the first order term vanishes because we are expanding around the equilibrium positions
120601120601120572120572120572120572prime 119899119899119899119899prime119904119904 119904119904prime represents the component of the force acted on the sth atom in the nth unit cell along 120572120572
direction when the atom 119904119904prime in the unit cell 119899119899prime moves a unit displacement along 120572120572prime direction The
symmetries of 120601120601120572120572120572120572prime 119899119899 119899119899prime119904119904 119904119904prime appearing in Eq (14) arise from the intechangability of the second
derivative and the translational invariance of the interactions
12 Dynamical matrix
The equation of motion of the sth atom in the nth unit cell can be derived using the EulerndashLagrange
equation as follows
2
119872119872119899119899119906119899119899119899119899120572120572 = minussum 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119899119899prime119899119899prime120572120572prime 119906119906119899119899prime119899119899prime120572120572prime (15)
If we displace all atoms equally ie shifting 119906119906119899119899prime119899119899prime120572120572prime to 119906119906119899119899prime119899119899prime120572120572prime + 120575120575 the total force on the sth atom in
the nth unit cell does not change From the above equation we have
sum 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119899119899prime120572120572prime = 0 (16)
We are looking for normal modes (because any general solution can be written as a linear combination
of them) these are solutions where all atoms oscillate with the same frequency Moreover because
of the lattice structure we expect solutions to reflect this periodicity So we guess solutions of the
form
119906119906119899119899119899119899120572120572(119905119905) = 1119872119872119904119904
119890119890119899119899120572120572119890119890119894119894(119954119954∙119929119929119899119899minus120596120596119905119905) (17)
where 119890119890119899119899120572120572 are real-space solutions that will be determined later and 119954119954 is the wave-vector in
reciprocal space Substituting Eq (17) into Eq (15) we can get
1205961205962119890119890119899119899120572120572(119954119954) = sum 119863119863119899119899120572120572119899119899prime120572120572prime(119954119954)119890119890119899119899prime120572120572prime(119954119954)119899119899prime120572120572prime (18)
where
119863119863119899119899120572120572119899119899prime120572120572prime(119954119954) = sum 1
119872119872119904119904119872119872119904119904prime120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119890119890
minus119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime119899119899prime = sum 1
119872119872119904119904119872119872119904119904prime120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime 119890119890
minus119894119894119954119954∙119929119929119897119897119897119897 (19)
is called dynamical matrix (dimension 3119903119903 times 3119903119903) Note that we have defined the relative cell distance
vector 119929119929119897119897 equiv 119929119929119899119899 minus 119929119929119899119899prime and number index 119897119897 equiv 119899119899 minus 119899119899prime where the infinite sum over 119899119899prime can be
replaced by the sum over 119897119897 for any value of the index 119899119899 Note that dynamical matrix is Hermitian
symmetric (ie 119863119863119899119899120572120572119899119899prime120572120572prime(119954119954)
lowast= 119863119863119899119899prime120572120572prime
119899119899120572120572 (minus119954119954)) because ϕ is symmetric so all the eigenvalues of Eq (18)
(120596120596120582120582(119954119954) 120582120582 = 12⋯ 3119903119903) are real for each 119954119954 in the Brillouin zone (BZ) which is determined by
det1205961205961205821205822(119954119954)120575120575119899119899120572120572120575120575119899119899prime120572120572prime minus 119863119863119899119899120572120572119899119899prime120572120572prime(119954119954) = 0 (110)
Taking the conjugate of this equation we have
0 = det 1205961205961205821205822(119954119954)lowast120575120575119899119899120572120572120575120575119899119899prime120572120572prime minus 119863119863119899119899120572120572119899119899
prime120572120572prime(119954119954)lowast = det1205961205961205821205822(119954119954)120575120575119899119899120572120572120575120575119899119899prime120572120572prime minus 119863119863119899119899120572120572119899119899
prime120572120572prime(minus119954119954) (111)
while replacing 119954119954 by minus119954119954 in Eq (110) we have det1205961205961205821205822(minus119954119954)120575120575119899119899120572120572120575120575119899119899prime120572120572prime minus 119863119863119899119899120572120572119899119899prime120572120572prime(minus119954119954) = 0
Itrsquos clear that 1205961205961205821205822(119954119954) and 1205961205961205821205822(minus119954119954) obey the same equation thus we have
120596120596120582120582(minus119954119954) = 120596120596120582120582(119954119954) (112)
The corresponding eigenvectors are orthonormal
sum 119890119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582
prime119899119899120572120572 = 120575120575120582120582120582120582prime (113)
The complex conjugate of Eq (18) gives
1205961205962119890119890119899119899120572120572lowast (119954119954) = sum 119863119863119899119899120572120572119899119899prime120572120572prime(119954119954)
lowast119890119890119899119899prime120572120572primelowast (119954119954)119899119899prime120572120572prime = sum 119863119863119899119899prime120572120572prime
119899119899120572120572 (minus119954119954)119890119890119899119899prime120572120572primelowast (119954119954)119899119899prime120572120572prime (114)
3
While replacing 119954119954 by minus119954119954 in Eq (18) we get the following equation
1205961205962119890119890119899119899120572120572(minus119954119954) = sum 119863119863119899119899120572120572119899119899prime120572120572prime(minus119954119954)119890119890119899119899prime120572120572prime(minus119954119954)119899119899prime120572120572prime (115)
From Eq (114) and Eq (115) we see that eigenvectors 119890119890119899119899120572120572120582120582 (minus119954119954)lowast and 119890119890119899119899120572120572120582120582 (119954119954) obey the same
eigenvalue equation Since the eigenvectors are normalized we get the following property
119890119890119899119899120572120572120582120582 (minus119954119954)lowast
= 119890119890119899119899120572120572120582120582 (119954119954) (116)
2 Second quantization The general solution is a linear combination of all these normal modes thus we have
119906119906119899119899119899119899120572120572(119905119905) = sum 119862119862120582120582(119954119954)119873119873119872119872119904119904
119890119890119899119899120572120572120582120582 (119954119954)119890119890minus119894119894120596120596119954119954120582120582119905119905119890119890119894119894119954119954∙119929119929119899119899119954119954120582120582 = sum 119876119876120582120582(119954119954119905119905)119873119873119872119872119904119904
119890119890119899119899120572120572120582120582 (119954119954)119890119890119894119894119954119954∙119929119929119899119899119954119954120582120582 (21)
where the we define the normal coordinates as 119876119876120582120582(119954119954 119905119905) equiv 119862119862120582120582(119954119954)119890119890minus119894119894120596120596120582120582(119954119954)119905119905 in the eigenvectors
representation To ensure that the displacements 119906119906119899119899119899119899120572120572(119905119905) are real (namely 119906119906119899119899119899119899120572120572lowast (119905119905) = 119906119906119899119899119899119899120572120572(119905119905)) the
following relation on 119876119876120582120582(119954119954 119905119905) and 119890119890119899119899120572120572120582120582 is enforced
119876119876120582120582(119954119954 119905119905)119890119890119899119899120572120572120582120582 (119954119954)lowast
= 119876119876120582120582(minus119954119954 119905119905)119890119890119899119899120572120572120582120582 (minus119954119954) (22)
where we used the fact that the sum over 119954119954 runs symmetrically over both negative and positive
values Using Eq (116) we have
[119876119876120582120582(119954119954 119905119905)]lowast = 119876119876120582120582(minus119954119954 119905119905) (23)
21 Kinetic energy term
Using Eq (21) the kinetic energy of the system can be expressed as
119879119879 = 12sum 1198721198721198991198991199061198991198991198991198991205721205722 (119905119905)119899119899119899119899120572120572 = 1
2sum 119872119872119904119904
119873119873119872119872119904119904sum sum 119876120582120582(119954119954 119905119905)119876120582120582prime(119954119954prime 119905119905)119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582
prime(119954119954prime) sum 119890119890119894119894119954119954+119954119954prime∙119929119929119899119899119899119899119954119954prime120582120582prime119954119954120582120582119899119899120572120572 =
12sum sum 119876120582120582(119954119954 119905119905)119876120582120582prime(minus119954119954 119905119905)119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582
prime(minus119954119954)119954119954119954119954prime120582120582prime =119899119899120572120572
12sum 119876120582120582(119954119954 119905119905)119876120582120582prime
lowast (119954119954 119905119905) sum 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582prime(119954119954)
lowast119899119899120572120572 =119954119954120582120582120582120582prime
12sum 119876120582120582(119954119954 119905119905)119876120582120582lowast(119954119954 119905119905)119954119954120582120582 =
12sum 119876120582120582(119954119954 119905119905)119876120582120582(minus119954119954 119905119905)119954119954120582120582
ie
119879119879 = 12sum 119876120582120582(119954119954 119905119905)119876120582120582(minus119954119954 119905119905)119954119954120582120582 (24)
To derive Eq (24) we used Eqs (113) and (116) as well as the following equations
1119873119873sum 119890119890119894119894119954119954+119954119954prime∙119929119929119899119899119899119899 = 120575120575119954119954+119954119954prime120782120782 (25)
4
22 Potential energy term
Similarity the potential energy can be rewritten in terms of the normal coordinates as follows
119880119880 =12120601120601119899119899119899119899120572120572119899119899prime119899119899prime120572120572prime119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
120572120572120572120572prime119899119899119899119899prime119899119899119899119899prime=
12120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119899119899+119954119954prime∙119929119929119899119899prime
119873119873119872119872119899119899119872119872119899119899prime
[119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(119954119954prime 119905119905)] 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime120582120582prime (119954119954prime)
119954119954prime120582120582prime119954119954120582120582119899119899119899119899prime120572120572120572120572prime119899119899119899119899prime
=12120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894(119954119954+119954119954prime)∙119929119929119899119899119890119890minus119894119894119954119954prime∙119929119929119897119897
119873119873119872119872119899119899119872119872119899119899prime
[119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(119954119954prime 119905119905)] 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime120582120582prime (119954119954prime)
119954119954prime120582120582prime119954119954120582120582119899119899119897119897120572120572120572120572prime119899119899119899119899prime
=12120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890minus119894119894119954119954prime∙119929119929119897119897
119872119872119899119899119872119872119899119899prime[119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(119954119954prime 119905119905)] 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (119954119954prime)1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899
119899119899=120575120575119902119902+119902119902prime=0
119954119954prime120582120582prime119954119954120582120582119897119897120572120572120572120572prime119899119899119899119899prime
=12120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894119954119954119929119929119897119897
119872119872119899119899119872119872119899119899prime [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)] 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (minus119954119954)119954119954120582120582120582120582prime119897119897120572120572120572120572prime119899119899119899119899prime
=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]119890119890119899119899120572120572120582120582 (119954119954)
119899119899120572120572
119863119863119899119899120572120572119899119899prime120572120572prime(minus119954119954)119890119890119899119899prime120572120572prime
120582120582prime (minus119954119954)119899119899prime120572120572prime119954119954120582120582120582120582prime
=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]119890119890119899119899120572120572120582120582 (119954119954)
119899119899120572120572
1205961205961205821205822(minus119954119954)119890119890119899119899120572120572120582120582prime (minus119954119954)
119954119954120582120582120582120582prime
=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]1205961205961205821205822(119954119954)120575120575120582120582120582120582prime =119954119954120582120582120582120582prime
121205961205961205821205822(119954119954)119876119876120582120582(119954119954 119905119905)119876119876120582120582(minus119954119954 119905119905)
119954119954120582120582
ie
119880119880 = 12sum 1205961205961205821205822(119954119954)119876119876120582120582(119954119954 119905119905)119876119876120582120582(minus119954119954 119905119905)119954119954120582120582 (26)
where in above deriviation the orthogonality of its eigenvectores and the symmetry of its eigenvalues
are used Thus the Lagrangian reads as
ℒ = 119879119879 minus 119881119881 = 12sum 119876120582120582(minus119954119954 119905119905)119876120582120582(119954119954 119905119905) minus 120596120596120582120582
2(119954119954)119876119876120582120582(minus119954119954 119905119905)119876119876120582120582(119954119954 119905119905)119954119954120582120582 (27)
Using the relation 119875119875120582120582(119954119954 119905119905) = 120597120597ℒ part119876120582120582(119954119954 119905119905) the Hamiltonian of the system then can be written as
119867119867 = 119879119879 + 119881119881 = 12sum [119875119875120582120582(minus119954119954 119905119905)119875119875120582120582(119954119954 119905119905) + 120596120596120582120582
2(119954119954)119876119876120582120582(minus119954119954 119905119905)119876119876120582120582(119954119954 119905119905)]119954119954120582120582 (28)
Now we quantize H by asking the momenta and coordinates to be operators
119867119867 = 12sum 119875119875minus119954119954120582120582119875119875119954119954120582120582 + 120596120596119954119954120582120582
2 119876119876minus119954119954120582120582119876119876119954119954120582120582119954119954120582120582 (29)
here 119875119875119954119954120582120582 and 119876119876119954119954120582120582 obey the following commutation relations
119876119876119954119954120582120582119875119875119954119954prime120582120582prime = 119894119894ℏ120575120575119954119954119954119954prime120575120575120582120582120582120582prime
119876119876119954119954120582120582119876119876119954119954prime120582120582prime = 0 119875119875119954119954120582120582119875119875119954119954prime120582120582prime = 0 (210)
Similar to the case of ordinary quantum harmonic oscillators it is convenient to define ladder
operators for each mode as follows
119876119876119954119954120582120582 = ℏ
2120596120596119954119954120582120582119887119887119954119954120582120582 + 119887119887minus119954119954120582120582
dagger 119875119875119954119954120582120582 = 119894119894ℏ120596120596119954119954120582120582
2119887119887119954119954120582120582
dagger minus 119887119887minus119954119954120582120582 (211)
where 119887119887119954119954120582120582dagger and 119887119887119954119954120582120582 are Bosonic creation and annihilation operators for phonons with momentum
119954119954 branch index 120582120582 and frequency 120596120596119954119954120582120582 which obeys the Bosonic commutation relation
5
119887119887119954119954120582120582 119887119887119954119954prime120582120582prime
dagger = 120575120575119954119954119954119954prime120575120575120582120582120582120582prime
119887119887119954119954120582120582 119887119887119954119954prime120582120582prime = 0 119887119887119954119954120582120582dagger 119887119887119954119954prime120582120582prime
dagger = 0 (212)
Substituting Eqs (211) into (29) and using the properties Eq (210) and (212) we have
119867119867 =12119875119875minus119954119954120582120582119875119875119954119954120582120582 +120596120596119954119954120582120582
2 119876119876minus119954119954120582120582119876119876119954119954120582120582119954119954120582120582
=12minus
ℏ120596120596119954119954120582120582
2119887119887minus119954119954120582120582
dagger minus 119887119887119954119954120582120582119887119887119954119954120582120582dagger minus 119887119887minus119954119954120582120582+ 120596120596119954119954120582120582
2 ℏ2120596120596119954119954120582120582
119887119887minus119954119954120582120582 + 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 + 119887119887minus119954119954120582120582
dagger 119954119954120582120582
=ℏ4120596120596119954119954120582120582minus119887119887minus119954119954120582120582
dagger 119887119887119954119954120582120582dagger minus 119887119887minus119954119954120582120582
dagger 119887119887minus119954119954120582120582 minus 119887119887119954119954120582120582119887119887119954119954120582120582dagger + 119887119887119954119954120582120582119887119887minus119954119954120582120582
119954119954120582120582
+ 119887119887minus119954119954120582120582119887119887119954119954120582120582 + 119887119887minus119954119954120582120582119887119887minus119954119954120582120582dagger + 119887119887119954119954120582120582
dagger 119887119887119954119954120582120582 + 119887119887119954119954120582120582dagger 119887119887minus119954119954120582120582
dagger
=ℏ4120596120596119954119954120582120582119887119887minus119954119954120582120582
dagger 119887119887minus119954119954120582120582 + 119887119887minus119954119954120582120582119887119887minus119954119954120582120582dagger + 119887119887119954119954120582120582119887119887119954119954120582120582
dagger + 119887119887119954119954120582120582dagger 119887119887119954119954120582120582
119954119954120582120582
=ℏ41205961205961199541199541205821205822119887119887minus119954119954120582120582
dagger 119887119887minus119954119954120582120582 + 1+ 2119887119887119954119954120582120582dagger 119887119887119954119954120582120582 + 1
119954119954120582120582
= ℏ120596120596119954119954120582120582 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 +
12
119954119954120582120582
Therefore we finally get the quantized representation of non-interacting phonons
119867119867 = sum ℏ120596120596119954119954120582120582 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 + 1
2119954119954120582120582 (213)
The operator of atom displacements (Eq 114) is expressed in terms of the phonon operators by
119906119906119899119899119899119899120572120572 = sum ℏ
2119873119873119872119872119904119904120596120596119954119954120582120582119890119890119899119899120572120572120582120582 119890119890119894119894119954119954∙119929119929119899119899119887119887119954119954120582120582 + 119887119887minus119954119954120582120582
dagger 119954119954120582120582 (214)
These equations will be used below
3 Inter-phonon coupling within harmonic approximation 31 Hamiltonian with inter-phonon coupling
In this section we consider systems that consist of two (or more) covalently bonded units that are
weakly coupled between them Unlike previous studies that considered phonon-phonon couplings
resulting from anharmonicity effects [1] here all phonon mode considered are Harmonic and the
couplings arise from the division of the entire system into subunits The Hamiltonian of the whole
system can be written as
119867119867 = 1198671198671 + 1198671198672 + 11986711986712 (31)
To derive the expressions of 1198671198671 1198671198672 and 11986711986712 in Eq (31) we consider the Hamiltonian of the
whole system written as the function of the atomic displacements
119867119867 = 119879119879 + 119881119881 = sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)2119899119899119899119899120572120572 + 1
2sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime120572120572120572120572prime119899119899119899119899prime119899119899119899119899prime (32)
6
Letrsquos assume that subsystem I and II contain atoms with indeices ranging from 119904119904 = 12⋯ 1199031199032 and
119904119904 = 1199031199032
+ 1 1199031199032
+ 2⋯ 119903119903 respectively Then the kinetic energy term in Eq (32) can be rewritten as
119879119879 = sum sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)2
1199031199032119899119899=1 + sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)
2119903119903119899119899=1199031199032+1
119899119899120572120572 (33)
While the potential energy term is
119881119881 =12 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
1199031199032
119899119899119899119899prime=1
+ 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
119903119903
119899119899119899119899prime=1199031199032+1
+ 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime119899119899119899119899prime
+ 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
1199031199032
119899119899prime=1
119903119903
119899119899=1199031199032+1
= 11988111988111 + 11988111988122 + 11988111988112 + 11988111988121
Or equivalently
⎩⎪⎪⎨
⎪⎪⎧ 11988111988111 = 1
2sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
1199031199032119899119899119899119899prime=1120572120572120572120572prime119899119899119899119899prime
11988111988122 = 12sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
119903119903119899119899119899119899prime=1199031199032+1120572120572120572120572prime119899119899119899119899prime
11988111988112 = 12sum sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
119903119903119899119899prime=1199031199032+1
1199031199032119899119899=1120572120572120572120572prime119899119899119899119899prime
11988111988121 = 12sum sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
1199031199032119899119899prime=1
119903119903119899119899=1199031199032+1120572120572120572120572prime119899119899119899119899prime
(34)
32 Second quantization
We may now quantize this Hamiltonian and write it in the basis of the eigenstates of the coupled
subunits In second quantization the atomic displacement operators of the two subunits are given in
the following form
119906119906119899119899119899119899120572120572 =
⎩⎨
⎧ sum ℏ
2119873119873119872119872119904119904ω119954119954120582120582119890119890119899119899120572120572120582120582 119890119890119894119894119954119954∙119929119929119899119899119886119886119954119954120582120582 + 119886119886minus119954119954120582120582
dagger 119954119954120582120582 119904119904 isin 1 1199031199032
sum ℏ
2119873119873119872119872119904119904120596120596119954119954prime120582120582prime119890119899119899120572120572120582120582
prime119890119890119894119894119954119954prime∙119929119929119899119899 119886119886 119954119954prime120582120582prime + 119886119886minus119954119954prime120582120582primedagger 119954119954prime120582120582prime 119904119904 isin 119903119903
2+ 1 119903119903
(35)
Here we use different notations for the creation and annihilation operators (119886119886119954119954120582120582119886119886minus119954119954120582120582dagger and 119886119886119954119954120582120582119886119886minus119954119954120582120582
dagger )
eigenvalues (ω119954119954120582120582120596120596119954119954120582120582) and eigenvectors (119890119890119899119899120572120572120582120582 119890119899119899120572120572120582120582 ) for the two subunits Note that 119890119890119899119899120572120572120582120582 and 119890119899119899120572120572120582120582 are
of the dimensions of the whole system however their non-zero elements appear only on the relevant
subunits such that they obey the following relations sum 119890119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582
prime119899119899120572120572 = 120575120575120582120582120582120582prime sum 119890119899119899120572120572120582120582
lowast119890119899119899120572120572120582120582
prime119899119899120572120572 =
120575120575120582120582120582120582prime sum 119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582
prime119899119899120572120572 = sum 119890119890119899119899120572120572120582120582
lowast119890119899119899120572120572120582120582
prime119899119899120572120572 = 0 Defining the normal coordinates of the two subunits as
119876119876119954119954120582120582 equiv ℏ
2ω119954119954120582120582119886119886119954119954120582120582 + 119886119886minus119954119954120582120582
dagger 119876119876119954119954120582120582 equiv ℏ
2ω119954119954120582120582119886119886119954119954120582120582 + 119886119886minus119954119954120582120582
dagger (36)
and the corresponding momenta operators as
7
119875119875119954119954120582120582 = 119894119894ℏω119954119954120582120582
2119886119886119954119954120582120582
dagger minus 119886119886minus119954119954120582120582 119875119875119954119954120582120582 = 119894119894ℏω1199541199541205821205822
119886119886119954119954120582120582dagger minus 119886119886minus119954119954120582120582 (37)
Eq (35) reads
119906119906119899119899119899119899120572120572 = sum 119890119890119904119904119899119899120582120582 (119954119954)
119873119873119872119872119904119904119890119890119894119894119954119954∙119929119929119899119899119876119876119954119954120582120582119954119954120582120582 119904119904 isin 1 119903119903
2
sum 119890119904119904119899119899120582120582 (119954119954)119873119873119872119872119904119904
119890119890119894119894119954119954∙119929119929119899119899119876119876119954119954120582120582119954119954120582120582 119904119904 isin 1199031199032
+ 1 119903119903 (38)
The second quantized kinetic energy operator is then written as
119879119879 = 1198791198791 + 1198791198792 = 12sum 119875119875119954119954120582120582119875119875minus119954119954120582120582119954119954120582120582 + 1
2sum 119875119875119954119954prime120582120582prime119875119875minus119954119954prime120582120582prime119954119954prime120582120582prime (39)
Correspondingly the various potential energy terms are obtained by substituting Eq (38) in Eq (34)
11988111988111 =12 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119899119899119899119899prime
119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime120582120582prime (119954119954prime)
119872119872119899119899119872119872119899119899prime
1119873119873119890119890119894119894119954119954∙119929119929119899119899+119894119894119954119954prime∙119929119929119899119899prime119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954120582120582119954119954prime120582120582prime
1199031199032
119899119899119899119899prime=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954119954119954prime120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (119954119954prime)1119873119873 119890119890
119894119894(119954119954+119954119954prime)119929119929119899119899prime
119899119899prime120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime
119872119872119899119899119872119872119899119899prime119899119899
1199031199032
119899119899119899119899prime=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954119954119954prime120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (119954119954prime) 120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime
1199031199032
119899119899119899119899prime=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime
119954119954120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (minus119954119954)120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
1199031199032
119899119899119899119899prime=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime
119954119954120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954) 119863119863119899119899120572120572119899119899
prime120572120572prime(minus119954119954)119890119890119899119899prime120572120572prime120582120582prime (minus119954119954)
120572120572prime
1199031199032
119899119899prime=1
1199031199032
119899119899=1120572120572
=12 ω119954119954120582120582prime
2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582120582120582prime
119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582prime (119954119954)
lowast
1199031199032
119899119899=1
=120572120572
12ω119954119954120582120582
2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582
ie
11988111988111 = 12sum ω119954119954120582120582
2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582 (310)
Going from the second line to the third line we used the fact that the sum over 119899119899prime runs between plusmninfin
and the summand depends only on the difference between 119899119899 and 119899119899prime hence the sum is independent
of the value of the index 119899119899 Therefore we can replace the sum over 119899119899prime by a sum over 119897119897 equiv 119899119899 minus 119899119899prime
amd define 119929119929119897119897 equiv 119929119929119899119899 minus 119929119929119899119899prime Following the same procedure we can get the corresponding
expressions for the second diagonal term
11988111988122 = 12sum 1205961205961199541199541205821205822 119876119876119954119954120582120582119876119876minus119954119954120582120582119954119954120582120582 (311)
Similarly for the off-diagonal terms we get
8
11988111988112 =12 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119899119899119899119899prime
119903119903
119899119899prime=1199031199032+1
119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime
120582120582prime (119954119954prime)
119872119872119899119899119872119872119899119899prime
1119873119873119890119890119894119894119954119954∙119929119929119899119899+119894119894119954119954prime∙119929119929119899119899prime119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954120582120582119954119954prime120582120582prime
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954119954119954prime120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime
120582120582prime (119954119954prime)1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime
119872119872119899119899119872119872119899119899prime119899119899
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954119954119954prime120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime
120582120582prime (119954119954prime) 120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime
119954119954120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime
120582120582prime (minus119954119954)120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954
119890119899119899prime120572120572prime120582120582prime (119954119954)
lowast120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119890119890119899119899120572120572120582120582 (119954119954)119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954
Where we have defined
119881119881120582120582120582120582prime(119954119954) equiv sum sum sum 119890119899119899prime120572120572prime120582120582prime (119954119954)
lowast119881119881119899119899120572120572119899119899
prime120572120572prime(119954119954)119890119890119899119899120572120572120582120582 (119954119954)119903119903119899119899prime=1199031199032+1
1199031199032119899119899=1120572120572120572120572prime (312)
where
119881119881119899119899120572120572119899119899prime120572120572prime(119954119954) equiv sum 119890119890119894119894119954119954∙119929119929119897119897
119872119872119904119904119872119872119904119904prime119897119897 120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime (313)
Itrsquos easy to show that 119881119881120582120582120582120582prime(119954119954) has the following property
119881119881120582120582120582120582primelowast (119954119954) = 119881119881120582120582120582120582prime(minus119954119954) (314)
Following the same procedure we have
9
11988111988121 =12 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119899119899119899119899prime
119903119903
119899119899=1199031199032+1
119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (119954119954prime)
119872119872119899119899119872119872119899119899prime
1119873119873119890119890119894119894119954119954∙119929119929119899119899+119894119894119954119954prime∙119929119929119899119899prime119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954120582120582119954119954prime120582120582prime
1199031199032
119899119899prime=1120572120572120572120572prime
=12 120601120601120572120572prime120572120572
119899119899prime minus 119899119899119904119904prime 119904119904
119899119899prime119899119899
119903119903
119899119899prime=1199031199032+1
119890119899119899prime120572120572prime120582120582prime (119954119954prime)119890119890119899119899120572120572120582120582 (119954119954)
119872119872119899119899prime119872119872119899119899
1119873119873 119890119890
119894119894119954119954prime∙119929119929119899119899prime+119894119894119954119954∙119929119929119899119899119876119876119954119954prime120582120582prime119876119876119954119954120582120582119954119954prime120582120582prime119954119954120582120582
1199031199032
119899119899=1120572120572prime120572120572
=12 119876119876119954119954prime120582120582prime119876119876119954119954120582120582
119954119954119954119954prime120582120582120582120582prime119890119899119899prime120572120572prime120582120582prime (119954119954prime)119890119890119899119899120572120572120582120582 (119954119954)
1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime
119872119872119899119899119872119872119899119899prime119899119899
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954prime120582120582prime119876119876119954119954120582120582
119954119954119954119954prime120582120582120582120582prime119890119899119899prime120572120572prime120582120582prime (119954119954prime)119890119890119899119899120572120572120582120582 (119954119954) 120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954prime120582120582prime119876119876minus119954119954prime120582120582119890119899119899prime120572120572prime
120582120582prime (119954119954prime)119890119890119899119899120572120572120582120582 (minus119954119954prime)120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890minus119894119894119954119954prime∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897119954119954prime120582120582120582120582prime
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954prime120582120582prime119876119876119954119954prime120582120582
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954prime 119890119890119899119899120572120572120582120582 (119954119954prime)
lowast120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890minus119894119894119954119954prime∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119890119899119899prime120572120572prime120582120582prime (119954119954prime)
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582prime119876119876119954119954120582120582
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954
119890119890119899119899120572120572120582120582 (119954119954)lowast120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890minus119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119890119899119899prime120572120572prime120582120582prime (119954119954)
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger dagger
3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954
119890119899119899prime120572120572prime120582120582prime (119954119954)
lowast120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119890119890119899119899120572120572120582120582 (119954119954)119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
lowast
=
⎩⎨
⎧12 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954⎭⎬
⎫dagger
= 11988111988112dagger
Define
1198671198671 = 11987911987911 + 119881119881111198671198672 = 11987911987922 + 1198811198812211986711986712 = 11988111988112 + 11988111988121
(315)
we finally get the expressions of 1198671198671 1198671198672 and 11986711986712 in Eq (31) as follows
⎩⎪⎨
⎪⎧ 1198671198671 = sum sum ℏ120596120596119954119954120582120582 119886119886119954119954120582120582
dagger 119886119886119954119954120582120582 + 1231199031199032
120582120582=1119954119954
1198671198672 = sum sum ℏ120596120596119954119954120582120582prime 119886119886119954119954120582120582primedagger 119886119886119954119954120582120582prime + 1
23119903119903
120582120582prime=1+31199031199032119954119954
11986711986712 = 12sum sum sum 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903120582120582prime=31199031199032 +1
31199031199032120582120582=1 + h c 119954119954
(316)
Here hc means the Hermitian conjugate Since the indices of 119886119886119954119954120582120582 119886119886119954119954120582120582prime and 119876119876119954119954120582120582 119876119876119954119954120582120582prime belong to
the two system sections we can define an abbreviated notation 119886119886119954119954120582120582 and 119876119876119954119954120582120582 using the index 120582120582 to
identify which subsystem they belong to In this case the Hamiltonian of the coupled system can be
10
simplified as follows
119867119867 = 1198671198670 + 119867119867119862119862 (317)
where
1198671198670 = sum sum ℏ120596120596119954119954120582120582 119886119886119954119954120582120582
dagger 119886119886119954119954120582120582 + 123119903119903
120582120582=1119954119954
119867119867119862119862 = 12sum sum sum 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903120582120582prime=31199031199032 +1
31199031199032120582120582=1 + ℎ 119888119888 119954119954
(318)
4 Greenrsquos function 41 Dynamics of the ladder operators
In order to describe the dynamics of the ladder operators appearing in the Hamiltonian of Eq (317)
we express them in the Heisenberg picture as follows
119886119886119901119901(120591120591) = 119890119890119894119894ℏ119867119867119905119905119886119886119901119901119890119890
minus119894119894ℏ119867119867119905119905 = 119890119890
119867119867120591120591ℏ 119886119886119901119901119890119890
minus119867119867120591120591ℏ (41)
where we define the imaginary time 120591120591 equiv 119894119894119905119905 and introduce the notation 119953119953 equiv (119954119954 120582120582) and 119953119953 equiv (minus119954119954 120582120582)
The corresponding equation of motion for the ladder operators is given by
ℏ part119886119886119953119953(120591120591)
120597120597120591120591= 119867119867119886119886119953119953(120591120591) (42)
For the uncoupled system (119867119867119862119862 = 0) this gives
ℏpart119886119886119953119953(120591120591)120597120597120591120591
= 1198671198670119886119886119953119953(120591120591) = 1198671198670 1198901198901198671198670120591120591ℏ 119886119886119953119953119890119890
minus1198671198670120591120591ℏ = 1198671198670119890119890
1198671198670120591120591ℏ 119886119886119953119953119890119890
minus1198671198670120591120591ℏ minus 119890119890
1198671198670120591120591ℏ 119886119886119953119953119890119890
minus1198671198670120591120591ℏ 1198671198670
= 1198901198901198671198670120591120591ℏ 1198671198670119886119886119953119953 minus 1198861198861199531199531198671198670119890119890
minus1198671198670120591120591ℏ = 119890119890
1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890
minus1198671198670120591120591ℏ
ie
ℏ part119886119886119953119953(120591120591)
120597120597120591120591= 119890119890
1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890
minus1198671198670120591120591ℏ (43)
The commutator on the right-hand-side of Eq (43) can be evaluated as follows
1198671198670119886119886119953119953prime = sum ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953 + 1
2119953119953 119886119886119953119953prime = sum ℏ120596120596119953119953119886119886119953119953
dagger119886119886119953119953119886119886119953119953prime119953119953 = sum ℏ120596120596119953119953119886119886119953119953dagger119886119886119953119953119886119886119953119953prime minus 119886119886119953119953prime119886119886119953119953
dagger119886119886119953119953119953119953 =
sum ℏ120596120596119953119953119886119886119953119953dagger119886119886119953119953119886119886119953119953prime minus 120575120575119953119953prime119953119953 + 119886119886119953119953
dagger119886119886119953119953prime119886119886119953119953119953119953 = minusℏ120596120596119953119953prime119886119886119953119953prime + sum ℏ120596120596119953119953119886119886119953119953dagger119886119886119953119953119886119886119953119953prime minus 119886119886119953119953
dagger119886119886119953119953119886119886119953119953prime119953119953 = minusℏ120596120596119953119953prime119886119886119953119953prime
ie
1198671198670119886119886119953119953prime = minusℏ120596120596119953119953prime119886119886119953119953prime (44)
Therefore we have
ℏ part119886119886119953119953(120591120591)
120597120597120591120591= 119890119890
1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890
minus1198671198670120591120591ℏ = minusℏ120596120596119953119953119890119890
1198671198670120591120591ℏ 119886119886119953119953119890119890
minus1198671198670120591120591ℏ = minusℏ120596120596119953119953119886119886119953119953(120591120591) (45)
11
The solution of Eq (45) is 119886119886119953119953(120591120591) = 119886119886119953119953(0)119890119890minus120596120596119953119953120591120591 = 119886119886119953119953119890119890minus120596120596119953119953120591120591 For 119886119886119953119953dagger(120591120591) we have
1198671198670119886119886119953119953primedagger (0) = 1198671198670119886119886119953119953prime
dagger = ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953 +
12
119953119953
119886119886119953119953primedagger = ℏ120596120596119953119953119886119886119953119953
dagger119886119886119953119953119886119886119953119953primedagger
119953119953
= ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953119886119886119953119953prime
dagger minus 119886119886119953119953primedagger 119886119886119953119953
dagger119886119886119953119953119953119953
= ℏ120596120596119953119953 119886119886119953119953dagger 120575120575119953119953119953119953prime + 119886119886119953119953prime
dagger 119886119886119953119953 minus 119886119886119953119953primedagger 119886119886119953119953
dagger119886119886119953119953119953119953
= ℏ120596120596119953119953prime119886119886119953119953primedagger + ℏ120596120596119953119953 119886119886119953119953
dagger119886119886119953119953primedagger 119886119886119953119953 minus 119886119886119953119953prime
dagger 119886119886119953119953dagger119886119886119953119953
119953119953
= ℏ120596120596119953119953prime119886119886119953119953primedagger
In summary we have
119886119886119953119953(120591120591) = 119886119886119953119953(0)119890119890minus120596120596119953119953120591120591 = 119886119886119953119953119890119890minus120596120596119953119953120591120591
119886119886119953119953dagger(120591120591) = 119886119886119953119953
dagger(0)119890119890120596120596119953119953120591120591 = 119886119886119953119953dagger119890119890120596120596119953119953120591120591 (46)
42 Greenrsquos function
To describe thermal transport properties between different system sections we use the formalism of
thermal (or imaginary time) phonon Greenrsquos function [2] To this end we define the thermal Greenrsquos
function for phonons as
119866119866119953119953119953119953prime(120591120591 120591120591prime) = minuslang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime)rang = minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime) (47)
where 119879119879120591120591 is the time ordering operator and 120588120588119867119867 is the statistical operator for the grand canonical
ensemble (note that the chemical potential for phonons is zero)
120588120588119867119867 = 119890119890minus120573120573119867119867119885119885119867119867 (48)
where the partition function is given by 119885119885119867119867 equiv Tr119890119890minus120573120573119867119867 and 120573120573 = 1119896119896119861119861119879119879 with 119896119896119861119861 being
Boltzmannrsquos constant and 119879119879 the temperature For time independent Hamiltonians 119866119866119953119953119953119953prime(120591120591 120591120591prime)
depends only on 120591120591 minus 120591120591prime ie
119866119866119953119953119953119953prime(120591120591 120591120591prime) = 119866119866119953119953119953119953prime(120591120591 minus 120591120591prime 0) (49)
To show this we shall first assume that 120591120591 gt 120591120591prime such that
119866119866119953119953119953119953prime(120591120591 120591120591prime) = minuslang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime)rang = minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime) = minusTr 120588120588119867119867119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime)
= minusTr 120588120588119867119867119890119890119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591ℏ 119890119890
119867119867120591120591primeℏ 119886119886119953119953prime
dagger 119890119890minus119867119867120591120591primeℏ = minusTr 120588120588119867119867119890119890
minus119867119867120591120591primeℏ 119890119890
119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591minus120591120591prime
ℏ 119886119886119953119953primedagger
= minusTr 120588120588119867119867119890119890119867119867120591120591minus120591120591prime
ℏ 119886119886119953119953119890119890minus119867119867120591120591minus120591120591prime
ℏ 119886119886119953119953primedagger = minusTr 120588120588119867119867119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953prime
dagger (0)
= minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953primedagger (0) = minus lang119879119879120591120591119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953prime
dagger (0)rang = 119866119866119953119953119953119953prime(120591120591 minus 120591120591prime 0)
Where we have used the commutativity of 120588120588119867119867 and 119890119890plusmn119867119867120591120591prime
ℏ and the invariance of the trace operation
12
towards cyclic permutations Similarly for 120591120591 lt 120591120591prime we have
119866119866119953119953119953119953prime(120591120591 120591120591prime) = minuslang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime)rang = minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime) = minusTr 120588120588119867119867119886119886119953119953primedagger (120591120591prime)119886119886119953119953(120591120591)
= minusTr 120588120588119867119867119890119890119867119867120591120591primeℏ 119886119886119953119953prime
dagger 119890119890minus119867119867120591120591primeℏ 119890119890
119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591ℏ = minusTr 120588120588119867119867119886119886119953119953prime
dagger 119890119890119867119867120591120591minus120591120591prime
ℏ 119886119886119953119953119890119890minus119867119867120591120591ℏ 119890119890
119867119867120591120591primeℏ
= minusTr 120588120588119867119867119886119886119953119953primedagger 119890119890
119867119867120591120591minus120591120591primeℏ 119886119886119953119953119890119890
minus119867119867120591120591minus120591120591prime
ℏ = minusTr 120588120588119867119867119886119886119953119953primedagger (0)119886119886119953119953(120591120591 minus 120591120591prime)
= minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953primedagger (0)119886119886119953119953(120591120591 minus 120591120591prime) = minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953prime
dagger (0)
= minuslang119879119879120591120591119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953primedagger (0)rang = 119866119866119953119953119953119953prime(120591120591 minus 120591120591prime 0)
For simplicity we will introduce the notation 119866119866119953119953119953119953prime(120591120591 0) equiv 119866119866119953119953119953119953prime(120591120591) We further note that when using
imaginary time 119866119866119953119953119953119953prime(120591120591) is a periodic functions in the domain [minus120573120573ℏ120573120573ℏ] with a period of 120573120573ℏ (see
Page 236 of Ref [2])
119866119866119953119953119953119953prime(120591120591 + 120573120573ℏ 0) = 119866119866119953119953119953119953prime(120591120591 0) 120591120591 lt 0119866119866119953119953119953119953prime(120591120591 minus 120573120573ℏ 0) = 119866119866119953119953119953119953prime(120591120591 0) 120591120591 gt 0 (410)
To show this we shall again assume first that minus120573120573ℏ lt 120591120591 lt 0 to write that is
119866119866119953119953119953119953prime(120591120591 + 120573120573ℏ 0) = minus lang119879119879120591120591119886119886119953119953(120591120591 + 120573120573ℏ)119886119886119953119953primedagger (0)rang = minuslang119886119886119953119953(120591120591 + 120573120573ℏ)119886119886119953119953prime
dagger (0)rang
= minusTr 120588120588119867119867119890119890119867119867(120591120591+120573120573ℏ)
ℏ 119886119886119953119953119890119890minus119867119867(120591120591+120573120573ℏ)
ℏ 119886119886119953119953prime = minusTr 120588120588119867119867119890119890120573120573119867119867119890119890
119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591ℏ 119890119890minus120573120573119867119867119886119886119953119953prime
= minusTr120588120588119867119867119890119890120573120573119867119867119886119886119953119953(120591120591)119890119890minus120573120573119867119867119886119886119953119953prime(0) = minus
1119885119885119867119867
Tr119890119890minus120573120573119867119867119890119890120573120573119867119867119886119886119953119953(120591120591)119890119890minus120573120573119867119867119886119886119953119953prime(0)
= minus1119885119885119867119867
Tr119886119886119953119953(120591120591)119890119890minus120573120573119867119867119886119886119953119953prime(0) = minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119886119886119953119953prime(0)119886119886119953119953(120591120591)
= minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime(0) = minusTr120588120588119867119867119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime = 119866119866119953119953119953119953prime(120591120591 0)
Similarly for 120573120573ℏ gt 120591120591 gt 0 we have
119866119866119953119953119953119953prime(120591120591 minus 120573120573ℏ 0) = minus lang119879119879120591120591119886119886119953119953(120591120591 minus 120573120573ℏ)119886119886119953119953primedagger (0)rang = minuslang119886119886119953119953prime
dagger (0)119886119886119953119953(120591120591 minus 120573120573ℏ)rang
= minusTr 120588120588119867119867119886119886119953119953prime119890119890119867119867(120591120591minus120573120573ℏ)
ℏ 119886119886119953119953119890119890minus119867119867(120591120591minus120573120573ℏ)
ℏ = minusTr 120588120588119867119867119886119886119953119953prime119890119890minus120573120573119867119867119890119890
119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591ℏ 119890119890120573120573119867119867
= minusTr120588120588119867119867119886119886119953119953prime(0)119890119890minus120573120573119867119867119886119886119953119953(120591120591)119890119890120573120573119867119867 = minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119886119886119953119953prime(0)119890119890minus120573120573119867119867119886119886119953119953(120591120591)119890119890120573120573119867119867
= minus1119885119885119867119867
Tr119886119886119953119953prime(0)119890119890minus120573120573119867119867119886119886119953119953(120591120591) = minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119886119886119953119953(120591120591)119886119886119953119953prime(0)
= minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime(0) = minusTr120588120588119867119867119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime = 119866119866119953119953119953119953prime(120591120591 0)
13
Therefore 119866119866119953119953119953119953prime(120591120591) can be expanded as a Fourier series in the domain [0120573120573ℏ] as follows
119866119866119953119953119953119953prime(120591120591) = 1120573120573ℏsum 119890119890minus119894119894120596120596119899119899120591120591119866119866119953119953119953119953prime(119894119894120596120596119899119899)infinminusinfin (411)
where 120596120596119899119899 = 2120587120587119899119899120573120573ℏ
and the associated Fourier coefficient is given by
119866119866119953119953119953119953prime(119894119894120596120596119899119899) = int d120591120591119890119890119894119894120596120596119899119899120591120591119866119866119953119953119953119953prime(120591120591)120573120573ℏ0 120596120596119899119899 = 2119899119899120587120587
120573120573ℏ (412)
Having proven the translational time invariance and the periodicity of the Greenrsquos functios we can
now calculate it for the uncoupled system
119866119866119953119953119953119953prime0 (120591120591 120591120591prime) = minuslang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime)rang0 == minus lang119886119886119953119953(0)119890119890minus120596120596119953119953120591120591119886119886119953119953prime
dagger (0)119890119890120596120596119953119953prime120591120591primerang 120591120591 minus 120591120591prime gt 0
minus lang119886119886119953119953primedagger (0)119890119890120596120596119953119953prime120591120591
prime119886119886119953119953(0)119890119890minus120596120596119953119953120591120591rang 120591120591 minus 120591120591prime lt 0
= minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591
primelang119886119886119953119953(0)119886119886119953119953prime
dagger (0)rang 120591120591 minus 120591120591prime gt 0
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang 120591120591 minus 120591120591prime lt 0
= minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591
primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0) + 119886119886119953119953(0)119886119886119953119953primedagger (0)rang 120591120591 minus 120591120591prime gt 0
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang 120591120591 minus 120591120591prime lt 0
= minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591
primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang + 120575120575119953119953119953119953prime 120591120591 minus 120591120591prime gt 0
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang 120591120591 minus 120591120591prime lt 0
ie
119866119866119953119953119953119953prime0 (120591120591 120591120591prime) =
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang + 120575120575119953119953119953119953prime 120591120591 minus 120591120591prime gt 0
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang 120591120591 minus 120591120591prime lt 0 (413)
where we have used Eq (46) for 119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime) and Commutation relation 119886119886119953119953(0)119886119886119953119953prime
dagger (0) = 120575120575119953119953119953119953prime
To calculate lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang we first prove following equation
119890119890119860119860119861119861119890119890minus119860119860 = sum 1119899119899119860(119899119899)119861119861infin
119899119899=0 (414)
where 119860(119896119896)119861119861 equiv 119860 119860(119896119896minus1)119861119861 To prove this identity we define the operator 119891119891(119905119905) = 119890119890119905119905A119861119861119890119890minus119905119905119860119860
and taylor expand it around 119905119905 = 0
119891119891(119905119905) = 119891119891(0) + 119905119905119891119891prime(0) + 1199051199052
2119891119891primeprime(0) + ⋯ = sum 119905119905119899119899
119899119899119889119889119899119899119891119889119889119905119905119899119899
119905119905=0
infin119899119899=0 (415)
The corresponding derivatives are given by
⎩⎪⎨
⎪⎧ 119891119891prime(119905119905) = 119890119890119905119905119860119860119860119861119861119890119890minus119905119905119860119860 minus 119890119890119905119905119860119860119861119861119860119890119890minus119905119905119860119860 = 119890119890119905119905119860119860119860119861119861119890119890minus119905119905119860119860
119891119891primeprime(119905119905) = 119890119890119905119905119860119860119860119860119861119861 minus 119860119861119861119860119890119890minus119905119905119860119860 = 119890119890119905119905119860119860119860(2)119861119861119890119890minus119905119905119860119860
⋮119891119891(119899119899)(119905119905) = 119890119890119905119905119860119860119860(119899119899)119861119861119890119890minus119905119905119860119860
(416)
14
Substituting Eq (416) into Eq (415) we have
119890119890119905119905A119861119861119890119890minus119905119905119860119860 = sum 119905119905119899119899
119899119899119860(119899119899)119861119861infin
119899119899=0 (417)
Itrsquos clear that Eq (413) is a spectial case of Eq (416) with 119905119905 = 1 With this we can proceed as
follows
lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang =
11198851198851198671198670
Tr119890119890minus1205731205731198671198670119886119886119953119953primedagger (0)119886119886119953119953(0) =
11198851198851198671198670
Tr119890119890minus1205731205731198671198670119886119886119953119953primedagger (0)1198901198901205731205731198671198670119890119890minus1205731205731198671198670119886119886119953119953(0)
=11198851198851198671198670
Tr (minus120573120573)119899119899
1198991198991198671198670
(119899119899)119886119886119953119953primedagger (0)
infin
119899119899=0
119890119890minus1205731205731198671198670119886119886119953119953(0)
Note that 1198671198670119886119886119953119953primedagger (0) = ℏ120596120596119953119953prime119886119886119953119953prime
dagger (0) we have
1198671198670(119899119899)119886119886119953119953prime
dagger (0) = ℏ120596120596119953119953prime119899119899119886119886119953119953primedagger (0) (418)
such that
lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang =
11198851198851198671198670
Tr (minus120573120573)119899119899
1198991198991198671198670
(119899119899)119886119886119953119953primedagger (0)
infin
119899119899=0
119890119890minus1205731205731198671198670119886119886119953119953(0)
=11198851198851198671198670
Tr minus120573120573ℏ120596120596119953119953prime
119899119899
119899119899119886119886119953119953primedagger (0)
infin
119899119899=0
119890119890minus1205731205731198671198670119886119886119953119953(0)
= 119890119890minus120573120573ℏ120596120596119953119953prime11198851198851198671198670
Tr119890119890minus1205731205731198671198670119886119886119953119953(0)119886119886119953119953primedagger (0) = 119890119890minus120573120573ℏ120596120596119953119953primelang119886119886119953119953(0)119886119886119953119953prime
dagger (0)rang
= 119890119890minus120573120573ℏ120596120596119953119953primelang119886119886119953119953primedagger (0)119886119886119953119953(0)rang + 120575120575119953119953119953119953prime
therefore we obtain lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang 119890119890120573120573ℏ120596120596119953119953prime minus 1 = 120575120575119953119953119953119953prime For 119953119953prime ne 119953119953 and general 119890119890120573120573ℏ120596120596119953119953prime we have
lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang = 0 For 119953119953prime = 119953119953 we obtain lang119886119886119953119953
dagger(0)119886119886119953119953(0)rang = 119890119890120573120573ℏ120596120596119953119953 minus 1minus1
Hence we have
lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang = 120575120575119953119953prime119953119953
1119890119890120573120573ℏ120596120596119953119953minus1
equiv 120575120575119953119953prime119953119953119899119899119861119861120596120596119953119953 (419)
where 119899119899119861119861120596120596119953119953 is the Bose-Einstein distribution for phonons Substituting Eq (418) in Eq (413)
we obtain
119866119866119953119953119953119953prime0 (120591120591 120591120591prime) =
minus120575120575119953119953119953119953prime1 + 119899119899119861119861120596120596119953119953119890119890minus120596120596119953119953120591120591minus120591120591prime 120591120591 minus 120591120591prime gt 0
minus120575120575119953119953119953119953prime119899119899119861119861120596120596119953119953119890119890minus120596120596119953119953120591120591minus120591120591prime 120591120591 minus 120591120591prime lt 0 (420)
Note that only those Greenrsquos functions of the form 119866119866119953119953119953119953prime0 (120591120591 120591120591prime) = 1198661198661199531199530(120591120591 minus 120591120591prime)120575120575119953119953119953119953prime are non-zero due
to the orthogonality of normal modes Looking at the non-vanishing terms and setting 120591120591prime = 0
without loss of generality we can calculate the Fourier transform of 1198661198661199531199530(120591120591) from Eq (412)
1198661198661199531199530(119894119894120596120596119899119899) = int d1205911205911198901198901198941198941205961205961198991198991205911205911198661198661199531199530(120591120591)120573120573ℏ0 = minusint d120591120591119890119890119894119894120596120596119899119899120591120591120573120573ℏ
0 1 + 119899119899119861119861120596120596119953119953119890119890minus120596120596119953119953120591120591 = minus1+119899119899119861119861120596120596119953119953119894119894120596120596119899119899minus120596120596119953119953
119890119890119894119894120596120596119899119899minus120596120596119953119953120573120573ℏ minus
15
1 = minus1+ 1
119890119890120573120573ℏ120596120596119953119953minus1
119894119894120596120596119899119899minus120596120596119953119953119890119890119894119894
2120587120587119899119899120573120573ℏ 120573120573ℏ119890119890minus120573120573ℏ120596120596119953119953 minus 1 = minus 119890119890120573120573ℏ120596120596119953119953
119894119894120596120596119899119899minus120596120596119953119953
119890119890minus120573120573ℏ120596120596119953119953minus1
119890119890120573120573ℏ120596120596119953119953minus1= minus 1
119894119894120596120596119899119899minus120596120596119953119953
1minus119890119890120573120573ℏ120596120596119953119953
119890119890120573120573ℏ120596120596119953119953minus1= 1
119894119894120596120596119899119899minus120596120596119953119953
ie
1198661198661199531199530(119894119894120596120596119899119899) == 1119894119894120596120596119899119899minus120596120596119953119953
(421)
5 Fermirsquos golden rule 51 Interaction picture
Next we can proceed with calculating the Greenrsquos function of the coupled system To this end we
define the coupling Hamiltonian operator term in the interaction picture as
119867119867119862119862119868119868 (120591120591) = 1198901198901198671198670120591120591ℏ 119867119867119862119862119890119890
minus1198671198670120591120591ℏ (51)
The time evolution of 119867119867119862119862(120591120591) is then given by
ℏ part119867119867119862119862119868119868 (120591120591)120597120597120591120591
= 1198671198670119867119867119862119862119868119868 (120591120591) (52)
Note that the Greenrsquos function defined above was given in the Heisenberg picture To proceed we
need to transform it to the interaction picture
119866119866119953119953119953119953prime(120591120591 0) = minus lang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953primedagger (0)rang = minusTr 120588120588119867119867119879119879120591120591 119890119890
119867119867120591120591ℏ 119886119886119953119953(0)119890119890minus
119867119867120591120591ℏ 119886119886119953119953prime
dagger (0)
= minusTr 120588120588119867119867119879119879120591120591 119890119890119867119867120591120591ℏ 119890119890minus
1198671198670120591120591ℏ 119890119890
1198671198670120591120591ℏ 119886119886119953119953(0)119890119890minus
1198671198670120591120591ℏ 119890119890
1198671198670120591120591ℏ 119890119890minus
119867119867120591120591ℏ 119886119886119953119953prime
dagger (0)
= minusTr 120588120588119867119867119879119879120591120591 119880119880(0 120591120591)119886119886119953119953119868119868 (120591120591)119880119880(120591120591 0)119886119886119953119953primedagger (0)
where
119886119886119953119953119868119868 (120591120591) equiv 1198901198901198671198670120591120591ℏ 119886119886119953119953(0)119890119890minus
1198671198670120591120591ℏ (53)
is the operator in interaction picture and the operator 119880119880 is defined by
119880119880(1205911205911 1205911205912) equiv 11989011989011986711986701205911205911ℏ 119890119890minus
119867119867 (1205911205911minus1205911205912)ℏ 119890119890minus
11986711986701205911205912ℏ (54)
Note that while 119880119880 is not unitary it satisfies the following group property
119880119880(1205911205911 1205911205912)119880119880(1205911205912 1205911205913) = 119880119880(1205911205911 1205911205913) (55)
and the boundary condition 119880119880(1205911205911 1205911205911) = 1 In addition the 120591120591 derivative of 119880119880 is simply
ℏpart119880119880(120591120591 120591120591prime)
120597120597120591120591= 1198671198670119890119890
1198671198670120591120591ℏ 119890119890minus
119867119867120591120591minus120591120591primeℏ minus 119890119890
1198671198670120591120591ℏ 119867119867119890119890minus
119867119867120591120591minus120591120591primeℏ 119890119890minus
1198671198670120591120591primeℏ
= 1198901198901198671198670120591120591ℏ 1198671198670 minus 119867119867119890119890minus
1198671198670120591120591ℏ 119890119890
1198671198670120591120591ℏ 119890119890minus
119867119867120591120591minus120591120591primeℏ 119890119890minus
1198671198670120591120591primeℏ = 119890119890
1198671198670120591120591ℏ minus119867119867119862119862119890119890
minus1198671198670120591120591ℏ 119880119880(120591120591 120591120591prime)
= minus119867119867119862119862119868119868 (120591120591)119880119880(120591120591 120591120591prime)
16
ie
ℏ part119880119880120591120591120591120591prime120597120597120591120591
= minus119867119867119862119862119868119868 (120591120591)119880119880(120591120591 120591120591prime) (56)
The solution of Eq (56) is (see page 235 of Ref [2])
119880119880(120591120591 120591120591prime) = summinus1ℏ
119899119899
119899119899 int 1198891198891205911205911120591120591120591120591prime ⋯int 119889119889120591120591119899119899
120591120591120591120591prime 119879119879120591120591119867119867119862119862119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119899119899)infin
119899119899=0 (57)
The exact thermal Greens function now may be rewritten in the interaction picture as
119866119866119953119953119953119953prime(120591120591 0) = minus1119885119885119867119867
Tr 119890119890minus120573120573119867119867119879119879120591120591119880119880(0 120591120591)119886119886119953119953119868119868 (120591120591)119880119880(120591120591 0)119886119886119953119953primedagger (0)
= minusTr 119890119890minus120573120573119867119867119879119879120591120591 119880119880(0 120591120591)119886119886119953119953119868119868 (120591120591)119880119880(120591120591 0)119886119886119953119953prime
dagger (0)
Tr119890119890minus120573120573119867119867
= minusTr 119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)119879119879120591120591 119886119886119953119953119868119868 (120591120591)119880119880(0 120591120591)119880119880(120591120591 0)119886119886119953119953prime
dagger (0)
Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)
= minusTr 119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)119879119879120591120591 119886119886119953119953119868119868 (120591120591)119880119880(00)119886119886119953119953prime
dagger (0)
Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)=
= minusTr 119890119890minus1205731205731198671198670119879119879120591120591 119880119880(120573120573ℏ 0)119886119886119953119953119868119868 (120591120591)119886119886119953119953prime
dagger (0)
Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)
Where we used the fact that we are free to change the order of the operators within the time ordering
operation (see pages 241-242 of Ref [2])
52 Wickrsquos theorem
Thus the Greenrsquos function can be expanded as [2]
119866119866119953119953119953119953prime(120591120591 0) = minussum 1
119898119898minus1ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)119886119886119953119953119868119868 (120591120591)119886119886
119953119953primedagger (0)rang0infin
119898119898=0
sum 1119898119898minus
1ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)rang0infin
119898119898=0 (58)
where lang⋯ rang0 represents the ensemble average with respect to the non-interacting basis
Tr119890119890minus1205731205731198671198670(⋯ ) Or explicitly
119866119866119953119953119953119953prime(120591120591 0) = minus119866119866119953119953119953119953prime0 (120591120591)minus1ℏint 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953primedagger (0)rang0+
12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0+⋯
1minus1ℏint 1198891198891205911205911120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)rang0+12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0+⋯
(59)
For consiceness we introduce the following notation
119863119863119898119898 equiv 1119898119898minus 1
ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)rang0 (510)
To simplify the calculation in Eq (58) we adopt Wickrsquos theorem (see pages 237-241 of Ref
[2])which can be expressed as follows
lang119879119879120591120591119860119861119861119862119863119863 ⋯119884119884119885rang0 = lang119879119879120591120591119860119861119861rang0lang119879119879120591120591119862119863119863rang0⋯ lang119879119879120591120591119884119884119885rang0 + lang119879119879120591120591119860119862rang0lang119879119879120591120591119861119861119863119863rang0⋯ lang119879119879120591120591119883119883119885rang0 + ⋯ (511)
17
Here 119860119861119861 hellip 119884119884 119885 represent 119886119886119953119953(120591120591) or 119886119886119953119953dagger(120591120591) In Eq (511) each term corresponds to a particular
pairing of the operators 119860119861119861119862119863119863 ⋯119884119884119885 and all possible pairings are taken into account Here the only
non-vanishing propagators will have the form [see Eq (420)]
lang119879119879120591120591 119886119886119953119953primedagger (120591120591prime)119886119886119953119953(120591120591)rang0 = minuslang119879119879120591120591 119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime)rang0 = 119866119866119953119953119953119953prime0 (120591120591 120591120591prime) = 1198661198661199531199530(120591120591 minus 120591120591prime)120575120575119953119953119953119953prime (512)
Therefore the second term in the numerator of Eq (59) can be calculated as
1198681198682 = minus1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0
= minus1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 lang119879119879120591120591119886119886119953119953prime
dagger (0)119886119886119953119953(120591120591)rang0 minus1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
= 119866119866119953119953119953119953prime0 (120591120591 0) minus
1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 minus
1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
The first term in above equation is called disconnected part since the pairing is performed separately
on 119867119867119862119862(1205911205911) and 119886119886119953119953(120591120591)119886119886119953119953primedagger (0) All other terms have pairs that mix creation and annihilation operators
of the Hamiltonian with 119886119886119953119953(120591120591) or 119886119886119953119953primedagger (0) and are said to have connected party For simplicity we
include all these terms in the notation lang119879119879120591120591119867119867119862119862(1205911205911)119886119886119953119953(120591120591)119886119886119953119953primedagger (0)rang0119888119888 Furthermore we define 119866119866119953119953119953119953prime
(1) (120591120591) equiv
minus1ℏ int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862(1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 Then we have
1198681198683 = minus1ℏint 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 = 119866119866119953119953119953119953prime0 (120591120591 0)1198631198631 + 119866119866119953119953119953119953prime
(1) (120591120591) (513)
Using this method the third term in the numerator of Eq (59) can be calculated as
1198681198683 = 12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 = 12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0 lang119879119879120591120591119886119886119953119953prime
dagger (0)119886119886119953119953(120591120591)rang0 +
12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 + 12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 +
12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 = 119866119866119953119953119953119953prime0 (120591120591 0) 1
2ℏ2 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0 +
12ℏ2 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 + 12ℏ2 int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 +
12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
Here the last term is denoted as 119866119866119953119953119953119953prime(2) (120591120591)
Using Eq (513) the second and third terms on the right hand above equation can be simplified as
follows
18
12ℏ2
1198891198891205911205912120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 +1
2ℏ2 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
=12 minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0
minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
+12 minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0
minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
=12 minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0
minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
+12 minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0
minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
= minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0
minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 = 1198631198631119866119866119953119953119953119953prime(1) (120591120591)
Where we performed the following integration variables interchange 1205911205911 ⟷ 1205911205912 Thus we have
1198681198683 = 119866119866119953119953119953119953prime(2) (120591120591) + 119866119866119953119953119953119953prime
(1) (120591120591)1198631198631 + 119866119866119953119953119953119953prime0 (120591120591 0)1198631198632 (514)
Similarity we can calculate the fourth term of Eq (59) as
1198681198684 = minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 =
minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0 lang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 + 3 times
minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 + 3 times
minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 +
minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 =
119866119866119953119953119953119953prime0 (120591120591 0) minus1
6ℏ3 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0+
12ℏ2 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0
minus1ℏ int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888+
12ℏ2 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 minus1ℏ int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0+ 119866119866119953119953119953119953prime
(3) (120591120591) = 119866119866119953119953119953119953prime(3) (120591120591) +
119866119866119953119953119953119953prime(2) (120591120591)1198631198631 + 119866119866119953119953119953119953prime
(1) (120591120591)1198631198632 + 119866119866119953119953119953119953prime0 (120591120591 0)1198631198633
Higher order terms can be treated in the same manner (see pages 95-96 of Ref [2]) Then when
substituting 1198681198682 1198681198683 1198681198684 and all higher order terms into Eq (59) we can simplify the numerator as
follows
119866119866119953119953119953119953prime0 (120591120591) minus
1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0
+1
2ℏ2 1198891198891205911205911120573120573ℏ
0 1198891198891205911205912120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 + ⋯
= 119866119866119953119953119953119953prime0 (120591120591) + 119866119866119953119953119953119953prime
(1) (120591120591) + 119866119866119953119953119953119953prime(2) (120591120591) + ⋯ (1 + 1198631198631 + 1198631198632 + ⋯ )
Noting that the expression (1 + 1198631198631 + 1198631198632 + ⋯ ) is exactly canceled with the denominator the
Greenrsquos function can be simplified as
19
119866119866119953119953119953119953prime(120591120591) = 119866119866119953119953119953119953prime0 (120591120591) + 119866119866119953119953119953119953prime
(1) (120591120591) + 119866119866119953119953119953119953prime(2) (120591120591) + ⋯ (515)
where
119866119866119953119953119953119953prime
(1) (120591120591) = minus1ℏ int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888
119866119866119953119953119953119953prime(2) (120591120591) = 1
2ℏ2 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888 (516)
53 Calculation of Greenrsquos function
531 First order appriximation In Eq (516) we go back to the full notation 119886119886119954119954120582120582 to distinguish phonons of different branches
then 119866119866119953119953119953119953prime(1) (120591120591) is calculated as
119866119866119953119953119953119953prime(1) (120591120591) = minus
1ℏ
12 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591 119881119881119895119895119895119895prime(119948119948)119876119876119948119948119895119895(1205911205911)119876119876119948119948119895119895prime
dagger (1205911205911)3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ 119881119881119895119895119895119895primelowast (119948119948)119876119876119948119948119895119895prime(1205911205911)119876119876119948119948119895119895
dagger (1205911205911)3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= minus12ℏ
1198891198891205911205911120573120573ℏ
0lang119879119879120591120591
ℏ119881119881119895119895119895119895prime(119948119948)2120596120596119948119948119895119895120596120596119948119948119895119895prime
119886119886119948119948119895119895(1205911205911) + 119886119886minus119948119948119895119895dagger (1205911205911) 119886119886minus119948119948119895119895prime(1205911205911) + 119886119886119948119948119895119895prime
dagger (1205911205911)3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ ℏ119881119881119895119895119895119895prime
lowast (119948119948)
2120596120596119948119948119895119895120596120596119948119948119895119895prime119886119886119948119948119895119895prime(1205911205911) + 119886119886minus119948119948119895119895prime
dagger (1205911205911) 119886119886minus119948119948119895119895(1205911205911) + 119886119886119948119948119895119895dagger (1205911205911)
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
According to Wickrsquos theorem [2] the terms that contain 119886119886119948119948119895119895119886119886minus119948119948119895119895prime and 119886119886minus119948119948119895119895dagger 119886119886119948119948119895119895prime
dagger are equal to zero
thus the first term of above equation are calculated as
11989211989211 = minus14
119881119881119895119895119895119895prime(119948119948)
120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119886119886minus119948119948119895119895
dagger (1205911205911)119886119886minus119948119948119895119895prime(1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ lang119879119879120591120591119886119886119948119948119895119895(1205911205911)119886119886119948119948119895119895primedagger (1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888
= minus14
119881119881119895119895119895119895prime(119948119948)
120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119886119886minus119948119948119895119895
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus119948119948119895119895prime(1205911205911)rang0
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ lang119879119879120591120591119886119886119948119948119895119895primedagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)119886119886119948119948119895119895(1205911205911)rang0
= minus14
119881119881119895119895119895119895prime(119948119948)
120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
01198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954minus119948119948120575120575120582120582119895119895119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119954119954primeminus119948119948120575120575120582120582prime119895119895prime3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ 1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119948119948120575120575120582120582119895119895prime119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119954119954prime119948119948120575120575120582120582prime119895119895
20
Simplification of above equation gives
11989211989211 =
⎩⎪⎨
⎪⎧minus119881119881120582120582120582120582prime(minus119954119954)
4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954120582120582
0 (120591120591 1205911205911)119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903
2lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582(119954119954)
4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954120582120582
0 (120591120591 1205911205911)119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582prime le 3119903119903
2lt 120582120582 le 3119903119903
0 else
(517)
Similarity the second term of 119866119866119953119953119953119953prime(1) (120591120591) is calculated as
11989211989212 = minus14
minus119881119881119895119895119895119895primelowast (119948119948)
4120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119886119886119948119948119895119895prime(1205911205911)119886119886119948119948119895119895
dagger (1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ lang119879119879120591120591119886119886minus119948119948119895119895primedagger (1205911205911)119886119886minus119948119948119895119895(1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888
= minus14
minus119881119881119895119895119895119895primelowast (119948119948)
4120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)119886119886119948119948119895119895prime(1205911205911)rang0 lang119879119879120591120591119886119886119948119948119895119895dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus119948119948119895119895(1205911205911)rang0 lang119879119879120591120591119886119886minus119948119948119895119895prime
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
ie
11989211989212 =
⎩⎪⎨
⎪⎧
minus119881119881120582120582120582120582primelowast (119954119954)
4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954prime120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903
2lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582lowast (minus119954119954)
4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954prime120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119954119954prime 1 le 120582120582prime le 3119903119903
2lt 120582120582 le 3119903119903
0 else
(518)
Note that since 119881119881120582120582120582120582primelowast (119954119954) = 119881119881120582120582120582120582prime(minus119954119954) [Eq (314)] the first order approximation of Greenrsquos function
can be written as
119866119866119954119954120582120582119954119954prime120582120582prime(1) (120591120591) =
⎩⎪⎨
⎪⎧minus119881119881
120582120582120582120582prime(minus119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ
0 1198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582 le3119903119903
2lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582
(119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ
0 1198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582prime le3119903119903
2lt 120582120582 le 3119903119903
0 else
(519)
To derive the Greenrsquos function in frequency domain we use the expression for Fourier series then
we have
1198891198891205911205911120573120573ℏ
01198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0) = 1198891198891205911205911120573120573ℏ
0
1120573120573ℏ
119890119890minus119894119894120596120596119899119899(120591120591minus1205911205911)1198661198661199541199541205821205820 (119894119894120596120596119899119899)
119899119899
1120573120573ℏ
119890119890minus119894119894120596120596119899119899prime1205911205911119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899prime)
119899119899prime
=1
(120573120573ℏ)2119890119890minus119894119894120596120596119899119899120591120591 1198891198891205911205911120573120573ℏ
0119890119890119894119894120596120596119899119899minus120596120596119899119899prime1205911205911 119866119866119954119954120582120582
0 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899prime)
119899119899119899119899prime=
=1
(120573120573ℏ)2119890119890minus119894119894120596120596119899119899120591120591[120573120573ℏ120575120575119899119899119899119899prime]1198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899prime)119899119899119899119899prime
=1120573120573ℏ
119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899)119899119899
According to the definition of Fourier series [see Eq (412)] we get the Fourier coefficient of
21
119866119866119954119954120582120582119954119954prime120582120582prime(1) (120591120591)
119866119866119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) =
⎩⎪⎨
⎪⎧minus119881119881120582120582120582120582prime(minus119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582prime1198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime 1 le 120582120582 le 31199031199032
lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582(119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582prime1198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime 1 le 120582120582prime le 31199031199032
lt 120582120582 le 3119903119903
0 else
(520)
where 119866119866119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) = 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) times Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) times 119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899) and Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) is defined as
Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) =
⎩⎪⎨
⎪⎧minus119881119881120582120582120582120582prime(minus119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582prime120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903
2lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582(119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582prime120575120575119954119954119954119954prime 1 le 120582120582prime le 3119903119903
2lt 120582120582 le 3119903119903
0 else
(521)
532 second order approximation The second order approximation of the Greenrsquos function in Eq (516) 119866119866119954119954120582120582119954119954prime120582120582prime
(2) (120591120591) is calculated as
119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) =
18ℏ2 1198891198891205911205911
120573120573ℏ
0 1198891198891205911205912120573120573ℏ
0lang119879119879120591120591
⎣⎢⎢⎡ 11988111988111989511989511198951198951prime(119948119948120783120783)1198761198761199481199481207831207831198951198951(1205911205911)1198761198761199481199481207831207831198951198951prime
dagger (1205911205911)3119903119903
1198951198951prime=31199031199032 +1
31199031199032
1198951198951=1119948119948120783120783
+ 11988111988111989411989411198941198941primelowast (119948119948120783120783)1198761198761199481199481207831207831198941198941prime(1205911205911)1198761198761199481199481207831207831198941198941
dagger (1205911205911)3119903119903
1198941198941prime=31199031199032 +1
31199031199032
1198941198941=1119948119948120783120783 ⎦⎥⎥⎤
⎣⎢⎢⎡ 11988111988111989511989521198951198952prime(119948119948120784120784)1198761198761199481199481207841207841198951198952(1205911205912)1198761198761199481199481207841207841198951198952prime
dagger (1205911205912)3119903119903
1198951198952prime=31199031199032 +1
31199031199032
1198951198952=1119948119948120784120784
+ 11988111988111989411989421198941198942primelowast (119948119948120784120784)1198761198761199481199481207841207841198941198942prime(1205911205912)1198761198761199481199481207841207841198941198942
dagger (1205911205912)3119903119903
1198941198942prime=31199031199032 +1
31199031199032
1198941198942=1119948119948120784120784 ⎦⎥⎥⎤119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888 = 1198921198921 + 1198921198922 + 1198921198923 + 1198921198924
where
1198921198921 = 18ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0
sum sum 11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989511989521198951198952prime(119948119948120784120784)1le1198951198952le
31199031199032 lt1198951198952
primele311990311990311994811994812078412078411989511989521198951198952prime
1le1198951198951le31199031199032 lt1198951198951
primele311990311990311994811994812078312078311989511989511198951198951prime
times lang1198791198791205911205911198761198761199481199481207831207831198951198951(1205911205911)1198761198761199481199481207831207831198951198951primedagger (1205911205911)1198761198761199481199481207841207841198951198952(1205911205912)1198761198761199481199481207841207841198951198952prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(522)
1198921198922 = 18ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0
sum sum 11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942primelowast (119948119948120784120784)
1le1198941198942le31199031199032 lt1198941198942
primele311990311990311994811994812078412078411989411989421198941198942prime
1le1198951198951le31199031199032 lt1198951198951
primele311990311990311994811994812078312078311989511989511198951198951prime
times lang1198791198791205911205911198761198761199481199481207831207831198951198951(1205911205911)1198761198761199481199481207831207831198951198951primedagger (1205911205911)1198761198761199481199481207841207841198941198942prime (1205911205912)1198761198761199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(523)
1198921198923 = 18ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0
sum sum 11988111988111989411989411198941198941primelowast (119948119948120783120783)11988111988111989511989521198951198952prime(119948119948120784120784)
1le1198951198952le31199031199032 lt1198951198952
primele311990311990311994811994812078412078411989511989521198951198952prime
1le1198941198941le31199031199032 lt1198941198941
primele311990311990311994811994812078312078311989411989411198941198941prime
times lang1198791198791205911205911198761198761199481199481207831207831198941198941prime (1205911205911)1198761198761199481199481207831207831198941198941dagger (1205911205911)1198761198761199481199481207841207841198951198952(1205911205912)1198761198761199481199481207841207841198951198952prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(524)
22
1198921198924 = 18ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0
sum sum 11988111988111989411989411198941198941primelowast (119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)1le1198941198942le
31199031199032 lt1198941198942
primele311990311990311994811994812078412078411989411989421198941198942prime
1le1198941198941le31199031199032 lt1198941198941
primele311990311990311994811994812078312078311989411989411198941198941prime
times lang1198791198791205911205911198761198761199481199481207831207831198941198941prime (1205911205911)1198761198761199481199481207831207831198941198941dagger (1205911205911)1198761198761199481199481207841207841198941198942prime (1205911205912)1198761198761199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(525)
In what follows we will expand the term 1198921198922 The expansion of all other terms follows the same lines
and eventually results in the same expression Expanding the product 11987611987611994811994812078312078311989511989511198761198761199481199481207831207831198951198951primedagger 1198761198761199481199481207841207841198941198942prime1198761198761199481199481207841207841198941198942
dagger we can
rewrite Eq (523) as follows
1198921198922 = 132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)119881119881
11989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032 lt1198941198942
primele311990311990311994811994812078412078411989411989421198941198942prime
1le1198951198951le31199031199032 lt1198951198951
primele311990311990311994811994812078312078311989511989511198951198951prime
times 119868119868 (526)
Where
119868119868 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911) + 119886119886minus1199481199481207831207831198951198951dagger (1205911205911) 119886119886minus1199481199481207831207831198951198951prime(1205911205911) + 1198861198861199481199481207831207831198951198951prime
dagger (1205911205911) 1198861198861199481199481207841207841198941198942prime (1205911205912) + 119886119886minus1199481199481207841207841198941198942primedagger (1205911205912) 119886119886minus1199481199481207841207841198941198942(1205911205912) + 1198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0c
= lang119879119879120591120591 1198861198861199481199481207831207831198951198951119886119886minus1199481199481207831207831198951198951prime + 11988611988611994811994812078312078311989511989511198861198861199481199481207831207831198951198951primedagger + 119886119886minus1199481199481207831207831198951198951
dagger 119886119886minus1199481199481207831207831198951198951prime + 119886119886minus1199481199481207831207831198951198951dagger 1198861198861199481199481207831207831198951198951prime
dagger 12059112059111198861198861199481199481207841207841198941198942prime119886119886minus1199481199481207841207841198941198942 + 1198861198861199481199481207841207841198941198942prime1198861198861199481199481207841207841198941198942
dagger + 119886119886minus1199481199481207841207841198941198942primedagger 119886119886minus1199481199481207841207841198941198942
+ 119886119886minus1199481199481207841207841198941198942primedagger 1198861198861199481199481207841207841198941198942
dagger 1205911205912119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0c
= lang119879119879120591120591 11988611988611994811994812078312078311989511989511198861198861199481199481207831207831198951198951primedagger + 119886119886minus1199481199481207831207831198951198951
dagger 119886119886minus1199481199481207831207831198951198951prime12059112059111198861198861199481199481207841207841198941198942prime1198861198861199481199481207841207841198941198942
dagger + 119886119886minus1199481199481207841207841198941198942primedagger 119886119886minus11994811994812078412078411989411989421205911205912
119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
Where the symbol [hellip ]1205911205911 signifies that the operators in the brackets are given in the interaction
picture The last equality in above equation comes from the fact that the contractions of the product
of the operators are equal to zero when the number of creation and annihilation operators are not the
same (see Wickrsquos theorem in Ref [2]) Using Wickrsquos theorem [2] we are able to calculate the above
equation term by term as follows
⎩⎪⎨
⎪⎧ 1198681198681 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951prime
dagger (1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)1198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888
1198681198682 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
1198681198683 = lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)1198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
1198681198684 = lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(527)
We shall now calculate them term by term
23
1198681198681 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951primedagger (1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)1198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942
dagger (1205911205912)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
+ lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951prime
dagger (1205911205911)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0
+ lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)1198861198861199481199481207841207841198941198942prime(1205911205912)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0119888119888
= 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199481199481207841207841198951198952
0 (1205911205912 1205911205912)12057512057511989411989421198941198942prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582
+ 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 1198661198661199481199481207831207831198951198951
0 (1205911205911 1205911205911)12057512057511989511989511198951198951prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582
+ 1198661198661199481199481207831207831198951198951prime0 (1205911205912 1205911205911)1205751205751199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951prime ∙ 119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582
= 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582
+ 1198661198661199481199481207831207831198951198951prime0 (1205911205912 1205911205911)1205751205751199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951prime ∙ 119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582
The last equality results from the fact that 1198951198951 and 1198951198951prime are indices of different ranges (belonging to
different subsystems) and the same holds for 1198941198942 and 1198941198942prime Similarity
1198681198684 = lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0
= 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 119866119866minus11994811994812078412078411989411989420 (1205911205912 1205911205912)12057512057511989411989421198941198942prime
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 119866119866minus1199481199481207831207831198951198951prime 0 (1205911205911 1205911205912)1205751205751199481199481207831207831199481199481207841207841205751205751198951198951prime1198941198942prime
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205911 1205911205911)12057512057511989511989511198951198951prime
+ 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205912 1205911205911)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942
= 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 119866119866minus1199481199481207831207831198951198951prime 0 (1205911205911 1205911205912)1205751205751199481199481207831207831199481199481207841207841205751205751198951198951prime1198941198942prime
+ 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205912 1205911205911)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942
Where again the last equality results from the fact that 1198951198951 and 1198951198951prime are indices of different ranges
(belonging to different subsystems) and the same holds for 1198941198942 and 1198941198942prime Similarly we obtain 1198681198682 =
1198681198683 = 0 for the same reason as shown below
24
1198681198682 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951prime
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0
+ lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)1198861198861199481199481207831207831198951198951(1205911205911)rang0
+ lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0
= 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582 ∙ 119866119866minus11994811994812078412078411989411989420 (1205911205912 1205911205912)12057512057511989411989421198941198942prime
+ 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)120575120575minus11994811994812078412078411994811994812078312078312057512057511989511989511198941198942prime ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205911)12057512057511989511989511198951198951prime
+ 1198661198661199481199481207831207831198951198951prime0 (1205911205912 1205911205911)120575120575minus11994811994812078412078411994811994812078312078312057512057511989411989421198951198951prime ∙ 119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 = 0
1198681198683 = lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)1198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0
+ lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0
+ lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0
= 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 11986611986611994811994812078412078411989411989420 (1205911205912 1205911205912)12057512057511989411989421198941198942prime
+ 11986611986611994811994812078412078411989411989420 (1205911205911 1205911205912)120575120575minus1199481199481207831207831199481199481207841207841205751205751198951198951prime1198941198942 ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205911 1205911205911)12057512057511989511989511198951198951prime
+ 1198661198661199481199481207841207841198941198942prime0 (1205911205912 1205911205911)120575120575minus1199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951 ∙ 119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 = 0
The two non-vanishing terms (1198681198681 and 1198681198684) produce the following contributions to 1198921198922 of Eq (526)
11989211989221 = 132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime1198681198681 =
132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime1198661198661199481199481207831207831198951198951
0 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙
1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582 + 1
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime1198661198661199481199481207831207831198951198951prime
0 (1205911205912 1205911205911)1205751205751199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951prime ∙
119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 =
⎩⎪⎨
⎪⎧
120575120575119954119954119954119954prime
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast (119954119954prime)
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (1205911205911 1205911205912) ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
120575120575119954119954119954119954prime
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032+1 1198661198661199541199541198951198951prime
0 (1205911205912 1205911205911) ∙ 119866119866119954119954prime120582120582prime0 (1205911205911 0)119866119866119954119954120582120582
0 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032
0 else
and
25
11989211989224 = 132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime1198681198684 =
132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙
119866119866minus1199481199481207831207831198951198951prime 0 (1205911205911 1205911205912)1205751205751199481199481207831207831199481199481207841207841205751205751198951198951prime1198941198942prime + 1
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙
1198661198661199541199541205821205820 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 119866119866minus1199481199481207831207831198951198951
0 (1205911205912 1205911205911)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 =
⎩⎪⎨
⎪⎧120575120575119954119954119954119954prime
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032+1 1198661198661199541199541198951198951prime
0 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime0 (1205911205912 0)119866119866119954119954120582120582
0 (120591120591 1205911205911) 120582120582 120582120582prime isin 1 31199031199032
120575120575119954119954119954119954prime
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast (119954119954prime)
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
0 else
Here we used the following properties 120596120596minus119954119954120582120582 = 120596120596119954119954120582120582 and 1198811198811205821205821198951198951prime(minus119954119954) = 1198811198811205821205821198951198951primelowast (119954119954) [see Eqs (112) and
(314)]
The equivalence of 11989211989221 and 11989211989224 can be seen by changing the integration variables 1205911205912 harr 1205911205911
Substituting 11989211989221 and 11989211989224 into Eq (526) 1198921198922 is given by
1198921198922 =
⎩⎪⎨
⎪⎧
120575120575119954119954119954119954prime
16 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast (119954119954prime)
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime
0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
120575120575119954119954119954119954prime
16 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032+1 1198661198661199541199541198951198951prime
0 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime0 (1205911205911 0)119866119866119954119954120582120582
0 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032
0 else
(528)
By changing the indices and integration variables itrsquos straightforward to show that the other three
terms (119892119892111989211989231198921198924 ) are identical to 1198921198922 thus the final expression of 119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) [Eq (516)) can be
calculated as
119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) = 41198921198922 =
⎩⎪⎨
⎪⎧120575120575119954119954119954119954prime
4 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032
120575120575119954119954119954119954prime
4 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast 119954119954prime
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime
0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 31199031199032
+ 13119903119903
0 else
(529)
To calculate 119866119866119954119954120582120582119954119954prime120582120582prime(2) in frequency space we expand 119866119866119954119954120582120582119954119954prime120582120582prime
(2) (120591120591) in a Fourier series
119866119866119954119954120582120582119954119954prime120582120582prime
(2) (120591120591) = 1120573120573ℏsum 119890119890minus119894119894120596120596119899119899120591120591119866119866119954119954120582120582119954119954prime120582120582prime
(2) (119894119894120596120596119899119899)119899119899
119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = int d120591120591119890119890119894119894120596120596119899119899120591120591119866119866119954119954120582120582119954119954prime120582120582prime
(2) (120591120591)120573120573ℏ0
(530)
Using Eq (421) we get for 120582120582 120582120582prime isin 1 31199031199032
26
119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) =
120575120575119954119954119954119954prime4
1198891198891205911205911120573120573ℏ
0 1198891198891205911205912120573120573ℏ
0
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1198661198661199541199541198951198951prime0 (1205911205912 1205911205911)119866119866119954119954120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912)
=120575120575119954119954119954119954prime
4
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1198891198891205911205911120573120573ℏ
0 1198891198891205911205912120573120573ℏ
0
1120573120573ℏ
119890119890minus1198941198941205961205961198991198992(1205911205912minus1205911205911)1198661198661199541199541198951198951prime0 1198941198941205961205961198991198992
1198991198992
∙1120573120573ℏ
119890119890minus11989411989412059612059611989911989911205911205911119866119866119954119954120582120582prime0 1198941198941205961205961198991198991
1198991198991
∙1120573120573ℏ
119890119890minus119894119894120596120596119899119899(120591120591minus1205911205912)1198661198661199541199541205821205820 (119894119894120596120596119899119899)
119899119899
=120575120575119954119954119954119954prime
4
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1120573120573ℏ
119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954120582120582prime
0 11989411989412059612059611989911989911198661198661199541199541198951198951prime0 1198941198941205961205961198991198992
11989911989911989911989911198991198992⎩⎪⎨
⎪⎧ 1120573120573ℏ
1198891198891205911205911120573120573ℏ
0119890119890minus1198941198941205961205961198991198991minus12059612059611989911989921205911205911
times1120573120573ℏ
1198891198891205911205912120573120573ℏ
0119890119890minus1198941198941205961205961198991198992minus1205961205961198991198991205911205912
⎭⎪⎬
⎪⎫
=120575120575119954119954119954119954prime
4
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1120573120573ℏ
119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)
11989911989911989911989911198991198992
119866119866119954119954120582120582prime0 11989411989412059612059611989911989911198661198661199541199541198951198951prime
0 1198941198941205961205961198991198992120575120575119899119899111989911989921205751205751198991198991198991198992
=120575120575119954119954119954119954prime120573120573ℏ
119890119890minus119894119894120596120596119899119899120591120591
119899119899
1198661198661199541199541205821205820 (119894119894120596120596119899119899)
14
119881119881120582120582prime1198951198951prime(119954119954
prime)1198811198811205821205821198951198951primelowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899)119866119866119954119954120582120582prime
0 (119894119894120596120596119899119899)
Comparing above expression with Eq (530) the Fourier transform of 119866119866119954119954120582120582119954119954120582120582prime(2) (120591120591) reads as
119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) 120575120575119954119954119954119954prime
4sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 119866119866119954119954120582120582prime
0 (119894119894120596120596119899119899) (531)
where we have introduced the notation
Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) equiv
120575120575119954119954119954119954prime
4sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) (532)
and 119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954120582120582prime(2) (119894119894120596120596119899119899) ∙ 119866119866119954119954120582120582prime
0 (119894119894120596120596119899119899) Similarly for 120582120582 120582120582prime isin 31199031199032
+ 13119903119903 we
have
Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) =
120575120575119954119954119954119954prime
4sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast 119954119954prime
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (119894119894120596120596119899119899) (533)
and all together we have for Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899)
Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) =
⎩⎪⎪⎨
⎪⎪⎧120575120575119954119954119954119954prime
4sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 1 3119903119903
2
120575120575119954119954119954119954prime
4sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast 119954119954prime
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
0 else
(534)
27
533 Dysonrsquos equation Substituting Eqs (521) and (534) into Eq (516) and performing a Fourier transform we have
119866119866119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) = 119866119866119954119954120582120582119954119954prime120582120582prime0 (119894119894120596120596119899119899) + 119866119866119954119954120582120582119954119954prime120582120582prime
(1) (119894119894120596120596119899119899) + 119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) + ⋯
= 1198661198661199541199541205821205820 (119894119894120596120596119899119899)120575120575119954119954119954119954prime120575120575120582120582120582120582prime + 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) ∙ 119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899) + 1198661198661199541199541205821205820 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime
(2) (119894119894120596120596119899119899)
∙ 119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899) + ⋯ = 119866119866119954119954120582120582
0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime120575120575120582120582120582120582prime + 1198661198661199541199541205821205820 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) ∙ 119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899)
or
119866119866119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) = 1198661198661199541199541205821205820 (119894119894120596120596119899119899)120575120575119954119954119954119954prime120575120575120582120582120582120582prime + 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) ∙ 119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899) (535)
Eq (535) is the so-called Dysonrsquos equation (see Ref [2]) where
Σ119954119954120582120582119954119954120582120582prime(119894119894120596120596119899119899) = Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) + Σ119954119954120582120582119954119954prime120582120582prime
(2) (119894119894120596120596119899119899) + ⋯
is called self-energy and Σ119954119954120582120582119954119954120582120582prime(119899119899) (119894119894120596120596119899119899)119899119899 = 12⋯ is the self-energy of nth order approximation Up
to the second order approximation the self-energy is written as
Σ119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) =
⎩⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎧ minus
120575120575119954119954119954119954prime
2
119881119881120582120582120582120582primelowast (119954119954)
120596120596119954119954120582120582120596120596119954119954120582120582prime 1 le 120582120582 le 3119903119903
2lt 120582120582prime le 3119903119903
minus120575120575119954119954119954119954prime
2
119881119881120582120582prime120582120582(119954119954)
120596120596119954119954120582120582120596120596119954119954120582120582prime 1 le 120582120582prime le 3119903119903
2lt 120582120582 le 3119903119903
120575120575119954119954119954119954prime
4sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 1 3119903119903
2
120575120575119954119954119954119954prime
4sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast 119954119954prime
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
(536)
54 Fermirsquos golden Rule
To get the Fermirsquos golden Rule we need to calculate the retarded Greenrsquos function which can be
calculated from 119866119866119954119954120582120582119954119954120582120582prime(119894119894120596120596119899119899) by analytic continuation to the real axis via 119894119894120596120596119899119899 rarr 120596120596 + 119894119894119894119894 with an
infinitesimal positive 119894119894 [12]
119866119866119954119954120582120582119954119954120582120582prime119903119903 (120596120596) equiv 119866119866119954119954120582120582119954119954120582120582prime(119894119894120596120596119899119899 rarr 120596120596 + 119894119894119894119894) 119894119894 rarr 0 (537)
Similarity the retarded self-energy is defined as
Σ119954119954120582120582119954119954120582120582prime119903119903 (120596120596) equiv Σ119954119954120582120582119954119954120582120582prime(119894119894120596120596119899119899 rarr 120596120596 + 119894119894119894119894) 119894119894 rarr 0 (538)
The transition rate of phonons of branch 120582120582 at 119954119954 Γ119954119954120582120582(120596120596) is related to the retarded self energy
Σ119954119954120582120582119903119903 (120596120596) via [13]
Γ119954119954120582120582(120596120596) = minus2ImΣ119954119954120582120582119954119954120582120582119903119903 (120596120596) (539)
Using the expression for 1198661198661199541199541205821205820 (119894119894120596120596119899119899) [Eq (421)] and focused on the second order approximation [13]
we have
28
Σ119954119954120582120582119954119954120582120582119903119903 (119894119894120596120596119899119899) =
⎩⎪⎨
⎪⎧14sum
1198811198811205821205821198951198951prime(119954119954)
2
1205961205961199541199541198951198951prime120596120596119954119954120582120582
31199031199031198951198951prime=
31199031199032 +1
1
120596120596minus1205961205961199541199541198951198951prime +119894119894119894119894
120582120582 isin 1 31199031199032
14sum 1198811198811198951198951120582120582(119954119954)2
1205961205961199541199541198951198951120596120596119954119954120582120582
311990311990321198951198951=1
1120596120596minus1205961205961199541199541198951198951+119894119894119894119894
120582120582 isin 31199031199032
+ 13119903119903
(540)
Using the relation
Im 1
120596120596minus1205961205961199541199541198951198951prime +119894119894119894119894
= minus120587120587120575120575120596120596 minus 1205961205961199541199541198951198951prime (541)
The transition rate reads as
Γ119954119954120582120582(120596120596) =
⎩⎪⎨
⎪⎧1205871205872sum
1198811198811205821205821198951198951prime(119954119954)
2
1205961205961199541199541198951198951prime120596120596119954119954120582120582
31199031199031198951198951prime=
31199031199032 +1
120575120575120596120596 minus 1205961205961199541199541198951198951prime 120582120582 isin 1 31199031199032
1205871205872sum 1198811198811198951198951120582120582(119954119954)2
1205961205961199541199541198951198951120596120596119954119954120582120582
311990311990321198951198951=1
120575120575120596120596 minus 1205961205961199541199541198951198951 120582120582 isin 31199031199032
+ 13119903119903
(542)
Or equivalently
Γ119954119954120582120582(119864119864 = ℏ120596120596) =
⎩⎪⎨
⎪⎧120587120587ℏ3
2sum
1198811198811205821205821198951198951prime(119954119954)
2
1198641198641199541199541198951198951prime119864119864119954119954120582120582
31199031199031198951198951prime=
31199031199032 +1
120575120575119864119864 minus 1198641198641199541199541198951198951prime 120582120582 isin 1 31199031199032
120587120587ℏ3
2sum 1198811198811198951198951120582120582(119954119954)2
1198641198641199541199541198951198951119864119864119954119954120582120582
311990311990321198951198951=1
120575120575119864119864 minus 1198641198641199541199541198951198951 120582120582 isin 31199031199032
+ 13119903119903
(543)
Γ119954119954120582120582(119864119864) represents the probability per unit time of a transition with energy 119864119864 from phonon branch 120582120582
at wave number 119954119954 in subsystem I (II) to a set of phonon branches in subsystem II (I) with the same
wave number Here state 119954119954120582120582 in one subsystem is coupled to state 1199541199541198951198951prime in the other subsystem via
1198811198811205821205821198951198951prime(119954119954) Since the two expressions in Eq (543) are completely equivalent just representing
transitions of opposite directions we consider only the first one in what follows Assuming that
subsystem II sufficiently large such that its density of states is nearly continuous the sum over its
states in the above expression can be replaced by an integral over the energy sum (⋯ )119895119895 rarr
int(⋯ )120588120588119864119864119954119954d119864119864119954119954 giving
Γ119954119954120582120582(119864119864) = 120587120587ℏ3
2 intd119864119864119954119954prime120588120588119864119864119954119954prime 119881119881120582120582119895119895prime(119954119954)
2119864119864119954119954119895119895prime=119864119864119954119954prime
119864119864119954119954prime119864119864119954119954120582120582120575120575119864119864 minus 119864119864119954119954prime = 120587120587ℏ3
2120588120588(119864119864)
119881119881120582120582119895119895prime(119954119954)2119864119864119954119954119895119895prime=119864119864
119864119864119864119864119954119954120582120582 (544)
Eq (544) represents the transition rate of the phonons with energy 119864119864 from branch 120582120582 and wave
number 119954119954 in subsystem I to the continuous manifold of states in subsystem II The total transition
rate between the two subsystems is then given by the sum of transition rates from all states in
subsystem I weighted by their phonon populations to the manifold of states in subsystem II
Assuming that the two subsystems are weakly coupled such that the width of state 119954119954120582120582 in subsystem
I is small and the probability to leave it at an energy other than 119864119864119954119954120582120582 is negligible we can now write
29
total transition rate as
Γtot = 1119885119885119890119890minus120573120573119864119864119954119954120582120582Γ119954119954120582120582119864119864119954119954120582120582
119954119954120582120582
=120587120587ℏ3
2
119890119890minus120573120573119864119864119954119954120582120582
119885119885
120588120588119864119864119954119954120582120582 119881119881120582120582119895119895prime(119954119954)2119864119864119954119954119895119895prime=119864119864119954119954120582120582
1198641198641199541199541205821205822119954119954120582120582
=120587120587ℏ3
2
119890119890minus120573120573119864119864119954119954120582120582
119885119885
120588120588119864119864119954119954120582120582 119881119881120582120582120582120582+31199031199032(119954119954)
2
1198641198641199541199541205821205822119954119954120582120582
Finally we get
Γtot = Γ119954119954120582120582(119864119864) =120587120587ℏ3
2sum 119890119890minus120573120573119864119864119954119954120582120582
119885119885
120588120588119864119864119954119954120582120582119881119881120582120582120582120582+31199031199032(119954119954)
2
1198641198641199541199541205821205822119954119954120582120582 (545)
where 119885119885 = sum 119890119890minus120573120573119864119864119954119954120582120582119954119954120582120582 is the partition function Here we choose the index of phonon branches 120582120582
such that the phonon branch with lower energy has a smaller index ie 1198641198641199541199541205821205821 lt 1198641198641199541199541205821205822 for index 1205821205821 lt
1205821205822 Note that phonon branches 120582120582 and 1198951198951prime belong to the subsystem I (with index from 1 to 31199031199032) and
subsystem II (with index from 1 to 1 + 31199031199032 to 3119903119903) respectively we choose 119895119895prime = 120582120582 + 31199031199032
to make
sure that 119864119864119954119954119895119895prime = 119864119864119954119954120582120582 since phonon energy of two uncoupled system is identical Eq (545) is the
Fermirsquos golden rule for inter-phonon coupling
References
[1] Y Xu J-S Wang W Duan B-L Gu and B Li Nonequilibrium Greens function method for phonon-phonon interactions and ballistic-diffusive thermal transport Phys Rev B 78 224303 (2008) [2] A L Fetter and J D Walecka Quantum theory of many-particle systems (Courier Corporation 2012) [3] G E Stedman Fermis Golden RulemdashAn Exercise in Quantum Field Theory Am J Phys 39 205 (1971)
6
implemented in LAMMPS [33] The simulations allow us to evaluate the dynamical matrix from
which the phonon-phonon couplings can be extracted (see details in Sec 3 of SI) and the overall heat
transfer rate calculated Figure 3 shows the resulting heat transfer rate (normalized to is value for the
aligned contact (120579120579 = 0) ) as a function of the misfit angle compared to the interfacial thermal
conductivity defined as the inverse of the ITR presented in Figure 2(c) ITC equiv 1ITR The
remarkable agreement between the calculated interfacial thermal conductivity and Fermirsquos golden
rule results indicate that the dependence of the interlayer phonon-phonon couplings on the misfit
angle is responsible for the strong angle dependence of the interfacial conductivity Notably the sharp
heat conductivity drop at misfit angles in the range of 0deg-5deg as well as the small conductivity for larger
misfit angles are well captured by Fermirsquos golden rule
Figure 3 Comparison between Fermirsquos golden rule results (open blue squares) for the interfacial heat-
transfer rate of a tBLG and the calculated interfacial thermal conductivity at various misfit angles
ITC simulation results are presented for both 8 layers (open red circles) and 16 layers (open black
triangles) showing similar behavior For comparison purposes all data sets are normalized to their
value obtained for the aligned contact
To correlate our results with experimentally measured thermal conductivities that are often obtained
for thick samples we repeated our calculations for increasing stack thicknesses at fixed misfit angles
Figure 4 presents results for the calculated heat conductivity of (a) graphite and (b) h-BN stacks either
aligned (open red circles) or twisted by 120579120579 = 3016deg (open black diamond symbols) as a function of
number of layers in the stack As discussed above for both systems the misoriented stack exhibits
lower heat conductivity compared to the aligned system however its thickness dependence is
considerably stronger This can be attributed to the significantly higher interface resistance of the
twisted interface that when plugged in Eq (S2) of the SI for the overall conductivity induces stronger
7
thickness dependence
Comparing our calculated heat conductivities for the aligned contact (open red circles) to available
experimental data for ~35 nm thick graphite slabs [34] (dashed green line) we find that at the thickest
model system considered of 104 graphene layers (~34 nm thick) the calculated value of 085plusmn005
W m sdot Kfrasl is in remarkable agreement with the measured value of ~07 W m sdot Kfrasl Furthermore
experimental values for bulk graphite [5] indicate that the thermal conductivity continues to grow up
to ~68 W m sdot Kfrasl (black dash-dotted line) which is consistent with the general trend of the calculated
heat conductivity that does not saturate for the thickest model system considered These results
strongly enforce the validity of our force-field and model systems to model the heat conductivity of
twisted layered materials interfaces Available experimental results for the heat conductivity of bulk
h-BN are marked by the dashed-dotted black and dashed-green lines in Figure 4(b) In line with our
findings for the graphitic interface our calculated finite slab heat conductivities for the aligned
interface (open red circles) continue to grow with the number of layers and are consistently below the
bulk value
Figure 4 Thickness dependence of the thermal conductivity 120581120581CP of aligned (open red circles) and
twisted by 3016deg (open black diamond symbols) graphite (a) and h-BN (b) stacks Blue squares
represent results obtained using the isotropic Lennard-Jones potential for the aligned contacts The
green dashed and black dash-dotted lines represent experimental results measured for graphite (a)
(Refs [534]) and bulk h-BN (b) (Refs [1415]) Note that both axis scales are logarithmic Error bar
estimation procedure is discussed in Sec 1 of the SI
Another important factor that may affect the interlayer thermal transport properties of 2D material
stacks is the average temperature of the system which was taken to be ~300K in all abovementioned
simulations To evaluate the sensitivity of our results towards this parameter we repeated the heat
conductivity and interfacial resistance calculations of optimally stacked graphite and h-BN stacks for
an average temperature of 400 K The results presented in see Sec 4 of the SI indicate that 120581120581CP and
8
ITR weakly depend on the average temperature over the entire thickness range considered This
conclusion is consistent with recent experimental findings [11] We may attribute this to the fact that
the thickness of the graphite and h-BN stack model considered is much smaller than their phonon
mean-free path (~200 nm for graphite [1011133536] and ~100 nm for bulk h-BN [15]) such that
the phonon transport is dominated by phonon-boundary scattering which weakly depends on
temperature in the range considered Therefore temperature dependent Umklapp processes have only
marginal contribution to our results [11]
Finally we note that previous calculations of the heat conductivity of twisted graphitic interfaces
relied on Lennard-Jones (LJ) potentials describing the interlayer interactions [8-1013] To
demonstrate the importance of using registry-dependent interlayer potentials we have repeated our
calculations of the heat conductivity of graphitic slabs with the REBO intralayer potential augmented
by LJ interlayer interactions [37] (120576120576 = 284 meV120590120590 = 34 Å ) We find that the calculated heat
conductivities obtained using the LJ interlayer potential are consistently higher than those obtained
by our ILP and that the difference between them grows with the model system thickness Notably the
heat conductivity obtained using the LJ potential for a graphitic slab of thickness ~34 nm is 154
W m sdot Kfrasl overestimating the experimental value by more than a factor of 2
The excellent agreement of our ILP calculations with experimental data of nanoscale graphitic stacks
therefore demonstrates the reliability of our predictions for the strong interfacial misfit angle
dependence of cross-layer thermal conductivity in graphite and h-BN The observed sharp
conductivity decrease of twisted graphitic interfaces at misfit angles lt 5deg opens the way to control
the thermal evacuation rate and thermal isolation of active layers in graphene-based electronic and
mechanical devices The revealed underlying mechanism suggests that design rules can be obtained
by carefully tailoring the phonon-phonon couplings across the twisted interface While the misfit
angle dependence of h-BN is found to be weaker than that of graphite the overall thermal
conductivity of the former is found to be higher This may be utilized to achieve higher conductivity
and controllability in twisted heterogeneous junctions of layered materials
ASSOCIATED CONTENT
Supporting Information
The supporting Information section includes a description of the methodology thermal conductivity
of twisted bilayer graphene theory for calculating the phonon coupling of twisted bilayer graphene
9
derivation of Fermirsquos golden rule and temperature dependence of the cross-plane thermal
conductivity
AUTHOR INFORMATION
Corresponding Author
E-mail urbakhtauextauacil
Notes
The authors declare no competing financial interest
ACKNOWLEDGMENTS
The authors would like to thank Prof Abraham Nitzan Dr Guy Cohen and Dr Yiming Pan for
helpful discussions WO acknowledges the financial support from a fellowship program for
outstanding postdoctoral researchers from China and India in Israeli Universities and the support from
the National Natural Science Foundation of China (Nos 11890673 and 11890674) HQ
acknowledges the financial support from the National Natural Science Foundation of China (No
11890674) MU acknowledges the financial support of the Israel Science Foundation Grant
No 114118 and the ISF-NSFC joint grant 319119 OH is grateful for the generous financial
support of the Israel Science Foundation under grant no 158617 and the Naomi Foundation for
generous financial support via the 2017 Kadar Award This work is supported in part by COST Action
MP1303
10
References
[1] A A Balandin Thermal properties of graphene and nanostructured carbon materials Nat Mater 10 569 (2011) [2] S Ghosh I Calizo D Teweldebrhan E P Pokatilov D L Nika A A Balandin W Bao F Miao and C N Lau Extremely high thermal conductivity of graphene Prospects for thermal management applications in nanoelectronic circuits Appl Phys Lett 92 151911 (2008) [3] A A Balandin S Ghosh W Bao I Calizo D Teweldebrhan F Miao and C N Lau Superior thermal conductivity of single-layer graphene Nano Lett 8 902 (2008) [4] J H Seol et al Two-Dimensional Phonon Transport in Supported Graphene Science 328 213 (2010) [5] R Taylor The thermal conductivity of pyrolytic graphite J Philosophical Magazine 13 157 (1966) [6] S Ghosh W Bao D L Nika S Subrina E P Pokatilov C N Lau and A A Balandin Dimensional crossover of thermal transport in few-layer graphene Nat Mater 9 555 (2010) [7] Z Wang R Xie C T Bui D Liu X Ni B Li and J T L Thong Thermal Transport in Suspended and Supported Few-Layer Graphene Nano Lett 11 113 (2011) [8] Z Wei Z Ni K Bi M Chen and Y Chen Interfacial thermal resistance in multilayer graphene structures Phys Lett A 375 1195 (2011) [9] Y Ni Y Chalopin and S Volz Significant thickness dependence of the thermal resistance between few-layer graphenes Appl Phys Lett 103 061906 (2013) [10] J Chen J H Walther and P Koumoutsakos Strain Engineering of Kapitza Resistance in Few-Layer Graphene Nano Lett 14 819 (2014) [11] Q Fu J Yang Y Chen D Li and D Xu Experimental evidence of very long intrinsic phonon mean free path along the c-axis of graphite Appl Phys Lett 106 031905 (2015) [12] G Fugallo A Cepellotti L Paulatto M Lazzeri N Marzari and F Mauri Thermal Conductivity of Graphene and Graphite Collective Excitations and Mean Free Paths Nano Lett 14 6109 (2014) [13] Z Wei J Yang W Chen K Bi D Li and Y Chen Phonon mean free path of graphite along the c-axis Appl Phys Lett 104 081903 (2014) [14] E K Sichel R E Miller M S Abrahams and C J Buiocchi Heat capacity and thermal conductivity of hexagonal pyrolytic boron nitride Phys Rev B 13 4607 (1976) [15] P Jiang X Qian R Yang and L Lindsay Anisotropic thermal transport in bulk hexagonal boron nitride Phys Rev Mater 2 064005 (2018) [16] M-H Wang Y-E Xie and Y-P Chen Thermal transport in twisted few-layer graphene Chinese Physics B 26 116503 (2017) [17] X Nie L Zhao S Deng Y Zhang and Z Du How interlayer twist angles affect in-plane and cross-plane thermal conduction of multilayer graphene A non-equilibrium molecular dynamics study Int J Heat Mass Tran 137 161 (2019) [18] A N Kolmogorov and V H Crespi Registry-dependent interlayer potential for graphitic systems Phys Rev B 71 235415 (2005) [19] M Reguzzoni A Fasolino E Molinari and M C Righi Potential energy surface for graphene on graphene Ab initio derivation analytical description and microscopic interpretation Phys Rev B 86 (2012) [20] M M van Wijk A Schuring M I Katsnelson and A Fasolino Moire Patterns as a Probe of Interplanar Interactions for Graphene on h-BN Phys Rev Lett 113 135504 (2014) [21] D Mandelli W Ouyang M Urbakh and O Hod The Princess and the Nanoscale Pea Long-Range Penetration of Surface Distortions into Layered Materials Stacks ACS Nano 13 7603 (2019) [22] E Koren E Loumlrtscher C Rawlings A W Knoll and U Duerig Adhesion and friction in mesoscopic graphite contacts Science 348 679 (2015) [23] E Koren I Leven E Loumlrtscher A Knoll O Hod and U Duerig Coherent commensurate electronic states at the interface between misoriented graphene layers Nat Nanotechnol 11 752 (2016)
11
[24] W Ouyang I Azuri D Mandelli A Tkatchenko L Kronik M Urbakh and O Hod Mechanical and Tribological Properties of Layered Materials under High Pressure Assessing the Importance of Many-Body Dispersion Effects J Chem Theory Comput 16 666 (2020) [25] W Ouyang D Mandelli M Urbakh and O Hod Nanoserpents Graphene Nanoribbon Motion on Two-Dimensional Hexagonal Materials Nano Lett 18 6009 (2018) [26] I Leven T Maaravi I Azuri L Kronik and O Hod Interlayer Potential for Grapheneh-BN Heterostructures J Chem Theory Comput 12 2896 (2016) [27] T Maaravi I Leven I Azuri L Kronik and O Hod Interlayer Potential for Homogeneous Graphene and Hexagonal Boron Nitride Systems Reparametrization for Many-Body Dispersion Effects J Phys Chem C 121 22826 (2017) [28] L H Liang and B Li Size-dependent thermal conductivity of nanoscale semiconducting systems Phys Rev B 73 153303 (2006) [29] P K Schelling S R Phillpot and P Keblinski Comparison of atomic-level simulation methods for computing thermal conductivity Phys Rev B 65 144306 (2002) [30] J Shiomi Nonequilirium molecular dynamics methods for lattice heat conduction calculations Annual Review of Heat Transfer 17 (2014) [31] G E Stedman Fermis Golden RulemdashAn Exercise in Quantum Field Theory Am J Phys 39 205 (1971) [32] L T Kong G Bartels C Campantildeaacute C Denniston and M H Muumlser Implementation of Greens function molecular dynamics An extension to LAMMPS Comput Phys Commun 180 1004 (2009) [33] S Plimpton Fast Parallel Algorithms for Short-Range Molecular Dynamics J Comput Phys 117 1 (1995) [34] M Harb et al The c-axis thermal conductivity of graphite film of nanometer thickness measured by time resolved X-ray diffraction Appl Phys Lett 101 233108 (2012) [35] H Zhang X Chen Y-D Jho and A J Minnich Temperature-Dependent Mean Free Path Spectra of Thermal Phonons Along the c-Axis of Graphite Nano Lett 16 1643 (2016) [36] D P Sellan E S Landry J E Turney A J H McGaughey and C H Amon Size effects in molecular dynamics thermal conductivity predictions Phys Rev B 81 214305 (2010) [37] S J Stuart A B Tutein and J A Harrison A reactive potential for hydrocarbons with intermolecular interactions J Chem Phys 112 6472 (2000)
S1
Supporting information for ldquoControllable Thermal Conductivity of
Twisted Graphene and Hexagonal Boron Nitride Interfacesrdquo
Wengen Ouyang1 Huasong Qin2 Michael Urbakh1 and Oded Hod1
1Department of Physical Chemistry School of Chemistry The Raymond and Beverly
Sackler Faculty of Exact Sciences and The Sackler Center for Computational
Molecular and Materials Science Tel Aviv University Tel Aviv 6997801 Israel
2State Key Laboratory for Strength and Vibration of Mechanical Structures School of
Aerospace Xirsquoan Jiaotong University Xirsquoan 710049 China
The Supporting information includes following sections
1 Methodology
2 Thermal conductivity of twisted bilayer graphene
3 Theory for calculating the phonon coupling of twisted bilayer graphene
4 Derivation of Fermirsquos golden rule
5 Temperature dependence of cross-plane thermal conductivity
S2
1 Methodology
11 Model system
The initial interlayer distance across the layered stack was set equal to 34 Aring and 33 Aring for graphite
and bulk h-BN respectively Periodic boundary conditions were applied in all directions It should be
noted that the lattice structure is rigorously periodic only at some specific twist angles the values of
which are listed in Table S1 in section 3 below While the cross-sectional area for each misfit angle
θ is different all systems considered have a contact area exceeding 12 nm2 which was shown to
provide converged results with respect to unit-cell dimensions [1] The intralayer interactions within
each graphene and h-BN layer were modeled via the second generation REBO potential [2] and
Tersoff potential [3] respectively The interlayer interactions between the layers of graphite and bulk
h-BN were described via our dedicated interlayer potential (ILP) [4] which is implemented in the
LAMMPS [5] suite of codes [6]
Fig S1 Schematic representation of the simulation setup (a) and steady-state temperature profile (b) respectively In panel (a) two identical AB-stacked graphite slabs (gray and orange respectively) are twisted with respect to each other to create a stacking fault of misfit angle θ A thermal bias is induced by applying Langevin thermostats to the two layers marked by dashed red (Thot) and green (Tcold) rectangles The arrows indicate the direction of the vertical heat flux Since periodic boundary conditions are applied also in the vertical direction two twisted interfaces are shown across which heat flows in opposite directions The steady-state temperature profiles are illustrated in panel (b) where N is the total number of layers in the model system and RAB dAB and Rθ and dθ mark the interfacial Kapitza resistance [89] and interlayer distance for contacting graphene layers with AB-stacking and misfit angle θ respectively The red lines in panel (b) mark the temperature variation across the twisted interface where the vertical axis corresponds to the position of the various layers along the stack and the horizontal axis marks the temperature of the various layers
S3
12 Simulation Protocol
All MD simulations were performed with the LAMMPS simulation package [5] The velocity-Verlet
algorithm with a time-step of 05 fs was used to propagate the equations of motion A Noseacute-Hoover
thermostat with a time constant of 025 ps was used for constant temperature simulations To maintain
a specified hydrostatic pressure the three translational vectors of the simulation cell were adjusted
independently by a Noseacute-Hoover barostat with a time constant of 10 ps [7] To relax the box we first
equilibrated the systems in the NPT ensemble at a temperature of T = 300 K and zero pressure for
250 ps (see Fig S2) After equilibration Langevin thermostats with damping coefficients 10 ps-1
were applied to the bottom and middle layer of the graphene stack (see Fig S1) with target
temperature Thot = 375 K (hot reservoir) and Tcold = 225 K (cold reservoir) respectively Then the
system was allowed to reach steady-state over a subsequent simulation period of 750 ps (see Fig S2)
during which the dynamics of all non-thermostated layers followed the NVE ensemble For the larger
model systems the length of the NPT and Langevin stages was doubled (for the 32 and 48 layers
systems) or tripled (for the 104 layers graphitic system) to ensure convergence of the obtained steady-
state Once steady-state was obtained the last 500 ps were used to calculate the thermal conductivity
of the twisted graphite and bulk h-BN The statistical errors were estimated using ten different data
sets each calculated over a time interval of 50 ps
Fig S2 Time evolution of the temperature of thermostated layers for 16 layers twisted graphite with misfit angle (a) θ = 0deg (b) θ = 509deg (c) θ = 1518deg and (d) θ = 3016deg Note that the thermal fluctuations increase with increasing the misfit angle due to the growing interfacial thermal resistance that enhances phonon back scattering at the twisted junction
S4
13 Calculation of the interfacial thermal resistance
According to Fourierrsquos law the cross-plane thermal conductivity (120581120581CP) of a twisted graphitic interface
of misfit angle θ can be calculated as
120581120581CP = 119876119860119860Δ119879119879Δ119911119911
(S1)
where 119876 is the heat flux A is the cross-section area and Δ119879119879Δ119911119911 is the temperature gradient along
the direction of heat flux (perpendicular to the basal plane in our case) Fig S1(b) shows a schematic
temperature profile along the z-direction where the vertical axis corresponds to the position of the
various layers along the stack and the horizontal axis marks the temperature of the various layers The
actual temperature profiles extracted from the NEMD simulations for twisted graphite with different
number of layers can be found in Fig S3 For Bernal-stacked graphite (ie θ = 0deg red circles in Fig
S3) only the linear region of the temperature profile was used to calculate 120581120581CP and the points
corresponding to the layers where the thermostats were applied were omitted (marked with green
triangle in Fig S3) The 120581120581CP of the system was calculated using Eq (S1) by averaging over the two
linear regions of the temperature profiles For the twisted case (θ ne 0deg) we found a sudden
temperature decrease Δ119879119879120579120579 at the position of the twisted interface (see black squares in Fig S3) 120581120581CP
in this case was calculated using the temperature gradient calculated for the same layer range as that
for θ = 0deg To characterize the thermal properties of the twisted interface the concept of interfacial
thermal resistance (ITR) ie Kapitza resistance [89] was introduced According the definition of
the Kapitza resistance [8] 119877119877 = 119860119860Δ119879119879119876 and noticing that Δ119879119879tot = (1198731198732 minus 3)Δ119879119879AB + Δ119879119879120579120579 and
Δ119911119911 = (1198731198732 minus 3)119889119889AB + 119889119889120579120579 Eq (1) can be rewritten considering two-resistors in series as [see Fig
S1(b)]
(1198731198732 minus 3)119877119877AB + 119877119877120579120579 = [(1198731198732 minus 3)119889119889AB + 119889119889120579120579] 120581120581CPfrasl (S2)
Here 119877119877AB 119889119889AB Δ119879119879AB and 119877119877120579120579 119889119889120579120579 Δ119879119879120579120579 are the ITR interlayer distance and temperature
difference for adjacent AB-stacked and twisted graphene layers respectively For the aligned contact
(120579120579 = 0deg) the ITR can be simply calculated as 119877119877AB = 119889119889AB120581120581AB Once 119877119877AB is known 119877119877120579120579 can be
calculated from Eq (S2) We note that the sharp temperature drop at the twisted interface indicates
that 119877119877120579120579 should be much larger than 119877119877AB
S5
Fig S3 Temperature profiles for graphitic stacks consisting of (a) 8 and (b) 16 layers The red circles and black squares represent the temperature profiles for the aligned (120579120579 = 0deg) and twisted (120579120579 =3016deg) junctions respectively Green triangles represent data points that were omitted in the 120581120581CP calculation
2 Thermal conductivity of twisted bilayer graphene
As comparison we also calculated the interfacial thermal conductivity (ITC) and ITR of twisted
bilayer graphene (tBLG) with the transient MD simulation approach [15-17] since the NEMD
simulation protocol used in the main text becomes invalid in this case In this protocol the system
was first equilibrated within the NPT ensemble at T = 200 K and zero pressure for 100 ps which was
followed by a 100 ps NVT ensemble equilibration stage and a 100 ps of NVE ensemble equilibration
stage After the system reached steady-state an ultrafast heat impulse was imposed on the top layer
of the t-BLG for 50 fs to increase the temperature of the top layer from 200 K to 400 K while that of
bottom layer of tBLG remained unchanged After the external heat source was removed thermal
energy flowed from the top layer to the bottom layer due to the temperature difference and the
temperature of both layers approached 300 K when quasi-steady-state was reached During the
thermal relaxation time interval (500 ps) the temperature and energy of the system sections were
recorded The ITR could then be extracted using the following equation [15-17]
part119864119864t120597120597120597120597
= 119860119860119877119877119879119879bot(119905119905) minus 119879119879top(119905119905) (S3)
where 119864119864t is the total energy of the top graphene layer R is the ITR of the tBLG A is the interfacial
cross-section area and 119879119879bot and 119879119879top are the instantaneous temperatures measured for the bottom
and top layers of the tBLG respectively Note that in Eq S3 we assume a linear dependence of the
heat flux on the temperature difference between the layers The ITC of the tBLG is simply defined as
ITC equiv 119889119889ITR where d is the average interlayer distance
S6
The ITC and ITR of tBLG as functions of misfit angle calculated with the transient MD simulation
protocol are illustrated in Fig S4 demonstrating similar misfit-angle dependence as that for the
NEMD protocol with Langevin thermostat exercised to obtain the results presented in the main text
This further validates the reliability of the simulation protocol adopted in the main text which is more
suitable to treat thick slabs and allows to obtain a true stead-state
Fig S4 Misfit-angle dependence of (a) ITC and (b) ITR for a twisted bilayer graphene obtained using the transient MD simulation approach
3 Theory for calculating the phonon-phonon coupling of twisted bilayer graphene
31 Brillouin Zone of supercell in tBLG
For tBLG the lattice structure is rigorously periodic only at some specific misfit angles θ where the
lattice vector 1199231199231 = 11989911989911199381199381 + 11989911989921199381199382 in the bottom layer equals the vector 1199231199232 = 11989811989811199381199381 + 11989811989821199381199382 in the
top layer with certain integers m1 m2 and n1 n2 Here 1199381199381 = 119886119886(10) and 1199381199382 = 11988611988612radic32 are
the primitive lattice vectors of the bottom layer and a is the lattice constant of monolayer graphene
Thus the exact superlattice period is then given by [18]
119871119871 = |11989911989911199381199381 + 11989911989921199381199382| = 11988611988611989911989912 + 11989911989922 + 11989911989911198991198992 = |1198991198991minus1198991198992|1198861198862 sin(1205791205792) (S4)
where θ is the angle between two lattice vectors 1199231199231 and 1199231199232 In the simulations below we always
rotated the supercell such that its lattice vector is 1199231199231 = 119871119871(10) and 1199231199232 = 11987111987112radic32 In this case
the corresponding reciprocal lattice vector of the moireacute superlattice satisfies the relation 119918119918119894119894 ∙ 119923119923119895119895 =
2120587120587120575120575119894119894119895119895 such that
1199181199181 = 4120587120587radic3119871119871
radic32
minus12 1199181199182 = 4120587120587
radic3119871119871(01) (S5)
S7
Both the lattice vectors and the corresponding reciprocal lattice vectors of the superlattice of the tBLG
are presented in Fig S5 Table S1 reports the parameters used to construct rhombus periodic
supercells of different misfit angles that can be duplicated to construct a rectangular periodic supercell
Table S1 The parameters used to construct periodic supercells of various misfit angles
120579120579 (deg) 119860119860 (nm2) 1198991198991 1198991198992 1198981198981 1198981198982
0 60383688 24 0 24 0
0696407 709613169 48 47 47 48
1121311 273718420 30 29 29 30
2000628 85648391 17 16 16 17
3006558 152322047 23 21 21 23
4048894 189013524 26 23 23 26
5085849 53255058 14 12 12 14
7926470 49376245 14 11 11 14
9998709 86277388 19 14 14 19
15178179 31554670 15 4 4 15
19932013 42876612 15 8 8 15
25039660 13942761 9 4 4 9
30158276 18974735 11 4 4 11
32204228 21805221 12 4 4 12
32 Special points for Brillouin Zone integration
The calculation of the sum over wave vector q in Eq (4) in the main text can be transformed to an
integral using the relation sum (⋯ )119954119954 = 1119881119881119887119887int (⋯ )BZ d119954119954 where 119881119881119887119887 = (2120587120587)3119881119881 is the volume of the
Brillouin Zone (BZ) and V is volume of the real-space unit-cell The calculation of integral is usually
inefficient since it requires calculating the value of the function over a large set of k points in the first
BZ To calculate such integrations more efficiently simple k-point meshes can be replaced by a
carefully selected set of special points in the BZ 119954119954119894119894 [19-22] over which the function is evaluated
The integral can then be estimated via
119868119868 = 1119881119881119887119887int 119891119891(119954119954)BZ d119954119954 asymp 1
119873119873sum 119908119908119894119894119891119891(119954119954119894119894)119894119894 (S6)
S8
where 119881119881119887119887 is the volume of the BZ 119908119908119894119894 is the weight of the ith data point and N normalizes the
weighting factors to unity 119873119873 = sum 119908119908119894119894119894119894 The set of selected 119954119954119894119894 forms a grid in the irreducible
Brillouin zone (IBZ) as is illustrated by the red points in Fig S5(b) The coordinates of these points
for a hexagonal lattice are presented in Eq (S7)
Fig S5 (a) Twisted bilayer graphene of misfit angle θ = 509deg L1 and L2 are the superlattice vectors (b) The corresponding first Brillouin Zone of (a) G1 and G2 are the reciprocal lattice vectors of the superlattice The triangle ΔΓMK represents the irreducible Brillouin zone Red circles mark the position of the special points used to evaluate the integral over the first Brillouin Zone
⎩⎪⎨
⎪⎧ 1199541199541 = 1
9 1205971205973 1199541199542 = 2
9 1205971205973 1199541199543 = 1
18 512059712059718
1199541199544 = 518
512059712059718 1199541199545 = 1
9 21205971205979 1199541199546 = 2
9 21205971205979
1199541199547 = 118
312059712059718 1199541199548 = 1
9 1205971205979 1199541199549 = 1
18 12059712059718
(S7)
Here 119905119905 = radic3 and the unit of the coordinates is 2120587120587119871119871 The weighting factors 119908119908119894119894 are [19]
119908119908124689 = 112
119908119908357 = 16 (S8)
Using Eqs (S6-S8) Eq (4) in the main text can be evaluated as follows
Γtot = 120587120587ℏ3
2sum 119908119908119896119896 sum
119890119890minus120573120573119864119864119954119954119948119948120582120582
119885119885
120588120588119864119864119954119954119948119948120582120582119881119881120582120582120582120582+31199031199032(119954119954119948119948)
2
1198641198641199541199541199481199481205821205822120582120582
9119896119896=1 (S9)
This equation was used to calculate the transition rate presented in Fig 3 in the main text
4 Derivation of Fermirsquos golden rule
The derivation of Fermirsquos golden rule is provided in a separate file (Fermirsquos Golden Rule for
phononspdf) due to its extent
S9
5 Temperature dependence of interfacial thermal conductivity
In the main text the target temperatures of the Langevin thermostats for the bottom and middle layers
of graphene and h-BN were set to 225 K and 375 K respectively After reaching the steady-state the
average temperature of the system was found to be ~300 K To check the effect of average temperature
on our results we calculated the cross-plane thermal conductivity (120581120581CP ) and the corresponding
interfacial thermal resistance (ITR) at a different temperature gradient (325 K ndash 475 K) resulting the
average steady-state temperature of ~400K The protocol described in Section 1 above was used to
perform these calculations as well Both ILP and Lennard-Jones (LJ) potential were tested for
graphite whereas for the bulk h-BN simulations only the ILP was used The results for graphite and
bulk h-BN are illustrated in Fig S6 and Fig S7 respectively For the ILP we find that the overall
values of 120581120581CP (ITR) decrease (increase) slightly with increasing average temperature which is
consistent with a recent experiment [10] The LJ potential calculations as well exhibit very week
dependence on average temperature within the range studied Altogether the layer dependences of
both quantities remain mostly insensitive to the average temperature The reason is that the thickness
of the graphite and h-BN model systems was chosen to be much smaller than their phonon mean free
path (~200 nm for graphite [110-13] and ~100 nm for bulk h-BN [14]) such that phonon transport is
dominated by phonon-boundary scattering and the Umklapp process only makes marginal
contribution [10]
S10
Fig S6 Layer dependence of 120581120581CP (a c) and ITR (b d) for Bernal-stacked graphite at average steady-state temperatures of 300 K (red circles) and 400 K (black squares) The left and right columns correspond to the 120581120581CP and ITR calculated with ILP and LJ potential respectively
Fig S7 Layer dependence of 120581120581CP (a) and ITR (b) for AArsquo-stacked h-BN at average temperatures of 300 K (red circles) and 400 K (blue squares)
S11
References [1] Z Wei J Yang W Chen K Bi D Li and Y Chen Phonon mean free path of graphite along the c-axis Appl Phys Lett 104 081903 (2014) [2] D W Brenner O A Shenderova J A Harrison S J Stuart B Ni and S B Sinnott A second-generation reactive empirical bond order (REBO) potential energy expression for hydrocarbons J Phys Condens Matter 14 783 (2002) [3] A Kınacı J B Haskins C Sevik and T Ccedilağın Thermal conductivity of BN-C nanostructures Phys Rev B 86 (2012) [4] W Ouyang I Azuri D Mandelli A Tkatchenko L Kronik M Urbakh and O Hod Mechanical and Tribological Properties of Layered Materials under High Pressure Assessing the Importance of Many-Body Dispersion Effects J Chem Theory Comput 16 666 (2020) [5] S Plimpton Fast Parallel Algorithms for Short-Range Molecular Dynamics J Comput Phys 117 1 (1995) [6] W Ouyang D Mandelli M Urbakh and O Hod Nanoserpents Graphene Nanoribbon Motion on Two-Dimensional Hexagonal Materials Nano Lett 18 6009 (2018) [7] W Shinoda M Shiga and M Mikami Rapid estimation of elastic constants by molecular dynamics simulation under constant stress Phys Rev B 69 134103 (2004) [8] G L Pollack Kapitza Resistance Rev Mod Phys 41 48 (1969) [9] H-S Yang G R Bai L J Thompson and J A Eastman Interfacial thermal resistance in nanocrystalline yttria-stabilized zirconia Acta Mater 50 2309 (2002) [10] Q Fu J Yang Y Chen D Li and D Xu Experimental evidence of very long intrinsic phonon mean free path along the c-axis of graphite Appl Phys Lett 106 031905 (2015) [11] J Chen J H Walther and P Koumoutsakos Strain Engineering of Kapitza Resistance in Few-Layer Graphene Nano Lett 14 819 (2014) [12] H Zhang X Chen Y-D Jho and A J Minnich Temperature-Dependent Mean Free Path Spectra of Thermal Phonons Along the c-Axis of Graphite Nano Lett 16 1643 (2016) [13] D P Sellan E S Landry J E Turney A J H McGaughey and C H Amon Size effects in molecular dynamics thermal conductivity predictions Phys Rev B 81 214305 (2010) [14] P Jiang X Qian R Yang and L Lindsay Anisotropic thermal transport in bulk hexagonal boron nitride Phys Rev Mater 2 064005 (2018) [15] J Zhang Y Hong and Y Yue Thermal transport across graphene and single layer hexagonal boron nitride J Appl Phys 117 134307 (2015) [16] B Liu F Meng C D Reddy J A Baimova N Srikanth S V Dmitriev and K Zhou Thermal transport in a graphenendashMoS2 bilayer heterostructure a molecular dynamics study RSC Advances 5 29193 (2015) [17] Z Ding Q-X Pei J-W Jiang W Huang and Y-W Zhang Interfacial thermal conductance in grapheneMoS2 heterostructures Carbon 96 888 (2016) [18] N N T Nam and M Koshino Lattice relaxation and energy band modulation in twisted bilayer graphene Phys Rev B 96 075311 (2017)
S12
[19] R Ramiacutearez and M C Boumlhm Simple geometric generation of special points in brillouin-zone integrations Two-dimensional bravais lattices Int J Quantum Chem 30 391 (1986) [20]S L Cunningham Special points in the two-dimensional Brillouin zone Phys Rev B 10 4988 (1974) [21] H J Monkhorst and J D Pack Special points for Brillouin-zone integrations Phys Rev B 13 5188 (1976) [22] D J Chadi Special points for Brillouin-zone integrations Phys Rev B 16 1746 (1977)
1
Fermirsquos golden rule for phonons
1 Basic theory for phonons 11 Basic notations
Let us consider a 3D crystal with a total of 119873119873 = 119873119873111987311987321198731198733 unit cells and periodic boundary conditions
To be specific let 119938119938119894119894 119894119894 = 123 be the lattice vectors that define the unit cell We index unit cells
with n = (n1n2n3) where each ni = 12 ∙∙∙ Ni and their locations are 119929119929119899119899 = sum 1198991198991198941198941199381199381198941198943119894119894=1 Assume that
there are r atoms in each unit cell which are indexed with 119904119904 = 1⋯ 119903119903 The mass and the equilibrium
distance of the sth atom are notated as 119872119872s and 119929119929s0 respectively Then the location of the sth atom in
the nth unit cell at time t can be expressed as
119955119955119899119899119899119899(119905119905) = 119929119929119899119899 + 119929119929s0 + 119958119958119899119899119899119899(119905119905) (11)
where 119958119958119899119899119899119899(119905119905) is its displacement from its equilibrium position
The Lagrangian for this classical problem can be written as
ℒ = sum sum 119872119872119904119904|119955119899119899119904119904|2(119905119905)2
119903119903119899119899=1
119873119873119899119899=1 minus 119881119881 (12)
where the second term is the sum of interactions between all pairs of atoms
Under the harmonic approximation ie expanding the total potential energy 119881119881 around the
equilibrium positions The Lagrangian can be simplified as
ℒ = sum sum 119872119872119904119904|119958119899119899119904119904|2(119905119905)2
119903119903119899119899=1
119873119873119899119899=1 minus 1
2sum sum sum 120601120601120572120572120572120572prime
119899119899119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime120572120572120572120572prime119899119899119899119899prime119899119899119899119899prime (13)
where 119906119906119899119899119899119899120572120572 120572120572 = 123 are the Cartesian coordinates of the displacement 119958119958119899119899119899119899(119905119905) and
120601120601120572120572120572120572prime 119899119899119899119899prime119904119904 119904119904prime = 1205971205972119881119881
120597120597119903119903119899119899119904119904119899119899119903119903119899119899prime119904119904prime119899119899primeeq
= 120601120601120572120572prime120572120572 119899119899prime119899119899119904119904prime 119904119904 = 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime (14)
Note that the first order term vanishes because we are expanding around the equilibrium positions
120601120601120572120572120572120572prime 119899119899119899119899prime119904119904 119904119904prime represents the component of the force acted on the sth atom in the nth unit cell along 120572120572
direction when the atom 119904119904prime in the unit cell 119899119899prime moves a unit displacement along 120572120572prime direction The
symmetries of 120601120601120572120572120572120572prime 119899119899 119899119899prime119904119904 119904119904prime appearing in Eq (14) arise from the intechangability of the second
derivative and the translational invariance of the interactions
12 Dynamical matrix
The equation of motion of the sth atom in the nth unit cell can be derived using the EulerndashLagrange
equation as follows
2
119872119872119899119899119906119899119899119899119899120572120572 = minussum 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119899119899prime119899119899prime120572120572prime 119906119906119899119899prime119899119899prime120572120572prime (15)
If we displace all atoms equally ie shifting 119906119906119899119899prime119899119899prime120572120572prime to 119906119906119899119899prime119899119899prime120572120572prime + 120575120575 the total force on the sth atom in
the nth unit cell does not change From the above equation we have
sum 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119899119899prime120572120572prime = 0 (16)
We are looking for normal modes (because any general solution can be written as a linear combination
of them) these are solutions where all atoms oscillate with the same frequency Moreover because
of the lattice structure we expect solutions to reflect this periodicity So we guess solutions of the
form
119906119906119899119899119899119899120572120572(119905119905) = 1119872119872119904119904
119890119890119899119899120572120572119890119890119894119894(119954119954∙119929119929119899119899minus120596120596119905119905) (17)
where 119890119890119899119899120572120572 are real-space solutions that will be determined later and 119954119954 is the wave-vector in
reciprocal space Substituting Eq (17) into Eq (15) we can get
1205961205962119890119890119899119899120572120572(119954119954) = sum 119863119863119899119899120572120572119899119899prime120572120572prime(119954119954)119890119890119899119899prime120572120572prime(119954119954)119899119899prime120572120572prime (18)
where
119863119863119899119899120572120572119899119899prime120572120572prime(119954119954) = sum 1
119872119872119904119904119872119872119904119904prime120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119890119890
minus119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime119899119899prime = sum 1
119872119872119904119904119872119872119904119904prime120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime 119890119890
minus119894119894119954119954∙119929119929119897119897119897119897 (19)
is called dynamical matrix (dimension 3119903119903 times 3119903119903) Note that we have defined the relative cell distance
vector 119929119929119897119897 equiv 119929119929119899119899 minus 119929119929119899119899prime and number index 119897119897 equiv 119899119899 minus 119899119899prime where the infinite sum over 119899119899prime can be
replaced by the sum over 119897119897 for any value of the index 119899119899 Note that dynamical matrix is Hermitian
symmetric (ie 119863119863119899119899120572120572119899119899prime120572120572prime(119954119954)
lowast= 119863119863119899119899prime120572120572prime
119899119899120572120572 (minus119954119954)) because ϕ is symmetric so all the eigenvalues of Eq (18)
(120596120596120582120582(119954119954) 120582120582 = 12⋯ 3119903119903) are real for each 119954119954 in the Brillouin zone (BZ) which is determined by
det1205961205961205821205822(119954119954)120575120575119899119899120572120572120575120575119899119899prime120572120572prime minus 119863119863119899119899120572120572119899119899prime120572120572prime(119954119954) = 0 (110)
Taking the conjugate of this equation we have
0 = det 1205961205961205821205822(119954119954)lowast120575120575119899119899120572120572120575120575119899119899prime120572120572prime minus 119863119863119899119899120572120572119899119899
prime120572120572prime(119954119954)lowast = det1205961205961205821205822(119954119954)120575120575119899119899120572120572120575120575119899119899prime120572120572prime minus 119863119863119899119899120572120572119899119899
prime120572120572prime(minus119954119954) (111)
while replacing 119954119954 by minus119954119954 in Eq (110) we have det1205961205961205821205822(minus119954119954)120575120575119899119899120572120572120575120575119899119899prime120572120572prime minus 119863119863119899119899120572120572119899119899prime120572120572prime(minus119954119954) = 0
Itrsquos clear that 1205961205961205821205822(119954119954) and 1205961205961205821205822(minus119954119954) obey the same equation thus we have
120596120596120582120582(minus119954119954) = 120596120596120582120582(119954119954) (112)
The corresponding eigenvectors are orthonormal
sum 119890119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582
prime119899119899120572120572 = 120575120575120582120582120582120582prime (113)
The complex conjugate of Eq (18) gives
1205961205962119890119890119899119899120572120572lowast (119954119954) = sum 119863119863119899119899120572120572119899119899prime120572120572prime(119954119954)
lowast119890119890119899119899prime120572120572primelowast (119954119954)119899119899prime120572120572prime = sum 119863119863119899119899prime120572120572prime
119899119899120572120572 (minus119954119954)119890119890119899119899prime120572120572primelowast (119954119954)119899119899prime120572120572prime (114)
3
While replacing 119954119954 by minus119954119954 in Eq (18) we get the following equation
1205961205962119890119890119899119899120572120572(minus119954119954) = sum 119863119863119899119899120572120572119899119899prime120572120572prime(minus119954119954)119890119890119899119899prime120572120572prime(minus119954119954)119899119899prime120572120572prime (115)
From Eq (114) and Eq (115) we see that eigenvectors 119890119890119899119899120572120572120582120582 (minus119954119954)lowast and 119890119890119899119899120572120572120582120582 (119954119954) obey the same
eigenvalue equation Since the eigenvectors are normalized we get the following property
119890119890119899119899120572120572120582120582 (minus119954119954)lowast
= 119890119890119899119899120572120572120582120582 (119954119954) (116)
2 Second quantization The general solution is a linear combination of all these normal modes thus we have
119906119906119899119899119899119899120572120572(119905119905) = sum 119862119862120582120582(119954119954)119873119873119872119872119904119904
119890119890119899119899120572120572120582120582 (119954119954)119890119890minus119894119894120596120596119954119954120582120582119905119905119890119890119894119894119954119954∙119929119929119899119899119954119954120582120582 = sum 119876119876120582120582(119954119954119905119905)119873119873119872119872119904119904
119890119890119899119899120572120572120582120582 (119954119954)119890119890119894119894119954119954∙119929119929119899119899119954119954120582120582 (21)
where the we define the normal coordinates as 119876119876120582120582(119954119954 119905119905) equiv 119862119862120582120582(119954119954)119890119890minus119894119894120596120596120582120582(119954119954)119905119905 in the eigenvectors
representation To ensure that the displacements 119906119906119899119899119899119899120572120572(119905119905) are real (namely 119906119906119899119899119899119899120572120572lowast (119905119905) = 119906119906119899119899119899119899120572120572(119905119905)) the
following relation on 119876119876120582120582(119954119954 119905119905) and 119890119890119899119899120572120572120582120582 is enforced
119876119876120582120582(119954119954 119905119905)119890119890119899119899120572120572120582120582 (119954119954)lowast
= 119876119876120582120582(minus119954119954 119905119905)119890119890119899119899120572120572120582120582 (minus119954119954) (22)
where we used the fact that the sum over 119954119954 runs symmetrically over both negative and positive
values Using Eq (116) we have
[119876119876120582120582(119954119954 119905119905)]lowast = 119876119876120582120582(minus119954119954 119905119905) (23)
21 Kinetic energy term
Using Eq (21) the kinetic energy of the system can be expressed as
119879119879 = 12sum 1198721198721198991198991199061198991198991198991198991205721205722 (119905119905)119899119899119899119899120572120572 = 1
2sum 119872119872119904119904
119873119873119872119872119904119904sum sum 119876120582120582(119954119954 119905119905)119876120582120582prime(119954119954prime 119905119905)119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582
prime(119954119954prime) sum 119890119890119894119894119954119954+119954119954prime∙119929119929119899119899119899119899119954119954prime120582120582prime119954119954120582120582119899119899120572120572 =
12sum sum 119876120582120582(119954119954 119905119905)119876120582120582prime(minus119954119954 119905119905)119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582
prime(minus119954119954)119954119954119954119954prime120582120582prime =119899119899120572120572
12sum 119876120582120582(119954119954 119905119905)119876120582120582prime
lowast (119954119954 119905119905) sum 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582prime(119954119954)
lowast119899119899120572120572 =119954119954120582120582120582120582prime
12sum 119876120582120582(119954119954 119905119905)119876120582120582lowast(119954119954 119905119905)119954119954120582120582 =
12sum 119876120582120582(119954119954 119905119905)119876120582120582(minus119954119954 119905119905)119954119954120582120582
ie
119879119879 = 12sum 119876120582120582(119954119954 119905119905)119876120582120582(minus119954119954 119905119905)119954119954120582120582 (24)
To derive Eq (24) we used Eqs (113) and (116) as well as the following equations
1119873119873sum 119890119890119894119894119954119954+119954119954prime∙119929119929119899119899119899119899 = 120575120575119954119954+119954119954prime120782120782 (25)
4
22 Potential energy term
Similarity the potential energy can be rewritten in terms of the normal coordinates as follows
119880119880 =12120601120601119899119899119899119899120572120572119899119899prime119899119899prime120572120572prime119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
120572120572120572120572prime119899119899119899119899prime119899119899119899119899prime=
12120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119899119899+119954119954prime∙119929119929119899119899prime
119873119873119872119872119899119899119872119872119899119899prime
[119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(119954119954prime 119905119905)] 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime120582120582prime (119954119954prime)
119954119954prime120582120582prime119954119954120582120582119899119899119899119899prime120572120572120572120572prime119899119899119899119899prime
=12120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894(119954119954+119954119954prime)∙119929119929119899119899119890119890minus119894119894119954119954prime∙119929119929119897119897
119873119873119872119872119899119899119872119872119899119899prime
[119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(119954119954prime 119905119905)] 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime120582120582prime (119954119954prime)
119954119954prime120582120582prime119954119954120582120582119899119899119897119897120572120572120572120572prime119899119899119899119899prime
=12120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890minus119894119894119954119954prime∙119929119929119897119897
119872119872119899119899119872119872119899119899prime[119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(119954119954prime 119905119905)] 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (119954119954prime)1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899
119899119899=120575120575119902119902+119902119902prime=0
119954119954prime120582120582prime119954119954120582120582119897119897120572120572120572120572prime119899119899119899119899prime
=12120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894119954119954119929119929119897119897
119872119872119899119899119872119872119899119899prime [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)] 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (minus119954119954)119954119954120582120582120582120582prime119897119897120572120572120572120572prime119899119899119899119899prime
=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]119890119890119899119899120572120572120582120582 (119954119954)
119899119899120572120572
119863119863119899119899120572120572119899119899prime120572120572prime(minus119954119954)119890119890119899119899prime120572120572prime
120582120582prime (minus119954119954)119899119899prime120572120572prime119954119954120582120582120582120582prime
=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]119890119890119899119899120572120572120582120582 (119954119954)
119899119899120572120572
1205961205961205821205822(minus119954119954)119890119890119899119899120572120572120582120582prime (minus119954119954)
119954119954120582120582120582120582prime
=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]1205961205961205821205822(119954119954)120575120575120582120582120582120582prime =119954119954120582120582120582120582prime
121205961205961205821205822(119954119954)119876119876120582120582(119954119954 119905119905)119876119876120582120582(minus119954119954 119905119905)
119954119954120582120582
ie
119880119880 = 12sum 1205961205961205821205822(119954119954)119876119876120582120582(119954119954 119905119905)119876119876120582120582(minus119954119954 119905119905)119954119954120582120582 (26)
where in above deriviation the orthogonality of its eigenvectores and the symmetry of its eigenvalues
are used Thus the Lagrangian reads as
ℒ = 119879119879 minus 119881119881 = 12sum 119876120582120582(minus119954119954 119905119905)119876120582120582(119954119954 119905119905) minus 120596120596120582120582
2(119954119954)119876119876120582120582(minus119954119954 119905119905)119876119876120582120582(119954119954 119905119905)119954119954120582120582 (27)
Using the relation 119875119875120582120582(119954119954 119905119905) = 120597120597ℒ part119876120582120582(119954119954 119905119905) the Hamiltonian of the system then can be written as
119867119867 = 119879119879 + 119881119881 = 12sum [119875119875120582120582(minus119954119954 119905119905)119875119875120582120582(119954119954 119905119905) + 120596120596120582120582
2(119954119954)119876119876120582120582(minus119954119954 119905119905)119876119876120582120582(119954119954 119905119905)]119954119954120582120582 (28)
Now we quantize H by asking the momenta and coordinates to be operators
119867119867 = 12sum 119875119875minus119954119954120582120582119875119875119954119954120582120582 + 120596120596119954119954120582120582
2 119876119876minus119954119954120582120582119876119876119954119954120582120582119954119954120582120582 (29)
here 119875119875119954119954120582120582 and 119876119876119954119954120582120582 obey the following commutation relations
119876119876119954119954120582120582119875119875119954119954prime120582120582prime = 119894119894ℏ120575120575119954119954119954119954prime120575120575120582120582120582120582prime
119876119876119954119954120582120582119876119876119954119954prime120582120582prime = 0 119875119875119954119954120582120582119875119875119954119954prime120582120582prime = 0 (210)
Similar to the case of ordinary quantum harmonic oscillators it is convenient to define ladder
operators for each mode as follows
119876119876119954119954120582120582 = ℏ
2120596120596119954119954120582120582119887119887119954119954120582120582 + 119887119887minus119954119954120582120582
dagger 119875119875119954119954120582120582 = 119894119894ℏ120596120596119954119954120582120582
2119887119887119954119954120582120582
dagger minus 119887119887minus119954119954120582120582 (211)
where 119887119887119954119954120582120582dagger and 119887119887119954119954120582120582 are Bosonic creation and annihilation operators for phonons with momentum
119954119954 branch index 120582120582 and frequency 120596120596119954119954120582120582 which obeys the Bosonic commutation relation
5
119887119887119954119954120582120582 119887119887119954119954prime120582120582prime
dagger = 120575120575119954119954119954119954prime120575120575120582120582120582120582prime
119887119887119954119954120582120582 119887119887119954119954prime120582120582prime = 0 119887119887119954119954120582120582dagger 119887119887119954119954prime120582120582prime
dagger = 0 (212)
Substituting Eqs (211) into (29) and using the properties Eq (210) and (212) we have
119867119867 =12119875119875minus119954119954120582120582119875119875119954119954120582120582 +120596120596119954119954120582120582
2 119876119876minus119954119954120582120582119876119876119954119954120582120582119954119954120582120582
=12minus
ℏ120596120596119954119954120582120582
2119887119887minus119954119954120582120582
dagger minus 119887119887119954119954120582120582119887119887119954119954120582120582dagger minus 119887119887minus119954119954120582120582+ 120596120596119954119954120582120582
2 ℏ2120596120596119954119954120582120582
119887119887minus119954119954120582120582 + 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 + 119887119887minus119954119954120582120582
dagger 119954119954120582120582
=ℏ4120596120596119954119954120582120582minus119887119887minus119954119954120582120582
dagger 119887119887119954119954120582120582dagger minus 119887119887minus119954119954120582120582
dagger 119887119887minus119954119954120582120582 minus 119887119887119954119954120582120582119887119887119954119954120582120582dagger + 119887119887119954119954120582120582119887119887minus119954119954120582120582
119954119954120582120582
+ 119887119887minus119954119954120582120582119887119887119954119954120582120582 + 119887119887minus119954119954120582120582119887119887minus119954119954120582120582dagger + 119887119887119954119954120582120582
dagger 119887119887119954119954120582120582 + 119887119887119954119954120582120582dagger 119887119887minus119954119954120582120582
dagger
=ℏ4120596120596119954119954120582120582119887119887minus119954119954120582120582
dagger 119887119887minus119954119954120582120582 + 119887119887minus119954119954120582120582119887119887minus119954119954120582120582dagger + 119887119887119954119954120582120582119887119887119954119954120582120582
dagger + 119887119887119954119954120582120582dagger 119887119887119954119954120582120582
119954119954120582120582
=ℏ41205961205961199541199541205821205822119887119887minus119954119954120582120582
dagger 119887119887minus119954119954120582120582 + 1+ 2119887119887119954119954120582120582dagger 119887119887119954119954120582120582 + 1
119954119954120582120582
= ℏ120596120596119954119954120582120582 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 +
12
119954119954120582120582
Therefore we finally get the quantized representation of non-interacting phonons
119867119867 = sum ℏ120596120596119954119954120582120582 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 + 1
2119954119954120582120582 (213)
The operator of atom displacements (Eq 114) is expressed in terms of the phonon operators by
119906119906119899119899119899119899120572120572 = sum ℏ
2119873119873119872119872119904119904120596120596119954119954120582120582119890119890119899119899120572120572120582120582 119890119890119894119894119954119954∙119929119929119899119899119887119887119954119954120582120582 + 119887119887minus119954119954120582120582
dagger 119954119954120582120582 (214)
These equations will be used below
3 Inter-phonon coupling within harmonic approximation 31 Hamiltonian with inter-phonon coupling
In this section we consider systems that consist of two (or more) covalently bonded units that are
weakly coupled between them Unlike previous studies that considered phonon-phonon couplings
resulting from anharmonicity effects [1] here all phonon mode considered are Harmonic and the
couplings arise from the division of the entire system into subunits The Hamiltonian of the whole
system can be written as
119867119867 = 1198671198671 + 1198671198672 + 11986711986712 (31)
To derive the expressions of 1198671198671 1198671198672 and 11986711986712 in Eq (31) we consider the Hamiltonian of the
whole system written as the function of the atomic displacements
119867119867 = 119879119879 + 119881119881 = sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)2119899119899119899119899120572120572 + 1
2sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime120572120572120572120572prime119899119899119899119899prime119899119899119899119899prime (32)
6
Letrsquos assume that subsystem I and II contain atoms with indeices ranging from 119904119904 = 12⋯ 1199031199032 and
119904119904 = 1199031199032
+ 1 1199031199032
+ 2⋯ 119903119903 respectively Then the kinetic energy term in Eq (32) can be rewritten as
119879119879 = sum sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)2
1199031199032119899119899=1 + sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)
2119903119903119899119899=1199031199032+1
119899119899120572120572 (33)
While the potential energy term is
119881119881 =12 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
1199031199032
119899119899119899119899prime=1
+ 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
119903119903
119899119899119899119899prime=1199031199032+1
+ 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime119899119899119899119899prime
+ 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
1199031199032
119899119899prime=1
119903119903
119899119899=1199031199032+1
= 11988111988111 + 11988111988122 + 11988111988112 + 11988111988121
Or equivalently
⎩⎪⎪⎨
⎪⎪⎧ 11988111988111 = 1
2sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
1199031199032119899119899119899119899prime=1120572120572120572120572prime119899119899119899119899prime
11988111988122 = 12sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
119903119903119899119899119899119899prime=1199031199032+1120572120572120572120572prime119899119899119899119899prime
11988111988112 = 12sum sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
119903119903119899119899prime=1199031199032+1
1199031199032119899119899=1120572120572120572120572prime119899119899119899119899prime
11988111988121 = 12sum sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
1199031199032119899119899prime=1
119903119903119899119899=1199031199032+1120572120572120572120572prime119899119899119899119899prime
(34)
32 Second quantization
We may now quantize this Hamiltonian and write it in the basis of the eigenstates of the coupled
subunits In second quantization the atomic displacement operators of the two subunits are given in
the following form
119906119906119899119899119899119899120572120572 =
⎩⎨
⎧ sum ℏ
2119873119873119872119872119904119904ω119954119954120582120582119890119890119899119899120572120572120582120582 119890119890119894119894119954119954∙119929119929119899119899119886119886119954119954120582120582 + 119886119886minus119954119954120582120582
dagger 119954119954120582120582 119904119904 isin 1 1199031199032
sum ℏ
2119873119873119872119872119904119904120596120596119954119954prime120582120582prime119890119899119899120572120572120582120582
prime119890119890119894119894119954119954prime∙119929119929119899119899 119886119886 119954119954prime120582120582prime + 119886119886minus119954119954prime120582120582primedagger 119954119954prime120582120582prime 119904119904 isin 119903119903
2+ 1 119903119903
(35)
Here we use different notations for the creation and annihilation operators (119886119886119954119954120582120582119886119886minus119954119954120582120582dagger and 119886119886119954119954120582120582119886119886minus119954119954120582120582
dagger )
eigenvalues (ω119954119954120582120582120596120596119954119954120582120582) and eigenvectors (119890119890119899119899120572120572120582120582 119890119899119899120572120572120582120582 ) for the two subunits Note that 119890119890119899119899120572120572120582120582 and 119890119899119899120572120572120582120582 are
of the dimensions of the whole system however their non-zero elements appear only on the relevant
subunits such that they obey the following relations sum 119890119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582
prime119899119899120572120572 = 120575120575120582120582120582120582prime sum 119890119899119899120572120572120582120582
lowast119890119899119899120572120572120582120582
prime119899119899120572120572 =
120575120575120582120582120582120582prime sum 119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582
prime119899119899120572120572 = sum 119890119890119899119899120572120572120582120582
lowast119890119899119899120572120572120582120582
prime119899119899120572120572 = 0 Defining the normal coordinates of the two subunits as
119876119876119954119954120582120582 equiv ℏ
2ω119954119954120582120582119886119886119954119954120582120582 + 119886119886minus119954119954120582120582
dagger 119876119876119954119954120582120582 equiv ℏ
2ω119954119954120582120582119886119886119954119954120582120582 + 119886119886minus119954119954120582120582
dagger (36)
and the corresponding momenta operators as
7
119875119875119954119954120582120582 = 119894119894ℏω119954119954120582120582
2119886119886119954119954120582120582
dagger minus 119886119886minus119954119954120582120582 119875119875119954119954120582120582 = 119894119894ℏω1199541199541205821205822
119886119886119954119954120582120582dagger minus 119886119886minus119954119954120582120582 (37)
Eq (35) reads
119906119906119899119899119899119899120572120572 = sum 119890119890119904119904119899119899120582120582 (119954119954)
119873119873119872119872119904119904119890119890119894119894119954119954∙119929119929119899119899119876119876119954119954120582120582119954119954120582120582 119904119904 isin 1 119903119903
2
sum 119890119904119904119899119899120582120582 (119954119954)119873119873119872119872119904119904
119890119890119894119894119954119954∙119929119929119899119899119876119876119954119954120582120582119954119954120582120582 119904119904 isin 1199031199032
+ 1 119903119903 (38)
The second quantized kinetic energy operator is then written as
119879119879 = 1198791198791 + 1198791198792 = 12sum 119875119875119954119954120582120582119875119875minus119954119954120582120582119954119954120582120582 + 1
2sum 119875119875119954119954prime120582120582prime119875119875minus119954119954prime120582120582prime119954119954prime120582120582prime (39)
Correspondingly the various potential energy terms are obtained by substituting Eq (38) in Eq (34)
11988111988111 =12 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119899119899119899119899prime
119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime120582120582prime (119954119954prime)
119872119872119899119899119872119872119899119899prime
1119873119873119890119890119894119894119954119954∙119929119929119899119899+119894119894119954119954prime∙119929119929119899119899prime119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954120582120582119954119954prime120582120582prime
1199031199032
119899119899119899119899prime=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954119954119954prime120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (119954119954prime)1119873119873 119890119890
119894119894(119954119954+119954119954prime)119929119929119899119899prime
119899119899prime120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime
119872119872119899119899119872119872119899119899prime119899119899
1199031199032
119899119899119899119899prime=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954119954119954prime120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (119954119954prime) 120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime
1199031199032
119899119899119899119899prime=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime
119954119954120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (minus119954119954)120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
1199031199032
119899119899119899119899prime=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime
119954119954120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954) 119863119863119899119899120572120572119899119899
prime120572120572prime(minus119954119954)119890119890119899119899prime120572120572prime120582120582prime (minus119954119954)
120572120572prime
1199031199032
119899119899prime=1
1199031199032
119899119899=1120572120572
=12 ω119954119954120582120582prime
2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582120582120582prime
119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582prime (119954119954)
lowast
1199031199032
119899119899=1
=120572120572
12ω119954119954120582120582
2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582
ie
11988111988111 = 12sum ω119954119954120582120582
2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582 (310)
Going from the second line to the third line we used the fact that the sum over 119899119899prime runs between plusmninfin
and the summand depends only on the difference between 119899119899 and 119899119899prime hence the sum is independent
of the value of the index 119899119899 Therefore we can replace the sum over 119899119899prime by a sum over 119897119897 equiv 119899119899 minus 119899119899prime
amd define 119929119929119897119897 equiv 119929119929119899119899 minus 119929119929119899119899prime Following the same procedure we can get the corresponding
expressions for the second diagonal term
11988111988122 = 12sum 1205961205961199541199541205821205822 119876119876119954119954120582120582119876119876minus119954119954120582120582119954119954120582120582 (311)
Similarly for the off-diagonal terms we get
8
11988111988112 =12 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119899119899119899119899prime
119903119903
119899119899prime=1199031199032+1
119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime
120582120582prime (119954119954prime)
119872119872119899119899119872119872119899119899prime
1119873119873119890119890119894119894119954119954∙119929119929119899119899+119894119894119954119954prime∙119929119929119899119899prime119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954120582120582119954119954prime120582120582prime
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954119954119954prime120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime
120582120582prime (119954119954prime)1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime
119872119872119899119899119872119872119899119899prime119899119899
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954119954119954prime120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime
120582120582prime (119954119954prime) 120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime
119954119954120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime
120582120582prime (minus119954119954)120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954
119890119899119899prime120572120572prime120582120582prime (119954119954)
lowast120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119890119890119899119899120572120572120582120582 (119954119954)119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954
Where we have defined
119881119881120582120582120582120582prime(119954119954) equiv sum sum sum 119890119899119899prime120572120572prime120582120582prime (119954119954)
lowast119881119881119899119899120572120572119899119899
prime120572120572prime(119954119954)119890119890119899119899120572120572120582120582 (119954119954)119903119903119899119899prime=1199031199032+1
1199031199032119899119899=1120572120572120572120572prime (312)
where
119881119881119899119899120572120572119899119899prime120572120572prime(119954119954) equiv sum 119890119890119894119894119954119954∙119929119929119897119897
119872119872119904119904119872119872119904119904prime119897119897 120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime (313)
Itrsquos easy to show that 119881119881120582120582120582120582prime(119954119954) has the following property
119881119881120582120582120582120582primelowast (119954119954) = 119881119881120582120582120582120582prime(minus119954119954) (314)
Following the same procedure we have
9
11988111988121 =12 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119899119899119899119899prime
119903119903
119899119899=1199031199032+1
119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (119954119954prime)
119872119872119899119899119872119872119899119899prime
1119873119873119890119890119894119894119954119954∙119929119929119899119899+119894119894119954119954prime∙119929119929119899119899prime119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954120582120582119954119954prime120582120582prime
1199031199032
119899119899prime=1120572120572120572120572prime
=12 120601120601120572120572prime120572120572
119899119899prime minus 119899119899119904119904prime 119904119904
119899119899prime119899119899
119903119903
119899119899prime=1199031199032+1
119890119899119899prime120572120572prime120582120582prime (119954119954prime)119890119890119899119899120572120572120582120582 (119954119954)
119872119872119899119899prime119872119872119899119899
1119873119873 119890119890
119894119894119954119954prime∙119929119929119899119899prime+119894119894119954119954∙119929119929119899119899119876119876119954119954prime120582120582prime119876119876119954119954120582120582119954119954prime120582120582prime119954119954120582120582
1199031199032
119899119899=1120572120572prime120572120572
=12 119876119876119954119954prime120582120582prime119876119876119954119954120582120582
119954119954119954119954prime120582120582120582120582prime119890119899119899prime120572120572prime120582120582prime (119954119954prime)119890119890119899119899120572120572120582120582 (119954119954)
1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime
119872119872119899119899119872119872119899119899prime119899119899
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954prime120582120582prime119876119876119954119954120582120582
119954119954119954119954prime120582120582120582120582prime119890119899119899prime120572120572prime120582120582prime (119954119954prime)119890119890119899119899120572120572120582120582 (119954119954) 120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954prime120582120582prime119876119876minus119954119954prime120582120582119890119899119899prime120572120572prime
120582120582prime (119954119954prime)119890119890119899119899120572120572120582120582 (minus119954119954prime)120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890minus119894119894119954119954prime∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897119954119954prime120582120582120582120582prime
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954prime120582120582prime119876119876119954119954prime120582120582
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954prime 119890119890119899119899120572120572120582120582 (119954119954prime)
lowast120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890minus119894119894119954119954prime∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119890119899119899prime120572120572prime120582120582prime (119954119954prime)
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582prime119876119876119954119954120582120582
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954
119890119890119899119899120572120572120582120582 (119954119954)lowast120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890minus119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119890119899119899prime120572120572prime120582120582prime (119954119954)
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger dagger
3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954
119890119899119899prime120572120572prime120582120582prime (119954119954)
lowast120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119890119890119899119899120572120572120582120582 (119954119954)119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
lowast
=
⎩⎨
⎧12 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954⎭⎬
⎫dagger
= 11988111988112dagger
Define
1198671198671 = 11987911987911 + 119881119881111198671198672 = 11987911987922 + 1198811198812211986711986712 = 11988111988112 + 11988111988121
(315)
we finally get the expressions of 1198671198671 1198671198672 and 11986711986712 in Eq (31) as follows
⎩⎪⎨
⎪⎧ 1198671198671 = sum sum ℏ120596120596119954119954120582120582 119886119886119954119954120582120582
dagger 119886119886119954119954120582120582 + 1231199031199032
120582120582=1119954119954
1198671198672 = sum sum ℏ120596120596119954119954120582120582prime 119886119886119954119954120582120582primedagger 119886119886119954119954120582120582prime + 1
23119903119903
120582120582prime=1+31199031199032119954119954
11986711986712 = 12sum sum sum 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903120582120582prime=31199031199032 +1
31199031199032120582120582=1 + h c 119954119954
(316)
Here hc means the Hermitian conjugate Since the indices of 119886119886119954119954120582120582 119886119886119954119954120582120582prime and 119876119876119954119954120582120582 119876119876119954119954120582120582prime belong to
the two system sections we can define an abbreviated notation 119886119886119954119954120582120582 and 119876119876119954119954120582120582 using the index 120582120582 to
identify which subsystem they belong to In this case the Hamiltonian of the coupled system can be
10
simplified as follows
119867119867 = 1198671198670 + 119867119867119862119862 (317)
where
1198671198670 = sum sum ℏ120596120596119954119954120582120582 119886119886119954119954120582120582
dagger 119886119886119954119954120582120582 + 123119903119903
120582120582=1119954119954
119867119867119862119862 = 12sum sum sum 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903120582120582prime=31199031199032 +1
31199031199032120582120582=1 + ℎ 119888119888 119954119954
(318)
4 Greenrsquos function 41 Dynamics of the ladder operators
In order to describe the dynamics of the ladder operators appearing in the Hamiltonian of Eq (317)
we express them in the Heisenberg picture as follows
119886119886119901119901(120591120591) = 119890119890119894119894ℏ119867119867119905119905119886119886119901119901119890119890
minus119894119894ℏ119867119867119905119905 = 119890119890
119867119867120591120591ℏ 119886119886119901119901119890119890
minus119867119867120591120591ℏ (41)
where we define the imaginary time 120591120591 equiv 119894119894119905119905 and introduce the notation 119953119953 equiv (119954119954 120582120582) and 119953119953 equiv (minus119954119954 120582120582)
The corresponding equation of motion for the ladder operators is given by
ℏ part119886119886119953119953(120591120591)
120597120597120591120591= 119867119867119886119886119953119953(120591120591) (42)
For the uncoupled system (119867119867119862119862 = 0) this gives
ℏpart119886119886119953119953(120591120591)120597120597120591120591
= 1198671198670119886119886119953119953(120591120591) = 1198671198670 1198901198901198671198670120591120591ℏ 119886119886119953119953119890119890
minus1198671198670120591120591ℏ = 1198671198670119890119890
1198671198670120591120591ℏ 119886119886119953119953119890119890
minus1198671198670120591120591ℏ minus 119890119890
1198671198670120591120591ℏ 119886119886119953119953119890119890
minus1198671198670120591120591ℏ 1198671198670
= 1198901198901198671198670120591120591ℏ 1198671198670119886119886119953119953 minus 1198861198861199531199531198671198670119890119890
minus1198671198670120591120591ℏ = 119890119890
1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890
minus1198671198670120591120591ℏ
ie
ℏ part119886119886119953119953(120591120591)
120597120597120591120591= 119890119890
1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890
minus1198671198670120591120591ℏ (43)
The commutator on the right-hand-side of Eq (43) can be evaluated as follows
1198671198670119886119886119953119953prime = sum ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953 + 1
2119953119953 119886119886119953119953prime = sum ℏ120596120596119953119953119886119886119953119953
dagger119886119886119953119953119886119886119953119953prime119953119953 = sum ℏ120596120596119953119953119886119886119953119953dagger119886119886119953119953119886119886119953119953prime minus 119886119886119953119953prime119886119886119953119953
dagger119886119886119953119953119953119953 =
sum ℏ120596120596119953119953119886119886119953119953dagger119886119886119953119953119886119886119953119953prime minus 120575120575119953119953prime119953119953 + 119886119886119953119953
dagger119886119886119953119953prime119886119886119953119953119953119953 = minusℏ120596120596119953119953prime119886119886119953119953prime + sum ℏ120596120596119953119953119886119886119953119953dagger119886119886119953119953119886119886119953119953prime minus 119886119886119953119953
dagger119886119886119953119953119886119886119953119953prime119953119953 = minusℏ120596120596119953119953prime119886119886119953119953prime
ie
1198671198670119886119886119953119953prime = minusℏ120596120596119953119953prime119886119886119953119953prime (44)
Therefore we have
ℏ part119886119886119953119953(120591120591)
120597120597120591120591= 119890119890
1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890
minus1198671198670120591120591ℏ = minusℏ120596120596119953119953119890119890
1198671198670120591120591ℏ 119886119886119953119953119890119890
minus1198671198670120591120591ℏ = minusℏ120596120596119953119953119886119886119953119953(120591120591) (45)
11
The solution of Eq (45) is 119886119886119953119953(120591120591) = 119886119886119953119953(0)119890119890minus120596120596119953119953120591120591 = 119886119886119953119953119890119890minus120596120596119953119953120591120591 For 119886119886119953119953dagger(120591120591) we have
1198671198670119886119886119953119953primedagger (0) = 1198671198670119886119886119953119953prime
dagger = ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953 +
12
119953119953
119886119886119953119953primedagger = ℏ120596120596119953119953119886119886119953119953
dagger119886119886119953119953119886119886119953119953primedagger
119953119953
= ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953119886119886119953119953prime
dagger minus 119886119886119953119953primedagger 119886119886119953119953
dagger119886119886119953119953119953119953
= ℏ120596120596119953119953 119886119886119953119953dagger 120575120575119953119953119953119953prime + 119886119886119953119953prime
dagger 119886119886119953119953 minus 119886119886119953119953primedagger 119886119886119953119953
dagger119886119886119953119953119953119953
= ℏ120596120596119953119953prime119886119886119953119953primedagger + ℏ120596120596119953119953 119886119886119953119953
dagger119886119886119953119953primedagger 119886119886119953119953 minus 119886119886119953119953prime
dagger 119886119886119953119953dagger119886119886119953119953
119953119953
= ℏ120596120596119953119953prime119886119886119953119953primedagger
In summary we have
119886119886119953119953(120591120591) = 119886119886119953119953(0)119890119890minus120596120596119953119953120591120591 = 119886119886119953119953119890119890minus120596120596119953119953120591120591
119886119886119953119953dagger(120591120591) = 119886119886119953119953
dagger(0)119890119890120596120596119953119953120591120591 = 119886119886119953119953dagger119890119890120596120596119953119953120591120591 (46)
42 Greenrsquos function
To describe thermal transport properties between different system sections we use the formalism of
thermal (or imaginary time) phonon Greenrsquos function [2] To this end we define the thermal Greenrsquos
function for phonons as
119866119866119953119953119953119953prime(120591120591 120591120591prime) = minuslang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime)rang = minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime) (47)
where 119879119879120591120591 is the time ordering operator and 120588120588119867119867 is the statistical operator for the grand canonical
ensemble (note that the chemical potential for phonons is zero)
120588120588119867119867 = 119890119890minus120573120573119867119867119885119885119867119867 (48)
where the partition function is given by 119885119885119867119867 equiv Tr119890119890minus120573120573119867119867 and 120573120573 = 1119896119896119861119861119879119879 with 119896119896119861119861 being
Boltzmannrsquos constant and 119879119879 the temperature For time independent Hamiltonians 119866119866119953119953119953119953prime(120591120591 120591120591prime)
depends only on 120591120591 minus 120591120591prime ie
119866119866119953119953119953119953prime(120591120591 120591120591prime) = 119866119866119953119953119953119953prime(120591120591 minus 120591120591prime 0) (49)
To show this we shall first assume that 120591120591 gt 120591120591prime such that
119866119866119953119953119953119953prime(120591120591 120591120591prime) = minuslang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime)rang = minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime) = minusTr 120588120588119867119867119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime)
= minusTr 120588120588119867119867119890119890119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591ℏ 119890119890
119867119867120591120591primeℏ 119886119886119953119953prime
dagger 119890119890minus119867119867120591120591primeℏ = minusTr 120588120588119867119867119890119890
minus119867119867120591120591primeℏ 119890119890
119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591minus120591120591prime
ℏ 119886119886119953119953primedagger
= minusTr 120588120588119867119867119890119890119867119867120591120591minus120591120591prime
ℏ 119886119886119953119953119890119890minus119867119867120591120591minus120591120591prime
ℏ 119886119886119953119953primedagger = minusTr 120588120588119867119867119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953prime
dagger (0)
= minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953primedagger (0) = minus lang119879119879120591120591119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953prime
dagger (0)rang = 119866119866119953119953119953119953prime(120591120591 minus 120591120591prime 0)
Where we have used the commutativity of 120588120588119867119867 and 119890119890plusmn119867119867120591120591prime
ℏ and the invariance of the trace operation
12
towards cyclic permutations Similarly for 120591120591 lt 120591120591prime we have
119866119866119953119953119953119953prime(120591120591 120591120591prime) = minuslang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime)rang = minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime) = minusTr 120588120588119867119867119886119886119953119953primedagger (120591120591prime)119886119886119953119953(120591120591)
= minusTr 120588120588119867119867119890119890119867119867120591120591primeℏ 119886119886119953119953prime
dagger 119890119890minus119867119867120591120591primeℏ 119890119890
119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591ℏ = minusTr 120588120588119867119867119886119886119953119953prime
dagger 119890119890119867119867120591120591minus120591120591prime
ℏ 119886119886119953119953119890119890minus119867119867120591120591ℏ 119890119890
119867119867120591120591primeℏ
= minusTr 120588120588119867119867119886119886119953119953primedagger 119890119890
119867119867120591120591minus120591120591primeℏ 119886119886119953119953119890119890
minus119867119867120591120591minus120591120591prime
ℏ = minusTr 120588120588119867119867119886119886119953119953primedagger (0)119886119886119953119953(120591120591 minus 120591120591prime)
= minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953primedagger (0)119886119886119953119953(120591120591 minus 120591120591prime) = minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953prime
dagger (0)
= minuslang119879119879120591120591119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953primedagger (0)rang = 119866119866119953119953119953119953prime(120591120591 minus 120591120591prime 0)
For simplicity we will introduce the notation 119866119866119953119953119953119953prime(120591120591 0) equiv 119866119866119953119953119953119953prime(120591120591) We further note that when using
imaginary time 119866119866119953119953119953119953prime(120591120591) is a periodic functions in the domain [minus120573120573ℏ120573120573ℏ] with a period of 120573120573ℏ (see
Page 236 of Ref [2])
119866119866119953119953119953119953prime(120591120591 + 120573120573ℏ 0) = 119866119866119953119953119953119953prime(120591120591 0) 120591120591 lt 0119866119866119953119953119953119953prime(120591120591 minus 120573120573ℏ 0) = 119866119866119953119953119953119953prime(120591120591 0) 120591120591 gt 0 (410)
To show this we shall again assume first that minus120573120573ℏ lt 120591120591 lt 0 to write that is
119866119866119953119953119953119953prime(120591120591 + 120573120573ℏ 0) = minus lang119879119879120591120591119886119886119953119953(120591120591 + 120573120573ℏ)119886119886119953119953primedagger (0)rang = minuslang119886119886119953119953(120591120591 + 120573120573ℏ)119886119886119953119953prime
dagger (0)rang
= minusTr 120588120588119867119867119890119890119867119867(120591120591+120573120573ℏ)
ℏ 119886119886119953119953119890119890minus119867119867(120591120591+120573120573ℏ)
ℏ 119886119886119953119953prime = minusTr 120588120588119867119867119890119890120573120573119867119867119890119890
119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591ℏ 119890119890minus120573120573119867119867119886119886119953119953prime
= minusTr120588120588119867119867119890119890120573120573119867119867119886119886119953119953(120591120591)119890119890minus120573120573119867119867119886119886119953119953prime(0) = minus
1119885119885119867119867
Tr119890119890minus120573120573119867119867119890119890120573120573119867119867119886119886119953119953(120591120591)119890119890minus120573120573119867119867119886119886119953119953prime(0)
= minus1119885119885119867119867
Tr119886119886119953119953(120591120591)119890119890minus120573120573119867119867119886119886119953119953prime(0) = minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119886119886119953119953prime(0)119886119886119953119953(120591120591)
= minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime(0) = minusTr120588120588119867119867119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime = 119866119866119953119953119953119953prime(120591120591 0)
Similarly for 120573120573ℏ gt 120591120591 gt 0 we have
119866119866119953119953119953119953prime(120591120591 minus 120573120573ℏ 0) = minus lang119879119879120591120591119886119886119953119953(120591120591 minus 120573120573ℏ)119886119886119953119953primedagger (0)rang = minuslang119886119886119953119953prime
dagger (0)119886119886119953119953(120591120591 minus 120573120573ℏ)rang
= minusTr 120588120588119867119867119886119886119953119953prime119890119890119867119867(120591120591minus120573120573ℏ)
ℏ 119886119886119953119953119890119890minus119867119867(120591120591minus120573120573ℏ)
ℏ = minusTr 120588120588119867119867119886119886119953119953prime119890119890minus120573120573119867119867119890119890
119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591ℏ 119890119890120573120573119867119867
= minusTr120588120588119867119867119886119886119953119953prime(0)119890119890minus120573120573119867119867119886119886119953119953(120591120591)119890119890120573120573119867119867 = minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119886119886119953119953prime(0)119890119890minus120573120573119867119867119886119886119953119953(120591120591)119890119890120573120573119867119867
= minus1119885119885119867119867
Tr119886119886119953119953prime(0)119890119890minus120573120573119867119867119886119886119953119953(120591120591) = minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119886119886119953119953(120591120591)119886119886119953119953prime(0)
= minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime(0) = minusTr120588120588119867119867119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime = 119866119866119953119953119953119953prime(120591120591 0)
13
Therefore 119866119866119953119953119953119953prime(120591120591) can be expanded as a Fourier series in the domain [0120573120573ℏ] as follows
119866119866119953119953119953119953prime(120591120591) = 1120573120573ℏsum 119890119890minus119894119894120596120596119899119899120591120591119866119866119953119953119953119953prime(119894119894120596120596119899119899)infinminusinfin (411)
where 120596120596119899119899 = 2120587120587119899119899120573120573ℏ
and the associated Fourier coefficient is given by
119866119866119953119953119953119953prime(119894119894120596120596119899119899) = int d120591120591119890119890119894119894120596120596119899119899120591120591119866119866119953119953119953119953prime(120591120591)120573120573ℏ0 120596120596119899119899 = 2119899119899120587120587
120573120573ℏ (412)
Having proven the translational time invariance and the periodicity of the Greenrsquos functios we can
now calculate it for the uncoupled system
119866119866119953119953119953119953prime0 (120591120591 120591120591prime) = minuslang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime)rang0 == minus lang119886119886119953119953(0)119890119890minus120596120596119953119953120591120591119886119886119953119953prime
dagger (0)119890119890120596120596119953119953prime120591120591primerang 120591120591 minus 120591120591prime gt 0
minus lang119886119886119953119953primedagger (0)119890119890120596120596119953119953prime120591120591
prime119886119886119953119953(0)119890119890minus120596120596119953119953120591120591rang 120591120591 minus 120591120591prime lt 0
= minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591
primelang119886119886119953119953(0)119886119886119953119953prime
dagger (0)rang 120591120591 minus 120591120591prime gt 0
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang 120591120591 minus 120591120591prime lt 0
= minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591
primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0) + 119886119886119953119953(0)119886119886119953119953primedagger (0)rang 120591120591 minus 120591120591prime gt 0
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang 120591120591 minus 120591120591prime lt 0
= minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591
primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang + 120575120575119953119953119953119953prime 120591120591 minus 120591120591prime gt 0
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang 120591120591 minus 120591120591prime lt 0
ie
119866119866119953119953119953119953prime0 (120591120591 120591120591prime) =
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang + 120575120575119953119953119953119953prime 120591120591 minus 120591120591prime gt 0
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang 120591120591 minus 120591120591prime lt 0 (413)
where we have used Eq (46) for 119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime) and Commutation relation 119886119886119953119953(0)119886119886119953119953prime
dagger (0) = 120575120575119953119953119953119953prime
To calculate lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang we first prove following equation
119890119890119860119860119861119861119890119890minus119860119860 = sum 1119899119899119860(119899119899)119861119861infin
119899119899=0 (414)
where 119860(119896119896)119861119861 equiv 119860 119860(119896119896minus1)119861119861 To prove this identity we define the operator 119891119891(119905119905) = 119890119890119905119905A119861119861119890119890minus119905119905119860119860
and taylor expand it around 119905119905 = 0
119891119891(119905119905) = 119891119891(0) + 119905119905119891119891prime(0) + 1199051199052
2119891119891primeprime(0) + ⋯ = sum 119905119905119899119899
119899119899119889119889119899119899119891119889119889119905119905119899119899
119905119905=0
infin119899119899=0 (415)
The corresponding derivatives are given by
⎩⎪⎨
⎪⎧ 119891119891prime(119905119905) = 119890119890119905119905119860119860119860119861119861119890119890minus119905119905119860119860 minus 119890119890119905119905119860119860119861119861119860119890119890minus119905119905119860119860 = 119890119890119905119905119860119860119860119861119861119890119890minus119905119905119860119860
119891119891primeprime(119905119905) = 119890119890119905119905119860119860119860119860119861119861 minus 119860119861119861119860119890119890minus119905119905119860119860 = 119890119890119905119905119860119860119860(2)119861119861119890119890minus119905119905119860119860
⋮119891119891(119899119899)(119905119905) = 119890119890119905119905119860119860119860(119899119899)119861119861119890119890minus119905119905119860119860
(416)
14
Substituting Eq (416) into Eq (415) we have
119890119890119905119905A119861119861119890119890minus119905119905119860119860 = sum 119905119905119899119899
119899119899119860(119899119899)119861119861infin
119899119899=0 (417)
Itrsquos clear that Eq (413) is a spectial case of Eq (416) with 119905119905 = 1 With this we can proceed as
follows
lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang =
11198851198851198671198670
Tr119890119890minus1205731205731198671198670119886119886119953119953primedagger (0)119886119886119953119953(0) =
11198851198851198671198670
Tr119890119890minus1205731205731198671198670119886119886119953119953primedagger (0)1198901198901205731205731198671198670119890119890minus1205731205731198671198670119886119886119953119953(0)
=11198851198851198671198670
Tr (minus120573120573)119899119899
1198991198991198671198670
(119899119899)119886119886119953119953primedagger (0)
infin
119899119899=0
119890119890minus1205731205731198671198670119886119886119953119953(0)
Note that 1198671198670119886119886119953119953primedagger (0) = ℏ120596120596119953119953prime119886119886119953119953prime
dagger (0) we have
1198671198670(119899119899)119886119886119953119953prime
dagger (0) = ℏ120596120596119953119953prime119899119899119886119886119953119953primedagger (0) (418)
such that
lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang =
11198851198851198671198670
Tr (minus120573120573)119899119899
1198991198991198671198670
(119899119899)119886119886119953119953primedagger (0)
infin
119899119899=0
119890119890minus1205731205731198671198670119886119886119953119953(0)
=11198851198851198671198670
Tr minus120573120573ℏ120596120596119953119953prime
119899119899
119899119899119886119886119953119953primedagger (0)
infin
119899119899=0
119890119890minus1205731205731198671198670119886119886119953119953(0)
= 119890119890minus120573120573ℏ120596120596119953119953prime11198851198851198671198670
Tr119890119890minus1205731205731198671198670119886119886119953119953(0)119886119886119953119953primedagger (0) = 119890119890minus120573120573ℏ120596120596119953119953primelang119886119886119953119953(0)119886119886119953119953prime
dagger (0)rang
= 119890119890minus120573120573ℏ120596120596119953119953primelang119886119886119953119953primedagger (0)119886119886119953119953(0)rang + 120575120575119953119953119953119953prime
therefore we obtain lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang 119890119890120573120573ℏ120596120596119953119953prime minus 1 = 120575120575119953119953119953119953prime For 119953119953prime ne 119953119953 and general 119890119890120573120573ℏ120596120596119953119953prime we have
lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang = 0 For 119953119953prime = 119953119953 we obtain lang119886119886119953119953
dagger(0)119886119886119953119953(0)rang = 119890119890120573120573ℏ120596120596119953119953 minus 1minus1
Hence we have
lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang = 120575120575119953119953prime119953119953
1119890119890120573120573ℏ120596120596119953119953minus1
equiv 120575120575119953119953prime119953119953119899119899119861119861120596120596119953119953 (419)
where 119899119899119861119861120596120596119953119953 is the Bose-Einstein distribution for phonons Substituting Eq (418) in Eq (413)
we obtain
119866119866119953119953119953119953prime0 (120591120591 120591120591prime) =
minus120575120575119953119953119953119953prime1 + 119899119899119861119861120596120596119953119953119890119890minus120596120596119953119953120591120591minus120591120591prime 120591120591 minus 120591120591prime gt 0
minus120575120575119953119953119953119953prime119899119899119861119861120596120596119953119953119890119890minus120596120596119953119953120591120591minus120591120591prime 120591120591 minus 120591120591prime lt 0 (420)
Note that only those Greenrsquos functions of the form 119866119866119953119953119953119953prime0 (120591120591 120591120591prime) = 1198661198661199531199530(120591120591 minus 120591120591prime)120575120575119953119953119953119953prime are non-zero due
to the orthogonality of normal modes Looking at the non-vanishing terms and setting 120591120591prime = 0
without loss of generality we can calculate the Fourier transform of 1198661198661199531199530(120591120591) from Eq (412)
1198661198661199531199530(119894119894120596120596119899119899) = int d1205911205911198901198901198941198941205961205961198991198991205911205911198661198661199531199530(120591120591)120573120573ℏ0 = minusint d120591120591119890119890119894119894120596120596119899119899120591120591120573120573ℏ
0 1 + 119899119899119861119861120596120596119953119953119890119890minus120596120596119953119953120591120591 = minus1+119899119899119861119861120596120596119953119953119894119894120596120596119899119899minus120596120596119953119953
119890119890119894119894120596120596119899119899minus120596120596119953119953120573120573ℏ minus
15
1 = minus1+ 1
119890119890120573120573ℏ120596120596119953119953minus1
119894119894120596120596119899119899minus120596120596119953119953119890119890119894119894
2120587120587119899119899120573120573ℏ 120573120573ℏ119890119890minus120573120573ℏ120596120596119953119953 minus 1 = minus 119890119890120573120573ℏ120596120596119953119953
119894119894120596120596119899119899minus120596120596119953119953
119890119890minus120573120573ℏ120596120596119953119953minus1
119890119890120573120573ℏ120596120596119953119953minus1= minus 1
119894119894120596120596119899119899minus120596120596119953119953
1minus119890119890120573120573ℏ120596120596119953119953
119890119890120573120573ℏ120596120596119953119953minus1= 1
119894119894120596120596119899119899minus120596120596119953119953
ie
1198661198661199531199530(119894119894120596120596119899119899) == 1119894119894120596120596119899119899minus120596120596119953119953
(421)
5 Fermirsquos golden rule 51 Interaction picture
Next we can proceed with calculating the Greenrsquos function of the coupled system To this end we
define the coupling Hamiltonian operator term in the interaction picture as
119867119867119862119862119868119868 (120591120591) = 1198901198901198671198670120591120591ℏ 119867119867119862119862119890119890
minus1198671198670120591120591ℏ (51)
The time evolution of 119867119867119862119862(120591120591) is then given by
ℏ part119867119867119862119862119868119868 (120591120591)120597120597120591120591
= 1198671198670119867119867119862119862119868119868 (120591120591) (52)
Note that the Greenrsquos function defined above was given in the Heisenberg picture To proceed we
need to transform it to the interaction picture
119866119866119953119953119953119953prime(120591120591 0) = minus lang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953primedagger (0)rang = minusTr 120588120588119867119867119879119879120591120591 119890119890
119867119867120591120591ℏ 119886119886119953119953(0)119890119890minus
119867119867120591120591ℏ 119886119886119953119953prime
dagger (0)
= minusTr 120588120588119867119867119879119879120591120591 119890119890119867119867120591120591ℏ 119890119890minus
1198671198670120591120591ℏ 119890119890
1198671198670120591120591ℏ 119886119886119953119953(0)119890119890minus
1198671198670120591120591ℏ 119890119890
1198671198670120591120591ℏ 119890119890minus
119867119867120591120591ℏ 119886119886119953119953prime
dagger (0)
= minusTr 120588120588119867119867119879119879120591120591 119880119880(0 120591120591)119886119886119953119953119868119868 (120591120591)119880119880(120591120591 0)119886119886119953119953primedagger (0)
where
119886119886119953119953119868119868 (120591120591) equiv 1198901198901198671198670120591120591ℏ 119886119886119953119953(0)119890119890minus
1198671198670120591120591ℏ (53)
is the operator in interaction picture and the operator 119880119880 is defined by
119880119880(1205911205911 1205911205912) equiv 11989011989011986711986701205911205911ℏ 119890119890minus
119867119867 (1205911205911minus1205911205912)ℏ 119890119890minus
11986711986701205911205912ℏ (54)
Note that while 119880119880 is not unitary it satisfies the following group property
119880119880(1205911205911 1205911205912)119880119880(1205911205912 1205911205913) = 119880119880(1205911205911 1205911205913) (55)
and the boundary condition 119880119880(1205911205911 1205911205911) = 1 In addition the 120591120591 derivative of 119880119880 is simply
ℏpart119880119880(120591120591 120591120591prime)
120597120597120591120591= 1198671198670119890119890
1198671198670120591120591ℏ 119890119890minus
119867119867120591120591minus120591120591primeℏ minus 119890119890
1198671198670120591120591ℏ 119867119867119890119890minus
119867119867120591120591minus120591120591primeℏ 119890119890minus
1198671198670120591120591primeℏ
= 1198901198901198671198670120591120591ℏ 1198671198670 minus 119867119867119890119890minus
1198671198670120591120591ℏ 119890119890
1198671198670120591120591ℏ 119890119890minus
119867119867120591120591minus120591120591primeℏ 119890119890minus
1198671198670120591120591primeℏ = 119890119890
1198671198670120591120591ℏ minus119867119867119862119862119890119890
minus1198671198670120591120591ℏ 119880119880(120591120591 120591120591prime)
= minus119867119867119862119862119868119868 (120591120591)119880119880(120591120591 120591120591prime)
16
ie
ℏ part119880119880120591120591120591120591prime120597120597120591120591
= minus119867119867119862119862119868119868 (120591120591)119880119880(120591120591 120591120591prime) (56)
The solution of Eq (56) is (see page 235 of Ref [2])
119880119880(120591120591 120591120591prime) = summinus1ℏ
119899119899
119899119899 int 1198891198891205911205911120591120591120591120591prime ⋯int 119889119889120591120591119899119899
120591120591120591120591prime 119879119879120591120591119867119867119862119862119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119899119899)infin
119899119899=0 (57)
The exact thermal Greens function now may be rewritten in the interaction picture as
119866119866119953119953119953119953prime(120591120591 0) = minus1119885119885119867119867
Tr 119890119890minus120573120573119867119867119879119879120591120591119880119880(0 120591120591)119886119886119953119953119868119868 (120591120591)119880119880(120591120591 0)119886119886119953119953primedagger (0)
= minusTr 119890119890minus120573120573119867119867119879119879120591120591 119880119880(0 120591120591)119886119886119953119953119868119868 (120591120591)119880119880(120591120591 0)119886119886119953119953prime
dagger (0)
Tr119890119890minus120573120573119867119867
= minusTr 119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)119879119879120591120591 119886119886119953119953119868119868 (120591120591)119880119880(0 120591120591)119880119880(120591120591 0)119886119886119953119953prime
dagger (0)
Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)
= minusTr 119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)119879119879120591120591 119886119886119953119953119868119868 (120591120591)119880119880(00)119886119886119953119953prime
dagger (0)
Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)=
= minusTr 119890119890minus1205731205731198671198670119879119879120591120591 119880119880(120573120573ℏ 0)119886119886119953119953119868119868 (120591120591)119886119886119953119953prime
dagger (0)
Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)
Where we used the fact that we are free to change the order of the operators within the time ordering
operation (see pages 241-242 of Ref [2])
52 Wickrsquos theorem
Thus the Greenrsquos function can be expanded as [2]
119866119866119953119953119953119953prime(120591120591 0) = minussum 1
119898119898minus1ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)119886119886119953119953119868119868 (120591120591)119886119886
119953119953primedagger (0)rang0infin
119898119898=0
sum 1119898119898minus
1ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)rang0infin
119898119898=0 (58)
where lang⋯ rang0 represents the ensemble average with respect to the non-interacting basis
Tr119890119890minus1205731205731198671198670(⋯ ) Or explicitly
119866119866119953119953119953119953prime(120591120591 0) = minus119866119866119953119953119953119953prime0 (120591120591)minus1ℏint 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953primedagger (0)rang0+
12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0+⋯
1minus1ℏint 1198891198891205911205911120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)rang0+12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0+⋯
(59)
For consiceness we introduce the following notation
119863119863119898119898 equiv 1119898119898minus 1
ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)rang0 (510)
To simplify the calculation in Eq (58) we adopt Wickrsquos theorem (see pages 237-241 of Ref
[2])which can be expressed as follows
lang119879119879120591120591119860119861119861119862119863119863 ⋯119884119884119885rang0 = lang119879119879120591120591119860119861119861rang0lang119879119879120591120591119862119863119863rang0⋯ lang119879119879120591120591119884119884119885rang0 + lang119879119879120591120591119860119862rang0lang119879119879120591120591119861119861119863119863rang0⋯ lang119879119879120591120591119883119883119885rang0 + ⋯ (511)
17
Here 119860119861119861 hellip 119884119884 119885 represent 119886119886119953119953(120591120591) or 119886119886119953119953dagger(120591120591) In Eq (511) each term corresponds to a particular
pairing of the operators 119860119861119861119862119863119863 ⋯119884119884119885 and all possible pairings are taken into account Here the only
non-vanishing propagators will have the form [see Eq (420)]
lang119879119879120591120591 119886119886119953119953primedagger (120591120591prime)119886119886119953119953(120591120591)rang0 = minuslang119879119879120591120591 119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime)rang0 = 119866119866119953119953119953119953prime0 (120591120591 120591120591prime) = 1198661198661199531199530(120591120591 minus 120591120591prime)120575120575119953119953119953119953prime (512)
Therefore the second term in the numerator of Eq (59) can be calculated as
1198681198682 = minus1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0
= minus1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 lang119879119879120591120591119886119886119953119953prime
dagger (0)119886119886119953119953(120591120591)rang0 minus1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
= 119866119866119953119953119953119953prime0 (120591120591 0) minus
1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 minus
1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
The first term in above equation is called disconnected part since the pairing is performed separately
on 119867119867119862119862(1205911205911) and 119886119886119953119953(120591120591)119886119886119953119953primedagger (0) All other terms have pairs that mix creation and annihilation operators
of the Hamiltonian with 119886119886119953119953(120591120591) or 119886119886119953119953primedagger (0) and are said to have connected party For simplicity we
include all these terms in the notation lang119879119879120591120591119867119867119862119862(1205911205911)119886119886119953119953(120591120591)119886119886119953119953primedagger (0)rang0119888119888 Furthermore we define 119866119866119953119953119953119953prime
(1) (120591120591) equiv
minus1ℏ int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862(1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 Then we have
1198681198683 = minus1ℏint 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 = 119866119866119953119953119953119953prime0 (120591120591 0)1198631198631 + 119866119866119953119953119953119953prime
(1) (120591120591) (513)
Using this method the third term in the numerator of Eq (59) can be calculated as
1198681198683 = 12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 = 12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0 lang119879119879120591120591119886119886119953119953prime
dagger (0)119886119886119953119953(120591120591)rang0 +
12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 + 12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 +
12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 = 119866119866119953119953119953119953prime0 (120591120591 0) 1
2ℏ2 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0 +
12ℏ2 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 + 12ℏ2 int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 +
12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
Here the last term is denoted as 119866119866119953119953119953119953prime(2) (120591120591)
Using Eq (513) the second and third terms on the right hand above equation can be simplified as
follows
18
12ℏ2
1198891198891205911205912120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 +1
2ℏ2 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
=12 minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0
minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
+12 minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0
minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
=12 minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0
minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
+12 minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0
minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
= minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0
minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 = 1198631198631119866119866119953119953119953119953prime(1) (120591120591)
Where we performed the following integration variables interchange 1205911205911 ⟷ 1205911205912 Thus we have
1198681198683 = 119866119866119953119953119953119953prime(2) (120591120591) + 119866119866119953119953119953119953prime
(1) (120591120591)1198631198631 + 119866119866119953119953119953119953prime0 (120591120591 0)1198631198632 (514)
Similarity we can calculate the fourth term of Eq (59) as
1198681198684 = minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 =
minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0 lang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 + 3 times
minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 + 3 times
minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 +
minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 =
119866119866119953119953119953119953prime0 (120591120591 0) minus1
6ℏ3 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0+
12ℏ2 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0
minus1ℏ int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888+
12ℏ2 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 minus1ℏ int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0+ 119866119866119953119953119953119953prime
(3) (120591120591) = 119866119866119953119953119953119953prime(3) (120591120591) +
119866119866119953119953119953119953prime(2) (120591120591)1198631198631 + 119866119866119953119953119953119953prime
(1) (120591120591)1198631198632 + 119866119866119953119953119953119953prime0 (120591120591 0)1198631198633
Higher order terms can be treated in the same manner (see pages 95-96 of Ref [2]) Then when
substituting 1198681198682 1198681198683 1198681198684 and all higher order terms into Eq (59) we can simplify the numerator as
follows
119866119866119953119953119953119953prime0 (120591120591) minus
1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0
+1
2ℏ2 1198891198891205911205911120573120573ℏ
0 1198891198891205911205912120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 + ⋯
= 119866119866119953119953119953119953prime0 (120591120591) + 119866119866119953119953119953119953prime
(1) (120591120591) + 119866119866119953119953119953119953prime(2) (120591120591) + ⋯ (1 + 1198631198631 + 1198631198632 + ⋯ )
Noting that the expression (1 + 1198631198631 + 1198631198632 + ⋯ ) is exactly canceled with the denominator the
Greenrsquos function can be simplified as
19
119866119866119953119953119953119953prime(120591120591) = 119866119866119953119953119953119953prime0 (120591120591) + 119866119866119953119953119953119953prime
(1) (120591120591) + 119866119866119953119953119953119953prime(2) (120591120591) + ⋯ (515)
where
119866119866119953119953119953119953prime
(1) (120591120591) = minus1ℏ int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888
119866119866119953119953119953119953prime(2) (120591120591) = 1
2ℏ2 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888 (516)
53 Calculation of Greenrsquos function
531 First order appriximation In Eq (516) we go back to the full notation 119886119886119954119954120582120582 to distinguish phonons of different branches
then 119866119866119953119953119953119953prime(1) (120591120591) is calculated as
119866119866119953119953119953119953prime(1) (120591120591) = minus
1ℏ
12 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591 119881119881119895119895119895119895prime(119948119948)119876119876119948119948119895119895(1205911205911)119876119876119948119948119895119895prime
dagger (1205911205911)3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ 119881119881119895119895119895119895primelowast (119948119948)119876119876119948119948119895119895prime(1205911205911)119876119876119948119948119895119895
dagger (1205911205911)3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= minus12ℏ
1198891198891205911205911120573120573ℏ
0lang119879119879120591120591
ℏ119881119881119895119895119895119895prime(119948119948)2120596120596119948119948119895119895120596120596119948119948119895119895prime
119886119886119948119948119895119895(1205911205911) + 119886119886minus119948119948119895119895dagger (1205911205911) 119886119886minus119948119948119895119895prime(1205911205911) + 119886119886119948119948119895119895prime
dagger (1205911205911)3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ ℏ119881119881119895119895119895119895prime
lowast (119948119948)
2120596120596119948119948119895119895120596120596119948119948119895119895prime119886119886119948119948119895119895prime(1205911205911) + 119886119886minus119948119948119895119895prime
dagger (1205911205911) 119886119886minus119948119948119895119895(1205911205911) + 119886119886119948119948119895119895dagger (1205911205911)
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
According to Wickrsquos theorem [2] the terms that contain 119886119886119948119948119895119895119886119886minus119948119948119895119895prime and 119886119886minus119948119948119895119895dagger 119886119886119948119948119895119895prime
dagger are equal to zero
thus the first term of above equation are calculated as
11989211989211 = minus14
119881119881119895119895119895119895prime(119948119948)
120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119886119886minus119948119948119895119895
dagger (1205911205911)119886119886minus119948119948119895119895prime(1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ lang119879119879120591120591119886119886119948119948119895119895(1205911205911)119886119886119948119948119895119895primedagger (1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888
= minus14
119881119881119895119895119895119895prime(119948119948)
120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119886119886minus119948119948119895119895
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus119948119948119895119895prime(1205911205911)rang0
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ lang119879119879120591120591119886119886119948119948119895119895primedagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)119886119886119948119948119895119895(1205911205911)rang0
= minus14
119881119881119895119895119895119895prime(119948119948)
120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
01198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954minus119948119948120575120575120582120582119895119895119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119954119954primeminus119948119948120575120575120582120582prime119895119895prime3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ 1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119948119948120575120575120582120582119895119895prime119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119954119954prime119948119948120575120575120582120582prime119895119895
20
Simplification of above equation gives
11989211989211 =
⎩⎪⎨
⎪⎧minus119881119881120582120582120582120582prime(minus119954119954)
4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954120582120582
0 (120591120591 1205911205911)119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903
2lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582(119954119954)
4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954120582120582
0 (120591120591 1205911205911)119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582prime le 3119903119903
2lt 120582120582 le 3119903119903
0 else
(517)
Similarity the second term of 119866119866119953119953119953119953prime(1) (120591120591) is calculated as
11989211989212 = minus14
minus119881119881119895119895119895119895primelowast (119948119948)
4120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119886119886119948119948119895119895prime(1205911205911)119886119886119948119948119895119895
dagger (1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ lang119879119879120591120591119886119886minus119948119948119895119895primedagger (1205911205911)119886119886minus119948119948119895119895(1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888
= minus14
minus119881119881119895119895119895119895primelowast (119948119948)
4120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)119886119886119948119948119895119895prime(1205911205911)rang0 lang119879119879120591120591119886119886119948119948119895119895dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus119948119948119895119895(1205911205911)rang0 lang119879119879120591120591119886119886minus119948119948119895119895prime
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
ie
11989211989212 =
⎩⎪⎨
⎪⎧
minus119881119881120582120582120582120582primelowast (119954119954)
4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954prime120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903
2lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582lowast (minus119954119954)
4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954prime120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119954119954prime 1 le 120582120582prime le 3119903119903
2lt 120582120582 le 3119903119903
0 else
(518)
Note that since 119881119881120582120582120582120582primelowast (119954119954) = 119881119881120582120582120582120582prime(minus119954119954) [Eq (314)] the first order approximation of Greenrsquos function
can be written as
119866119866119954119954120582120582119954119954prime120582120582prime(1) (120591120591) =
⎩⎪⎨
⎪⎧minus119881119881
120582120582120582120582prime(minus119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ
0 1198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582 le3119903119903
2lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582
(119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ
0 1198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582prime le3119903119903
2lt 120582120582 le 3119903119903
0 else
(519)
To derive the Greenrsquos function in frequency domain we use the expression for Fourier series then
we have
1198891198891205911205911120573120573ℏ
01198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0) = 1198891198891205911205911120573120573ℏ
0
1120573120573ℏ
119890119890minus119894119894120596120596119899119899(120591120591minus1205911205911)1198661198661199541199541205821205820 (119894119894120596120596119899119899)
119899119899
1120573120573ℏ
119890119890minus119894119894120596120596119899119899prime1205911205911119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899prime)
119899119899prime
=1
(120573120573ℏ)2119890119890minus119894119894120596120596119899119899120591120591 1198891198891205911205911120573120573ℏ
0119890119890119894119894120596120596119899119899minus120596120596119899119899prime1205911205911 119866119866119954119954120582120582
0 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899prime)
119899119899119899119899prime=
=1
(120573120573ℏ)2119890119890minus119894119894120596120596119899119899120591120591[120573120573ℏ120575120575119899119899119899119899prime]1198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899prime)119899119899119899119899prime
=1120573120573ℏ
119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899)119899119899
According to the definition of Fourier series [see Eq (412)] we get the Fourier coefficient of
21
119866119866119954119954120582120582119954119954prime120582120582prime(1) (120591120591)
119866119866119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) =
⎩⎪⎨
⎪⎧minus119881119881120582120582120582120582prime(minus119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582prime1198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime 1 le 120582120582 le 31199031199032
lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582(119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582prime1198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime 1 le 120582120582prime le 31199031199032
lt 120582120582 le 3119903119903
0 else
(520)
where 119866119866119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) = 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) times Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) times 119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899) and Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) is defined as
Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) =
⎩⎪⎨
⎪⎧minus119881119881120582120582120582120582prime(minus119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582prime120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903
2lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582(119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582prime120575120575119954119954119954119954prime 1 le 120582120582prime le 3119903119903
2lt 120582120582 le 3119903119903
0 else
(521)
532 second order approximation The second order approximation of the Greenrsquos function in Eq (516) 119866119866119954119954120582120582119954119954prime120582120582prime
(2) (120591120591) is calculated as
119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) =
18ℏ2 1198891198891205911205911
120573120573ℏ
0 1198891198891205911205912120573120573ℏ
0lang119879119879120591120591
⎣⎢⎢⎡ 11988111988111989511989511198951198951prime(119948119948120783120783)1198761198761199481199481207831207831198951198951(1205911205911)1198761198761199481199481207831207831198951198951prime
dagger (1205911205911)3119903119903
1198951198951prime=31199031199032 +1
31199031199032
1198951198951=1119948119948120783120783
+ 11988111988111989411989411198941198941primelowast (119948119948120783120783)1198761198761199481199481207831207831198941198941prime(1205911205911)1198761198761199481199481207831207831198941198941
dagger (1205911205911)3119903119903
1198941198941prime=31199031199032 +1
31199031199032
1198941198941=1119948119948120783120783 ⎦⎥⎥⎤
⎣⎢⎢⎡ 11988111988111989511989521198951198952prime(119948119948120784120784)1198761198761199481199481207841207841198951198952(1205911205912)1198761198761199481199481207841207841198951198952prime
dagger (1205911205912)3119903119903
1198951198952prime=31199031199032 +1
31199031199032
1198951198952=1119948119948120784120784
+ 11988111988111989411989421198941198942primelowast (119948119948120784120784)1198761198761199481199481207841207841198941198942prime(1205911205912)1198761198761199481199481207841207841198941198942
dagger (1205911205912)3119903119903
1198941198942prime=31199031199032 +1
31199031199032
1198941198942=1119948119948120784120784 ⎦⎥⎥⎤119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888 = 1198921198921 + 1198921198922 + 1198921198923 + 1198921198924
where
1198921198921 = 18ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0
sum sum 11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989511989521198951198952prime(119948119948120784120784)1le1198951198952le
31199031199032 lt1198951198952
primele311990311990311994811994812078412078411989511989521198951198952prime
1le1198951198951le31199031199032 lt1198951198951
primele311990311990311994811994812078312078311989511989511198951198951prime
times lang1198791198791205911205911198761198761199481199481207831207831198951198951(1205911205911)1198761198761199481199481207831207831198951198951primedagger (1205911205911)1198761198761199481199481207841207841198951198952(1205911205912)1198761198761199481199481207841207841198951198952prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(522)
1198921198922 = 18ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0
sum sum 11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942primelowast (119948119948120784120784)
1le1198941198942le31199031199032 lt1198941198942
primele311990311990311994811994812078412078411989411989421198941198942prime
1le1198951198951le31199031199032 lt1198951198951
primele311990311990311994811994812078312078311989511989511198951198951prime
times lang1198791198791205911205911198761198761199481199481207831207831198951198951(1205911205911)1198761198761199481199481207831207831198951198951primedagger (1205911205911)1198761198761199481199481207841207841198941198942prime (1205911205912)1198761198761199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(523)
1198921198923 = 18ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0
sum sum 11988111988111989411989411198941198941primelowast (119948119948120783120783)11988111988111989511989521198951198952prime(119948119948120784120784)
1le1198951198952le31199031199032 lt1198951198952
primele311990311990311994811994812078412078411989511989521198951198952prime
1le1198941198941le31199031199032 lt1198941198941
primele311990311990311994811994812078312078311989411989411198941198941prime
times lang1198791198791205911205911198761198761199481199481207831207831198941198941prime (1205911205911)1198761198761199481199481207831207831198941198941dagger (1205911205911)1198761198761199481199481207841207841198951198952(1205911205912)1198761198761199481199481207841207841198951198952prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(524)
22
1198921198924 = 18ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0
sum sum 11988111988111989411989411198941198941primelowast (119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)1le1198941198942le
31199031199032 lt1198941198942
primele311990311990311994811994812078412078411989411989421198941198942prime
1le1198941198941le31199031199032 lt1198941198941
primele311990311990311994811994812078312078311989411989411198941198941prime
times lang1198791198791205911205911198761198761199481199481207831207831198941198941prime (1205911205911)1198761198761199481199481207831207831198941198941dagger (1205911205911)1198761198761199481199481207841207841198941198942prime (1205911205912)1198761198761199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(525)
In what follows we will expand the term 1198921198922 The expansion of all other terms follows the same lines
and eventually results in the same expression Expanding the product 11987611987611994811994812078312078311989511989511198761198761199481199481207831207831198951198951primedagger 1198761198761199481199481207841207841198941198942prime1198761198761199481199481207841207841198941198942
dagger we can
rewrite Eq (523) as follows
1198921198922 = 132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)119881119881
11989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032 lt1198941198942
primele311990311990311994811994812078412078411989411989421198941198942prime
1le1198951198951le31199031199032 lt1198951198951
primele311990311990311994811994812078312078311989511989511198951198951prime
times 119868119868 (526)
Where
119868119868 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911) + 119886119886minus1199481199481207831207831198951198951dagger (1205911205911) 119886119886minus1199481199481207831207831198951198951prime(1205911205911) + 1198861198861199481199481207831207831198951198951prime
dagger (1205911205911) 1198861198861199481199481207841207841198941198942prime (1205911205912) + 119886119886minus1199481199481207841207841198941198942primedagger (1205911205912) 119886119886minus1199481199481207841207841198941198942(1205911205912) + 1198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0c
= lang119879119879120591120591 1198861198861199481199481207831207831198951198951119886119886minus1199481199481207831207831198951198951prime + 11988611988611994811994812078312078311989511989511198861198861199481199481207831207831198951198951primedagger + 119886119886minus1199481199481207831207831198951198951
dagger 119886119886minus1199481199481207831207831198951198951prime + 119886119886minus1199481199481207831207831198951198951dagger 1198861198861199481199481207831207831198951198951prime
dagger 12059112059111198861198861199481199481207841207841198941198942prime119886119886minus1199481199481207841207841198941198942 + 1198861198861199481199481207841207841198941198942prime1198861198861199481199481207841207841198941198942
dagger + 119886119886minus1199481199481207841207841198941198942primedagger 119886119886minus1199481199481207841207841198941198942
+ 119886119886minus1199481199481207841207841198941198942primedagger 1198861198861199481199481207841207841198941198942
dagger 1205911205912119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0c
= lang119879119879120591120591 11988611988611994811994812078312078311989511989511198861198861199481199481207831207831198951198951primedagger + 119886119886minus1199481199481207831207831198951198951
dagger 119886119886minus1199481199481207831207831198951198951prime12059112059111198861198861199481199481207841207841198941198942prime1198861198861199481199481207841207841198941198942
dagger + 119886119886minus1199481199481207841207841198941198942primedagger 119886119886minus11994811994812078412078411989411989421205911205912
119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
Where the symbol [hellip ]1205911205911 signifies that the operators in the brackets are given in the interaction
picture The last equality in above equation comes from the fact that the contractions of the product
of the operators are equal to zero when the number of creation and annihilation operators are not the
same (see Wickrsquos theorem in Ref [2]) Using Wickrsquos theorem [2] we are able to calculate the above
equation term by term as follows
⎩⎪⎨
⎪⎧ 1198681198681 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951prime
dagger (1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)1198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888
1198681198682 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
1198681198683 = lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)1198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
1198681198684 = lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(527)
We shall now calculate them term by term
23
1198681198681 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951primedagger (1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)1198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942
dagger (1205911205912)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
+ lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951prime
dagger (1205911205911)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0
+ lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)1198861198861199481199481207841207841198941198942prime(1205911205912)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0119888119888
= 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199481199481207841207841198951198952
0 (1205911205912 1205911205912)12057512057511989411989421198941198942prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582
+ 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 1198661198661199481199481207831207831198951198951
0 (1205911205911 1205911205911)12057512057511989511989511198951198951prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582
+ 1198661198661199481199481207831207831198951198951prime0 (1205911205912 1205911205911)1205751205751199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951prime ∙ 119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582
= 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582
+ 1198661198661199481199481207831207831198951198951prime0 (1205911205912 1205911205911)1205751205751199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951prime ∙ 119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582
The last equality results from the fact that 1198951198951 and 1198951198951prime are indices of different ranges (belonging to
different subsystems) and the same holds for 1198941198942 and 1198941198942prime Similarity
1198681198684 = lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0
= 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 119866119866minus11994811994812078412078411989411989420 (1205911205912 1205911205912)12057512057511989411989421198941198942prime
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 119866119866minus1199481199481207831207831198951198951prime 0 (1205911205911 1205911205912)1205751205751199481199481207831207831199481199481207841207841205751205751198951198951prime1198941198942prime
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205911 1205911205911)12057512057511989511989511198951198951prime
+ 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205912 1205911205911)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942
= 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 119866119866minus1199481199481207831207831198951198951prime 0 (1205911205911 1205911205912)1205751205751199481199481207831207831199481199481207841207841205751205751198951198951prime1198941198942prime
+ 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205912 1205911205911)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942
Where again the last equality results from the fact that 1198951198951 and 1198951198951prime are indices of different ranges
(belonging to different subsystems) and the same holds for 1198941198942 and 1198941198942prime Similarly we obtain 1198681198682 =
1198681198683 = 0 for the same reason as shown below
24
1198681198682 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951prime
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0
+ lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)1198861198861199481199481207831207831198951198951(1205911205911)rang0
+ lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0
= 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582 ∙ 119866119866minus11994811994812078412078411989411989420 (1205911205912 1205911205912)12057512057511989411989421198941198942prime
+ 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)120575120575minus11994811994812078412078411994811994812078312078312057512057511989511989511198941198942prime ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205911)12057512057511989511989511198951198951prime
+ 1198661198661199481199481207831207831198951198951prime0 (1205911205912 1205911205911)120575120575minus11994811994812078412078411994811994812078312078312057512057511989411989421198951198951prime ∙ 119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 = 0
1198681198683 = lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)1198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0
+ lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0
+ lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0
= 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 11986611986611994811994812078412078411989411989420 (1205911205912 1205911205912)12057512057511989411989421198941198942prime
+ 11986611986611994811994812078412078411989411989420 (1205911205911 1205911205912)120575120575minus1199481199481207831207831199481199481207841207841205751205751198951198951prime1198941198942 ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205911 1205911205911)12057512057511989511989511198951198951prime
+ 1198661198661199481199481207841207841198941198942prime0 (1205911205912 1205911205911)120575120575minus1199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951 ∙ 119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 = 0
The two non-vanishing terms (1198681198681 and 1198681198684) produce the following contributions to 1198921198922 of Eq (526)
11989211989221 = 132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime1198681198681 =
132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime1198661198661199481199481207831207831198951198951
0 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙
1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582 + 1
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime1198661198661199481199481207831207831198951198951prime
0 (1205911205912 1205911205911)1205751205751199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951prime ∙
119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 =
⎩⎪⎨
⎪⎧
120575120575119954119954119954119954prime
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast (119954119954prime)
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (1205911205911 1205911205912) ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
120575120575119954119954119954119954prime
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032+1 1198661198661199541199541198951198951prime
0 (1205911205912 1205911205911) ∙ 119866119866119954119954prime120582120582prime0 (1205911205911 0)119866119866119954119954120582120582
0 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032
0 else
and
25
11989211989224 = 132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime1198681198684 =
132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙
119866119866minus1199481199481207831207831198951198951prime 0 (1205911205911 1205911205912)1205751205751199481199481207831207831199481199481207841207841205751205751198951198951prime1198941198942prime + 1
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙
1198661198661199541199541205821205820 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 119866119866minus1199481199481207831207831198951198951
0 (1205911205912 1205911205911)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 =
⎩⎪⎨
⎪⎧120575120575119954119954119954119954prime
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032+1 1198661198661199541199541198951198951prime
0 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime0 (1205911205912 0)119866119866119954119954120582120582
0 (120591120591 1205911205911) 120582120582 120582120582prime isin 1 31199031199032
120575120575119954119954119954119954prime
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast (119954119954prime)
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
0 else
Here we used the following properties 120596120596minus119954119954120582120582 = 120596120596119954119954120582120582 and 1198811198811205821205821198951198951prime(minus119954119954) = 1198811198811205821205821198951198951primelowast (119954119954) [see Eqs (112) and
(314)]
The equivalence of 11989211989221 and 11989211989224 can be seen by changing the integration variables 1205911205912 harr 1205911205911
Substituting 11989211989221 and 11989211989224 into Eq (526) 1198921198922 is given by
1198921198922 =
⎩⎪⎨
⎪⎧
120575120575119954119954119954119954prime
16 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast (119954119954prime)
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime
0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
120575120575119954119954119954119954prime
16 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032+1 1198661198661199541199541198951198951prime
0 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime0 (1205911205911 0)119866119866119954119954120582120582
0 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032
0 else
(528)
By changing the indices and integration variables itrsquos straightforward to show that the other three
terms (119892119892111989211989231198921198924 ) are identical to 1198921198922 thus the final expression of 119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) [Eq (516)) can be
calculated as
119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) = 41198921198922 =
⎩⎪⎨
⎪⎧120575120575119954119954119954119954prime
4 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032
120575120575119954119954119954119954prime
4 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast 119954119954prime
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime
0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 31199031199032
+ 13119903119903
0 else
(529)
To calculate 119866119866119954119954120582120582119954119954prime120582120582prime(2) in frequency space we expand 119866119866119954119954120582120582119954119954prime120582120582prime
(2) (120591120591) in a Fourier series
119866119866119954119954120582120582119954119954prime120582120582prime
(2) (120591120591) = 1120573120573ℏsum 119890119890minus119894119894120596120596119899119899120591120591119866119866119954119954120582120582119954119954prime120582120582prime
(2) (119894119894120596120596119899119899)119899119899
119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = int d120591120591119890119890119894119894120596120596119899119899120591120591119866119866119954119954120582120582119954119954prime120582120582prime
(2) (120591120591)120573120573ℏ0
(530)
Using Eq (421) we get for 120582120582 120582120582prime isin 1 31199031199032
26
119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) =
120575120575119954119954119954119954prime4
1198891198891205911205911120573120573ℏ
0 1198891198891205911205912120573120573ℏ
0
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1198661198661199541199541198951198951prime0 (1205911205912 1205911205911)119866119866119954119954120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912)
=120575120575119954119954119954119954prime
4
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1198891198891205911205911120573120573ℏ
0 1198891198891205911205912120573120573ℏ
0
1120573120573ℏ
119890119890minus1198941198941205961205961198991198992(1205911205912minus1205911205911)1198661198661199541199541198951198951prime0 1198941198941205961205961198991198992
1198991198992
∙1120573120573ℏ
119890119890minus11989411989412059612059611989911989911205911205911119866119866119954119954120582120582prime0 1198941198941205961205961198991198991
1198991198991
∙1120573120573ℏ
119890119890minus119894119894120596120596119899119899(120591120591minus1205911205912)1198661198661199541199541205821205820 (119894119894120596120596119899119899)
119899119899
=120575120575119954119954119954119954prime
4
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1120573120573ℏ
119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954120582120582prime
0 11989411989412059612059611989911989911198661198661199541199541198951198951prime0 1198941198941205961205961198991198992
11989911989911989911989911198991198992⎩⎪⎨
⎪⎧ 1120573120573ℏ
1198891198891205911205911120573120573ℏ
0119890119890minus1198941198941205961205961198991198991minus12059612059611989911989921205911205911
times1120573120573ℏ
1198891198891205911205912120573120573ℏ
0119890119890minus1198941198941205961205961198991198992minus1205961205961198991198991205911205912
⎭⎪⎬
⎪⎫
=120575120575119954119954119954119954prime
4
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1120573120573ℏ
119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)
11989911989911989911989911198991198992
119866119866119954119954120582120582prime0 11989411989412059612059611989911989911198661198661199541199541198951198951prime
0 1198941198941205961205961198991198992120575120575119899119899111989911989921205751205751198991198991198991198992
=120575120575119954119954119954119954prime120573120573ℏ
119890119890minus119894119894120596120596119899119899120591120591
119899119899
1198661198661199541199541205821205820 (119894119894120596120596119899119899)
14
119881119881120582120582prime1198951198951prime(119954119954
prime)1198811198811205821205821198951198951primelowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899)119866119866119954119954120582120582prime
0 (119894119894120596120596119899119899)
Comparing above expression with Eq (530) the Fourier transform of 119866119866119954119954120582120582119954119954120582120582prime(2) (120591120591) reads as
119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) 120575120575119954119954119954119954prime
4sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 119866119866119954119954120582120582prime
0 (119894119894120596120596119899119899) (531)
where we have introduced the notation
Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) equiv
120575120575119954119954119954119954prime
4sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) (532)
and 119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954120582120582prime(2) (119894119894120596120596119899119899) ∙ 119866119866119954119954120582120582prime
0 (119894119894120596120596119899119899) Similarly for 120582120582 120582120582prime isin 31199031199032
+ 13119903119903 we
have
Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) =
120575120575119954119954119954119954prime
4sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast 119954119954prime
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (119894119894120596120596119899119899) (533)
and all together we have for Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899)
Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) =
⎩⎪⎪⎨
⎪⎪⎧120575120575119954119954119954119954prime
4sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 1 3119903119903
2
120575120575119954119954119954119954prime
4sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast 119954119954prime
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
0 else
(534)
27
533 Dysonrsquos equation Substituting Eqs (521) and (534) into Eq (516) and performing a Fourier transform we have
119866119866119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) = 119866119866119954119954120582120582119954119954prime120582120582prime0 (119894119894120596120596119899119899) + 119866119866119954119954120582120582119954119954prime120582120582prime
(1) (119894119894120596120596119899119899) + 119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) + ⋯
= 1198661198661199541199541205821205820 (119894119894120596120596119899119899)120575120575119954119954119954119954prime120575120575120582120582120582120582prime + 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) ∙ 119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899) + 1198661198661199541199541205821205820 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime
(2) (119894119894120596120596119899119899)
∙ 119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899) + ⋯ = 119866119866119954119954120582120582
0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime120575120575120582120582120582120582prime + 1198661198661199541199541205821205820 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) ∙ 119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899)
or
119866119866119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) = 1198661198661199541199541205821205820 (119894119894120596120596119899119899)120575120575119954119954119954119954prime120575120575120582120582120582120582prime + 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) ∙ 119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899) (535)
Eq (535) is the so-called Dysonrsquos equation (see Ref [2]) where
Σ119954119954120582120582119954119954120582120582prime(119894119894120596120596119899119899) = Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) + Σ119954119954120582120582119954119954prime120582120582prime
(2) (119894119894120596120596119899119899) + ⋯
is called self-energy and Σ119954119954120582120582119954119954120582120582prime(119899119899) (119894119894120596120596119899119899)119899119899 = 12⋯ is the self-energy of nth order approximation Up
to the second order approximation the self-energy is written as
Σ119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) =
⎩⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎧ minus
120575120575119954119954119954119954prime
2
119881119881120582120582120582120582primelowast (119954119954)
120596120596119954119954120582120582120596120596119954119954120582120582prime 1 le 120582120582 le 3119903119903
2lt 120582120582prime le 3119903119903
minus120575120575119954119954119954119954prime
2
119881119881120582120582prime120582120582(119954119954)
120596120596119954119954120582120582120596120596119954119954120582120582prime 1 le 120582120582prime le 3119903119903
2lt 120582120582 le 3119903119903
120575120575119954119954119954119954prime
4sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 1 3119903119903
2
120575120575119954119954119954119954prime
4sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast 119954119954prime
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
(536)
54 Fermirsquos golden Rule
To get the Fermirsquos golden Rule we need to calculate the retarded Greenrsquos function which can be
calculated from 119866119866119954119954120582120582119954119954120582120582prime(119894119894120596120596119899119899) by analytic continuation to the real axis via 119894119894120596120596119899119899 rarr 120596120596 + 119894119894119894119894 with an
infinitesimal positive 119894119894 [12]
119866119866119954119954120582120582119954119954120582120582prime119903119903 (120596120596) equiv 119866119866119954119954120582120582119954119954120582120582prime(119894119894120596120596119899119899 rarr 120596120596 + 119894119894119894119894) 119894119894 rarr 0 (537)
Similarity the retarded self-energy is defined as
Σ119954119954120582120582119954119954120582120582prime119903119903 (120596120596) equiv Σ119954119954120582120582119954119954120582120582prime(119894119894120596120596119899119899 rarr 120596120596 + 119894119894119894119894) 119894119894 rarr 0 (538)
The transition rate of phonons of branch 120582120582 at 119954119954 Γ119954119954120582120582(120596120596) is related to the retarded self energy
Σ119954119954120582120582119903119903 (120596120596) via [13]
Γ119954119954120582120582(120596120596) = minus2ImΣ119954119954120582120582119954119954120582120582119903119903 (120596120596) (539)
Using the expression for 1198661198661199541199541205821205820 (119894119894120596120596119899119899) [Eq (421)] and focused on the second order approximation [13]
we have
28
Σ119954119954120582120582119954119954120582120582119903119903 (119894119894120596120596119899119899) =
⎩⎪⎨
⎪⎧14sum
1198811198811205821205821198951198951prime(119954119954)
2
1205961205961199541199541198951198951prime120596120596119954119954120582120582
31199031199031198951198951prime=
31199031199032 +1
1
120596120596minus1205961205961199541199541198951198951prime +119894119894119894119894
120582120582 isin 1 31199031199032
14sum 1198811198811198951198951120582120582(119954119954)2
1205961205961199541199541198951198951120596120596119954119954120582120582
311990311990321198951198951=1
1120596120596minus1205961205961199541199541198951198951+119894119894119894119894
120582120582 isin 31199031199032
+ 13119903119903
(540)
Using the relation
Im 1
120596120596minus1205961205961199541199541198951198951prime +119894119894119894119894
= minus120587120587120575120575120596120596 minus 1205961205961199541199541198951198951prime (541)
The transition rate reads as
Γ119954119954120582120582(120596120596) =
⎩⎪⎨
⎪⎧1205871205872sum
1198811198811205821205821198951198951prime(119954119954)
2
1205961205961199541199541198951198951prime120596120596119954119954120582120582
31199031199031198951198951prime=
31199031199032 +1
120575120575120596120596 minus 1205961205961199541199541198951198951prime 120582120582 isin 1 31199031199032
1205871205872sum 1198811198811198951198951120582120582(119954119954)2
1205961205961199541199541198951198951120596120596119954119954120582120582
311990311990321198951198951=1
120575120575120596120596 minus 1205961205961199541199541198951198951 120582120582 isin 31199031199032
+ 13119903119903
(542)
Or equivalently
Γ119954119954120582120582(119864119864 = ℏ120596120596) =
⎩⎪⎨
⎪⎧120587120587ℏ3
2sum
1198811198811205821205821198951198951prime(119954119954)
2
1198641198641199541199541198951198951prime119864119864119954119954120582120582
31199031199031198951198951prime=
31199031199032 +1
120575120575119864119864 minus 1198641198641199541199541198951198951prime 120582120582 isin 1 31199031199032
120587120587ℏ3
2sum 1198811198811198951198951120582120582(119954119954)2
1198641198641199541199541198951198951119864119864119954119954120582120582
311990311990321198951198951=1
120575120575119864119864 minus 1198641198641199541199541198951198951 120582120582 isin 31199031199032
+ 13119903119903
(543)
Γ119954119954120582120582(119864119864) represents the probability per unit time of a transition with energy 119864119864 from phonon branch 120582120582
at wave number 119954119954 in subsystem I (II) to a set of phonon branches in subsystem II (I) with the same
wave number Here state 119954119954120582120582 in one subsystem is coupled to state 1199541199541198951198951prime in the other subsystem via
1198811198811205821205821198951198951prime(119954119954) Since the two expressions in Eq (543) are completely equivalent just representing
transitions of opposite directions we consider only the first one in what follows Assuming that
subsystem II sufficiently large such that its density of states is nearly continuous the sum over its
states in the above expression can be replaced by an integral over the energy sum (⋯ )119895119895 rarr
int(⋯ )120588120588119864119864119954119954d119864119864119954119954 giving
Γ119954119954120582120582(119864119864) = 120587120587ℏ3
2 intd119864119864119954119954prime120588120588119864119864119954119954prime 119881119881120582120582119895119895prime(119954119954)
2119864119864119954119954119895119895prime=119864119864119954119954prime
119864119864119954119954prime119864119864119954119954120582120582120575120575119864119864 minus 119864119864119954119954prime = 120587120587ℏ3
2120588120588(119864119864)
119881119881120582120582119895119895prime(119954119954)2119864119864119954119954119895119895prime=119864119864
119864119864119864119864119954119954120582120582 (544)
Eq (544) represents the transition rate of the phonons with energy 119864119864 from branch 120582120582 and wave
number 119954119954 in subsystem I to the continuous manifold of states in subsystem II The total transition
rate between the two subsystems is then given by the sum of transition rates from all states in
subsystem I weighted by their phonon populations to the manifold of states in subsystem II
Assuming that the two subsystems are weakly coupled such that the width of state 119954119954120582120582 in subsystem
I is small and the probability to leave it at an energy other than 119864119864119954119954120582120582 is negligible we can now write
29
total transition rate as
Γtot = 1119885119885119890119890minus120573120573119864119864119954119954120582120582Γ119954119954120582120582119864119864119954119954120582120582
119954119954120582120582
=120587120587ℏ3
2
119890119890minus120573120573119864119864119954119954120582120582
119885119885
120588120588119864119864119954119954120582120582 119881119881120582120582119895119895prime(119954119954)2119864119864119954119954119895119895prime=119864119864119954119954120582120582
1198641198641199541199541205821205822119954119954120582120582
=120587120587ℏ3
2
119890119890minus120573120573119864119864119954119954120582120582
119885119885
120588120588119864119864119954119954120582120582 119881119881120582120582120582120582+31199031199032(119954119954)
2
1198641198641199541199541205821205822119954119954120582120582
Finally we get
Γtot = Γ119954119954120582120582(119864119864) =120587120587ℏ3
2sum 119890119890minus120573120573119864119864119954119954120582120582
119885119885
120588120588119864119864119954119954120582120582119881119881120582120582120582120582+31199031199032(119954119954)
2
1198641198641199541199541205821205822119954119954120582120582 (545)
where 119885119885 = sum 119890119890minus120573120573119864119864119954119954120582120582119954119954120582120582 is the partition function Here we choose the index of phonon branches 120582120582
such that the phonon branch with lower energy has a smaller index ie 1198641198641199541199541205821205821 lt 1198641198641199541199541205821205822 for index 1205821205821 lt
1205821205822 Note that phonon branches 120582120582 and 1198951198951prime belong to the subsystem I (with index from 1 to 31199031199032) and
subsystem II (with index from 1 to 1 + 31199031199032 to 3119903119903) respectively we choose 119895119895prime = 120582120582 + 31199031199032
to make
sure that 119864119864119954119954119895119895prime = 119864119864119954119954120582120582 since phonon energy of two uncoupled system is identical Eq (545) is the
Fermirsquos golden rule for inter-phonon coupling
References
[1] Y Xu J-S Wang W Duan B-L Gu and B Li Nonequilibrium Greens function method for phonon-phonon interactions and ballistic-diffusive thermal transport Phys Rev B 78 224303 (2008) [2] A L Fetter and J D Walecka Quantum theory of many-particle systems (Courier Corporation 2012) [3] G E Stedman Fermis Golden RulemdashAn Exercise in Quantum Field Theory Am J Phys 39 205 (1971)
7
thickness dependence
Comparing our calculated heat conductivities for the aligned contact (open red circles) to available
experimental data for ~35 nm thick graphite slabs [34] (dashed green line) we find that at the thickest
model system considered of 104 graphene layers (~34 nm thick) the calculated value of 085plusmn005
W m sdot Kfrasl is in remarkable agreement with the measured value of ~07 W m sdot Kfrasl Furthermore
experimental values for bulk graphite [5] indicate that the thermal conductivity continues to grow up
to ~68 W m sdot Kfrasl (black dash-dotted line) which is consistent with the general trend of the calculated
heat conductivity that does not saturate for the thickest model system considered These results
strongly enforce the validity of our force-field and model systems to model the heat conductivity of
twisted layered materials interfaces Available experimental results for the heat conductivity of bulk
h-BN are marked by the dashed-dotted black and dashed-green lines in Figure 4(b) In line with our
findings for the graphitic interface our calculated finite slab heat conductivities for the aligned
interface (open red circles) continue to grow with the number of layers and are consistently below the
bulk value
Figure 4 Thickness dependence of the thermal conductivity 120581120581CP of aligned (open red circles) and
twisted by 3016deg (open black diamond symbols) graphite (a) and h-BN (b) stacks Blue squares
represent results obtained using the isotropic Lennard-Jones potential for the aligned contacts The
green dashed and black dash-dotted lines represent experimental results measured for graphite (a)
(Refs [534]) and bulk h-BN (b) (Refs [1415]) Note that both axis scales are logarithmic Error bar
estimation procedure is discussed in Sec 1 of the SI
Another important factor that may affect the interlayer thermal transport properties of 2D material
stacks is the average temperature of the system which was taken to be ~300K in all abovementioned
simulations To evaluate the sensitivity of our results towards this parameter we repeated the heat
conductivity and interfacial resistance calculations of optimally stacked graphite and h-BN stacks for
an average temperature of 400 K The results presented in see Sec 4 of the SI indicate that 120581120581CP and
8
ITR weakly depend on the average temperature over the entire thickness range considered This
conclusion is consistent with recent experimental findings [11] We may attribute this to the fact that
the thickness of the graphite and h-BN stack model considered is much smaller than their phonon
mean-free path (~200 nm for graphite [1011133536] and ~100 nm for bulk h-BN [15]) such that
the phonon transport is dominated by phonon-boundary scattering which weakly depends on
temperature in the range considered Therefore temperature dependent Umklapp processes have only
marginal contribution to our results [11]
Finally we note that previous calculations of the heat conductivity of twisted graphitic interfaces
relied on Lennard-Jones (LJ) potentials describing the interlayer interactions [8-1013] To
demonstrate the importance of using registry-dependent interlayer potentials we have repeated our
calculations of the heat conductivity of graphitic slabs with the REBO intralayer potential augmented
by LJ interlayer interactions [37] (120576120576 = 284 meV120590120590 = 34 Å ) We find that the calculated heat
conductivities obtained using the LJ interlayer potential are consistently higher than those obtained
by our ILP and that the difference between them grows with the model system thickness Notably the
heat conductivity obtained using the LJ potential for a graphitic slab of thickness ~34 nm is 154
W m sdot Kfrasl overestimating the experimental value by more than a factor of 2
The excellent agreement of our ILP calculations with experimental data of nanoscale graphitic stacks
therefore demonstrates the reliability of our predictions for the strong interfacial misfit angle
dependence of cross-layer thermal conductivity in graphite and h-BN The observed sharp
conductivity decrease of twisted graphitic interfaces at misfit angles lt 5deg opens the way to control
the thermal evacuation rate and thermal isolation of active layers in graphene-based electronic and
mechanical devices The revealed underlying mechanism suggests that design rules can be obtained
by carefully tailoring the phonon-phonon couplings across the twisted interface While the misfit
angle dependence of h-BN is found to be weaker than that of graphite the overall thermal
conductivity of the former is found to be higher This may be utilized to achieve higher conductivity
and controllability in twisted heterogeneous junctions of layered materials
ASSOCIATED CONTENT
Supporting Information
The supporting Information section includes a description of the methodology thermal conductivity
of twisted bilayer graphene theory for calculating the phonon coupling of twisted bilayer graphene
9
derivation of Fermirsquos golden rule and temperature dependence of the cross-plane thermal
conductivity
AUTHOR INFORMATION
Corresponding Author
E-mail urbakhtauextauacil
Notes
The authors declare no competing financial interest
ACKNOWLEDGMENTS
The authors would like to thank Prof Abraham Nitzan Dr Guy Cohen and Dr Yiming Pan for
helpful discussions WO acknowledges the financial support from a fellowship program for
outstanding postdoctoral researchers from China and India in Israeli Universities and the support from
the National Natural Science Foundation of China (Nos 11890673 and 11890674) HQ
acknowledges the financial support from the National Natural Science Foundation of China (No
11890674) MU acknowledges the financial support of the Israel Science Foundation Grant
No 114118 and the ISF-NSFC joint grant 319119 OH is grateful for the generous financial
support of the Israel Science Foundation under grant no 158617 and the Naomi Foundation for
generous financial support via the 2017 Kadar Award This work is supported in part by COST Action
MP1303
10
References
[1] A A Balandin Thermal properties of graphene and nanostructured carbon materials Nat Mater 10 569 (2011) [2] S Ghosh I Calizo D Teweldebrhan E P Pokatilov D L Nika A A Balandin W Bao F Miao and C N Lau Extremely high thermal conductivity of graphene Prospects for thermal management applications in nanoelectronic circuits Appl Phys Lett 92 151911 (2008) [3] A A Balandin S Ghosh W Bao I Calizo D Teweldebrhan F Miao and C N Lau Superior thermal conductivity of single-layer graphene Nano Lett 8 902 (2008) [4] J H Seol et al Two-Dimensional Phonon Transport in Supported Graphene Science 328 213 (2010) [5] R Taylor The thermal conductivity of pyrolytic graphite J Philosophical Magazine 13 157 (1966) [6] S Ghosh W Bao D L Nika S Subrina E P Pokatilov C N Lau and A A Balandin Dimensional crossover of thermal transport in few-layer graphene Nat Mater 9 555 (2010) [7] Z Wang R Xie C T Bui D Liu X Ni B Li and J T L Thong Thermal Transport in Suspended and Supported Few-Layer Graphene Nano Lett 11 113 (2011) [8] Z Wei Z Ni K Bi M Chen and Y Chen Interfacial thermal resistance in multilayer graphene structures Phys Lett A 375 1195 (2011) [9] Y Ni Y Chalopin and S Volz Significant thickness dependence of the thermal resistance between few-layer graphenes Appl Phys Lett 103 061906 (2013) [10] J Chen J H Walther and P Koumoutsakos Strain Engineering of Kapitza Resistance in Few-Layer Graphene Nano Lett 14 819 (2014) [11] Q Fu J Yang Y Chen D Li and D Xu Experimental evidence of very long intrinsic phonon mean free path along the c-axis of graphite Appl Phys Lett 106 031905 (2015) [12] G Fugallo A Cepellotti L Paulatto M Lazzeri N Marzari and F Mauri Thermal Conductivity of Graphene and Graphite Collective Excitations and Mean Free Paths Nano Lett 14 6109 (2014) [13] Z Wei J Yang W Chen K Bi D Li and Y Chen Phonon mean free path of graphite along the c-axis Appl Phys Lett 104 081903 (2014) [14] E K Sichel R E Miller M S Abrahams and C J Buiocchi Heat capacity and thermal conductivity of hexagonal pyrolytic boron nitride Phys Rev B 13 4607 (1976) [15] P Jiang X Qian R Yang and L Lindsay Anisotropic thermal transport in bulk hexagonal boron nitride Phys Rev Mater 2 064005 (2018) [16] M-H Wang Y-E Xie and Y-P Chen Thermal transport in twisted few-layer graphene Chinese Physics B 26 116503 (2017) [17] X Nie L Zhao S Deng Y Zhang and Z Du How interlayer twist angles affect in-plane and cross-plane thermal conduction of multilayer graphene A non-equilibrium molecular dynamics study Int J Heat Mass Tran 137 161 (2019) [18] A N Kolmogorov and V H Crespi Registry-dependent interlayer potential for graphitic systems Phys Rev B 71 235415 (2005) [19] M Reguzzoni A Fasolino E Molinari and M C Righi Potential energy surface for graphene on graphene Ab initio derivation analytical description and microscopic interpretation Phys Rev B 86 (2012) [20] M M van Wijk A Schuring M I Katsnelson and A Fasolino Moire Patterns as a Probe of Interplanar Interactions for Graphene on h-BN Phys Rev Lett 113 135504 (2014) [21] D Mandelli W Ouyang M Urbakh and O Hod The Princess and the Nanoscale Pea Long-Range Penetration of Surface Distortions into Layered Materials Stacks ACS Nano 13 7603 (2019) [22] E Koren E Loumlrtscher C Rawlings A W Knoll and U Duerig Adhesion and friction in mesoscopic graphite contacts Science 348 679 (2015) [23] E Koren I Leven E Loumlrtscher A Knoll O Hod and U Duerig Coherent commensurate electronic states at the interface between misoriented graphene layers Nat Nanotechnol 11 752 (2016)
11
[24] W Ouyang I Azuri D Mandelli A Tkatchenko L Kronik M Urbakh and O Hod Mechanical and Tribological Properties of Layered Materials under High Pressure Assessing the Importance of Many-Body Dispersion Effects J Chem Theory Comput 16 666 (2020) [25] W Ouyang D Mandelli M Urbakh and O Hod Nanoserpents Graphene Nanoribbon Motion on Two-Dimensional Hexagonal Materials Nano Lett 18 6009 (2018) [26] I Leven T Maaravi I Azuri L Kronik and O Hod Interlayer Potential for Grapheneh-BN Heterostructures J Chem Theory Comput 12 2896 (2016) [27] T Maaravi I Leven I Azuri L Kronik and O Hod Interlayer Potential for Homogeneous Graphene and Hexagonal Boron Nitride Systems Reparametrization for Many-Body Dispersion Effects J Phys Chem C 121 22826 (2017) [28] L H Liang and B Li Size-dependent thermal conductivity of nanoscale semiconducting systems Phys Rev B 73 153303 (2006) [29] P K Schelling S R Phillpot and P Keblinski Comparison of atomic-level simulation methods for computing thermal conductivity Phys Rev B 65 144306 (2002) [30] J Shiomi Nonequilirium molecular dynamics methods for lattice heat conduction calculations Annual Review of Heat Transfer 17 (2014) [31] G E Stedman Fermis Golden RulemdashAn Exercise in Quantum Field Theory Am J Phys 39 205 (1971) [32] L T Kong G Bartels C Campantildeaacute C Denniston and M H Muumlser Implementation of Greens function molecular dynamics An extension to LAMMPS Comput Phys Commun 180 1004 (2009) [33] S Plimpton Fast Parallel Algorithms for Short-Range Molecular Dynamics J Comput Phys 117 1 (1995) [34] M Harb et al The c-axis thermal conductivity of graphite film of nanometer thickness measured by time resolved X-ray diffraction Appl Phys Lett 101 233108 (2012) [35] H Zhang X Chen Y-D Jho and A J Minnich Temperature-Dependent Mean Free Path Spectra of Thermal Phonons Along the c-Axis of Graphite Nano Lett 16 1643 (2016) [36] D P Sellan E S Landry J E Turney A J H McGaughey and C H Amon Size effects in molecular dynamics thermal conductivity predictions Phys Rev B 81 214305 (2010) [37] S J Stuart A B Tutein and J A Harrison A reactive potential for hydrocarbons with intermolecular interactions J Chem Phys 112 6472 (2000)
S1
Supporting information for ldquoControllable Thermal Conductivity of
Twisted Graphene and Hexagonal Boron Nitride Interfacesrdquo
Wengen Ouyang1 Huasong Qin2 Michael Urbakh1 and Oded Hod1
1Department of Physical Chemistry School of Chemistry The Raymond and Beverly
Sackler Faculty of Exact Sciences and The Sackler Center for Computational
Molecular and Materials Science Tel Aviv University Tel Aviv 6997801 Israel
2State Key Laboratory for Strength and Vibration of Mechanical Structures School of
Aerospace Xirsquoan Jiaotong University Xirsquoan 710049 China
The Supporting information includes following sections
1 Methodology
2 Thermal conductivity of twisted bilayer graphene
3 Theory for calculating the phonon coupling of twisted bilayer graphene
4 Derivation of Fermirsquos golden rule
5 Temperature dependence of cross-plane thermal conductivity
S2
1 Methodology
11 Model system
The initial interlayer distance across the layered stack was set equal to 34 Aring and 33 Aring for graphite
and bulk h-BN respectively Periodic boundary conditions were applied in all directions It should be
noted that the lattice structure is rigorously periodic only at some specific twist angles the values of
which are listed in Table S1 in section 3 below While the cross-sectional area for each misfit angle
θ is different all systems considered have a contact area exceeding 12 nm2 which was shown to
provide converged results with respect to unit-cell dimensions [1] The intralayer interactions within
each graphene and h-BN layer were modeled via the second generation REBO potential [2] and
Tersoff potential [3] respectively The interlayer interactions between the layers of graphite and bulk
h-BN were described via our dedicated interlayer potential (ILP) [4] which is implemented in the
LAMMPS [5] suite of codes [6]
Fig S1 Schematic representation of the simulation setup (a) and steady-state temperature profile (b) respectively In panel (a) two identical AB-stacked graphite slabs (gray and orange respectively) are twisted with respect to each other to create a stacking fault of misfit angle θ A thermal bias is induced by applying Langevin thermostats to the two layers marked by dashed red (Thot) and green (Tcold) rectangles The arrows indicate the direction of the vertical heat flux Since periodic boundary conditions are applied also in the vertical direction two twisted interfaces are shown across which heat flows in opposite directions The steady-state temperature profiles are illustrated in panel (b) where N is the total number of layers in the model system and RAB dAB and Rθ and dθ mark the interfacial Kapitza resistance [89] and interlayer distance for contacting graphene layers with AB-stacking and misfit angle θ respectively The red lines in panel (b) mark the temperature variation across the twisted interface where the vertical axis corresponds to the position of the various layers along the stack and the horizontal axis marks the temperature of the various layers
S3
12 Simulation Protocol
All MD simulations were performed with the LAMMPS simulation package [5] The velocity-Verlet
algorithm with a time-step of 05 fs was used to propagate the equations of motion A Noseacute-Hoover
thermostat with a time constant of 025 ps was used for constant temperature simulations To maintain
a specified hydrostatic pressure the three translational vectors of the simulation cell were adjusted
independently by a Noseacute-Hoover barostat with a time constant of 10 ps [7] To relax the box we first
equilibrated the systems in the NPT ensemble at a temperature of T = 300 K and zero pressure for
250 ps (see Fig S2) After equilibration Langevin thermostats with damping coefficients 10 ps-1
were applied to the bottom and middle layer of the graphene stack (see Fig S1) with target
temperature Thot = 375 K (hot reservoir) and Tcold = 225 K (cold reservoir) respectively Then the
system was allowed to reach steady-state over a subsequent simulation period of 750 ps (see Fig S2)
during which the dynamics of all non-thermostated layers followed the NVE ensemble For the larger
model systems the length of the NPT and Langevin stages was doubled (for the 32 and 48 layers
systems) or tripled (for the 104 layers graphitic system) to ensure convergence of the obtained steady-
state Once steady-state was obtained the last 500 ps were used to calculate the thermal conductivity
of the twisted graphite and bulk h-BN The statistical errors were estimated using ten different data
sets each calculated over a time interval of 50 ps
Fig S2 Time evolution of the temperature of thermostated layers for 16 layers twisted graphite with misfit angle (a) θ = 0deg (b) θ = 509deg (c) θ = 1518deg and (d) θ = 3016deg Note that the thermal fluctuations increase with increasing the misfit angle due to the growing interfacial thermal resistance that enhances phonon back scattering at the twisted junction
S4
13 Calculation of the interfacial thermal resistance
According to Fourierrsquos law the cross-plane thermal conductivity (120581120581CP) of a twisted graphitic interface
of misfit angle θ can be calculated as
120581120581CP = 119876119860119860Δ119879119879Δ119911119911
(S1)
where 119876 is the heat flux A is the cross-section area and Δ119879119879Δ119911119911 is the temperature gradient along
the direction of heat flux (perpendicular to the basal plane in our case) Fig S1(b) shows a schematic
temperature profile along the z-direction where the vertical axis corresponds to the position of the
various layers along the stack and the horizontal axis marks the temperature of the various layers The
actual temperature profiles extracted from the NEMD simulations for twisted graphite with different
number of layers can be found in Fig S3 For Bernal-stacked graphite (ie θ = 0deg red circles in Fig
S3) only the linear region of the temperature profile was used to calculate 120581120581CP and the points
corresponding to the layers where the thermostats were applied were omitted (marked with green
triangle in Fig S3) The 120581120581CP of the system was calculated using Eq (S1) by averaging over the two
linear regions of the temperature profiles For the twisted case (θ ne 0deg) we found a sudden
temperature decrease Δ119879119879120579120579 at the position of the twisted interface (see black squares in Fig S3) 120581120581CP
in this case was calculated using the temperature gradient calculated for the same layer range as that
for θ = 0deg To characterize the thermal properties of the twisted interface the concept of interfacial
thermal resistance (ITR) ie Kapitza resistance [89] was introduced According the definition of
the Kapitza resistance [8] 119877119877 = 119860119860Δ119879119879119876 and noticing that Δ119879119879tot = (1198731198732 minus 3)Δ119879119879AB + Δ119879119879120579120579 and
Δ119911119911 = (1198731198732 minus 3)119889119889AB + 119889119889120579120579 Eq (1) can be rewritten considering two-resistors in series as [see Fig
S1(b)]
(1198731198732 minus 3)119877119877AB + 119877119877120579120579 = [(1198731198732 minus 3)119889119889AB + 119889119889120579120579] 120581120581CPfrasl (S2)
Here 119877119877AB 119889119889AB Δ119879119879AB and 119877119877120579120579 119889119889120579120579 Δ119879119879120579120579 are the ITR interlayer distance and temperature
difference for adjacent AB-stacked and twisted graphene layers respectively For the aligned contact
(120579120579 = 0deg) the ITR can be simply calculated as 119877119877AB = 119889119889AB120581120581AB Once 119877119877AB is known 119877119877120579120579 can be
calculated from Eq (S2) We note that the sharp temperature drop at the twisted interface indicates
that 119877119877120579120579 should be much larger than 119877119877AB
S5
Fig S3 Temperature profiles for graphitic stacks consisting of (a) 8 and (b) 16 layers The red circles and black squares represent the temperature profiles for the aligned (120579120579 = 0deg) and twisted (120579120579 =3016deg) junctions respectively Green triangles represent data points that were omitted in the 120581120581CP calculation
2 Thermal conductivity of twisted bilayer graphene
As comparison we also calculated the interfacial thermal conductivity (ITC) and ITR of twisted
bilayer graphene (tBLG) with the transient MD simulation approach [15-17] since the NEMD
simulation protocol used in the main text becomes invalid in this case In this protocol the system
was first equilibrated within the NPT ensemble at T = 200 K and zero pressure for 100 ps which was
followed by a 100 ps NVT ensemble equilibration stage and a 100 ps of NVE ensemble equilibration
stage After the system reached steady-state an ultrafast heat impulse was imposed on the top layer
of the t-BLG for 50 fs to increase the temperature of the top layer from 200 K to 400 K while that of
bottom layer of tBLG remained unchanged After the external heat source was removed thermal
energy flowed from the top layer to the bottom layer due to the temperature difference and the
temperature of both layers approached 300 K when quasi-steady-state was reached During the
thermal relaxation time interval (500 ps) the temperature and energy of the system sections were
recorded The ITR could then be extracted using the following equation [15-17]
part119864119864t120597120597120597120597
= 119860119860119877119877119879119879bot(119905119905) minus 119879119879top(119905119905) (S3)
where 119864119864t is the total energy of the top graphene layer R is the ITR of the tBLG A is the interfacial
cross-section area and 119879119879bot and 119879119879top are the instantaneous temperatures measured for the bottom
and top layers of the tBLG respectively Note that in Eq S3 we assume a linear dependence of the
heat flux on the temperature difference between the layers The ITC of the tBLG is simply defined as
ITC equiv 119889119889ITR where d is the average interlayer distance
S6
The ITC and ITR of tBLG as functions of misfit angle calculated with the transient MD simulation
protocol are illustrated in Fig S4 demonstrating similar misfit-angle dependence as that for the
NEMD protocol with Langevin thermostat exercised to obtain the results presented in the main text
This further validates the reliability of the simulation protocol adopted in the main text which is more
suitable to treat thick slabs and allows to obtain a true stead-state
Fig S4 Misfit-angle dependence of (a) ITC and (b) ITR for a twisted bilayer graphene obtained using the transient MD simulation approach
3 Theory for calculating the phonon-phonon coupling of twisted bilayer graphene
31 Brillouin Zone of supercell in tBLG
For tBLG the lattice structure is rigorously periodic only at some specific misfit angles θ where the
lattice vector 1199231199231 = 11989911989911199381199381 + 11989911989921199381199382 in the bottom layer equals the vector 1199231199232 = 11989811989811199381199381 + 11989811989821199381199382 in the
top layer with certain integers m1 m2 and n1 n2 Here 1199381199381 = 119886119886(10) and 1199381199382 = 11988611988612radic32 are
the primitive lattice vectors of the bottom layer and a is the lattice constant of monolayer graphene
Thus the exact superlattice period is then given by [18]
119871119871 = |11989911989911199381199381 + 11989911989921199381199382| = 11988611988611989911989912 + 11989911989922 + 11989911989911198991198992 = |1198991198991minus1198991198992|1198861198862 sin(1205791205792) (S4)
where θ is the angle between two lattice vectors 1199231199231 and 1199231199232 In the simulations below we always
rotated the supercell such that its lattice vector is 1199231199231 = 119871119871(10) and 1199231199232 = 11987111987112radic32 In this case
the corresponding reciprocal lattice vector of the moireacute superlattice satisfies the relation 119918119918119894119894 ∙ 119923119923119895119895 =
2120587120587120575120575119894119894119895119895 such that
1199181199181 = 4120587120587radic3119871119871
radic32
minus12 1199181199182 = 4120587120587
radic3119871119871(01) (S5)
S7
Both the lattice vectors and the corresponding reciprocal lattice vectors of the superlattice of the tBLG
are presented in Fig S5 Table S1 reports the parameters used to construct rhombus periodic
supercells of different misfit angles that can be duplicated to construct a rectangular periodic supercell
Table S1 The parameters used to construct periodic supercells of various misfit angles
120579120579 (deg) 119860119860 (nm2) 1198991198991 1198991198992 1198981198981 1198981198982
0 60383688 24 0 24 0
0696407 709613169 48 47 47 48
1121311 273718420 30 29 29 30
2000628 85648391 17 16 16 17
3006558 152322047 23 21 21 23
4048894 189013524 26 23 23 26
5085849 53255058 14 12 12 14
7926470 49376245 14 11 11 14
9998709 86277388 19 14 14 19
15178179 31554670 15 4 4 15
19932013 42876612 15 8 8 15
25039660 13942761 9 4 4 9
30158276 18974735 11 4 4 11
32204228 21805221 12 4 4 12
32 Special points for Brillouin Zone integration
The calculation of the sum over wave vector q in Eq (4) in the main text can be transformed to an
integral using the relation sum (⋯ )119954119954 = 1119881119881119887119887int (⋯ )BZ d119954119954 where 119881119881119887119887 = (2120587120587)3119881119881 is the volume of the
Brillouin Zone (BZ) and V is volume of the real-space unit-cell The calculation of integral is usually
inefficient since it requires calculating the value of the function over a large set of k points in the first
BZ To calculate such integrations more efficiently simple k-point meshes can be replaced by a
carefully selected set of special points in the BZ 119954119954119894119894 [19-22] over which the function is evaluated
The integral can then be estimated via
119868119868 = 1119881119881119887119887int 119891119891(119954119954)BZ d119954119954 asymp 1
119873119873sum 119908119908119894119894119891119891(119954119954119894119894)119894119894 (S6)
S8
where 119881119881119887119887 is the volume of the BZ 119908119908119894119894 is the weight of the ith data point and N normalizes the
weighting factors to unity 119873119873 = sum 119908119908119894119894119894119894 The set of selected 119954119954119894119894 forms a grid in the irreducible
Brillouin zone (IBZ) as is illustrated by the red points in Fig S5(b) The coordinates of these points
for a hexagonal lattice are presented in Eq (S7)
Fig S5 (a) Twisted bilayer graphene of misfit angle θ = 509deg L1 and L2 are the superlattice vectors (b) The corresponding first Brillouin Zone of (a) G1 and G2 are the reciprocal lattice vectors of the superlattice The triangle ΔΓMK represents the irreducible Brillouin zone Red circles mark the position of the special points used to evaluate the integral over the first Brillouin Zone
⎩⎪⎨
⎪⎧ 1199541199541 = 1
9 1205971205973 1199541199542 = 2
9 1205971205973 1199541199543 = 1
18 512059712059718
1199541199544 = 518
512059712059718 1199541199545 = 1
9 21205971205979 1199541199546 = 2
9 21205971205979
1199541199547 = 118
312059712059718 1199541199548 = 1
9 1205971205979 1199541199549 = 1
18 12059712059718
(S7)
Here 119905119905 = radic3 and the unit of the coordinates is 2120587120587119871119871 The weighting factors 119908119908119894119894 are [19]
119908119908124689 = 112
119908119908357 = 16 (S8)
Using Eqs (S6-S8) Eq (4) in the main text can be evaluated as follows
Γtot = 120587120587ℏ3
2sum 119908119908119896119896 sum
119890119890minus120573120573119864119864119954119954119948119948120582120582
119885119885
120588120588119864119864119954119954119948119948120582120582119881119881120582120582120582120582+31199031199032(119954119954119948119948)
2
1198641198641199541199541199481199481205821205822120582120582
9119896119896=1 (S9)
This equation was used to calculate the transition rate presented in Fig 3 in the main text
4 Derivation of Fermirsquos golden rule
The derivation of Fermirsquos golden rule is provided in a separate file (Fermirsquos Golden Rule for
phononspdf) due to its extent
S9
5 Temperature dependence of interfacial thermal conductivity
In the main text the target temperatures of the Langevin thermostats for the bottom and middle layers
of graphene and h-BN were set to 225 K and 375 K respectively After reaching the steady-state the
average temperature of the system was found to be ~300 K To check the effect of average temperature
on our results we calculated the cross-plane thermal conductivity (120581120581CP ) and the corresponding
interfacial thermal resistance (ITR) at a different temperature gradient (325 K ndash 475 K) resulting the
average steady-state temperature of ~400K The protocol described in Section 1 above was used to
perform these calculations as well Both ILP and Lennard-Jones (LJ) potential were tested for
graphite whereas for the bulk h-BN simulations only the ILP was used The results for graphite and
bulk h-BN are illustrated in Fig S6 and Fig S7 respectively For the ILP we find that the overall
values of 120581120581CP (ITR) decrease (increase) slightly with increasing average temperature which is
consistent with a recent experiment [10] The LJ potential calculations as well exhibit very week
dependence on average temperature within the range studied Altogether the layer dependences of
both quantities remain mostly insensitive to the average temperature The reason is that the thickness
of the graphite and h-BN model systems was chosen to be much smaller than their phonon mean free
path (~200 nm for graphite [110-13] and ~100 nm for bulk h-BN [14]) such that phonon transport is
dominated by phonon-boundary scattering and the Umklapp process only makes marginal
contribution [10]
S10
Fig S6 Layer dependence of 120581120581CP (a c) and ITR (b d) for Bernal-stacked graphite at average steady-state temperatures of 300 K (red circles) and 400 K (black squares) The left and right columns correspond to the 120581120581CP and ITR calculated with ILP and LJ potential respectively
Fig S7 Layer dependence of 120581120581CP (a) and ITR (b) for AArsquo-stacked h-BN at average temperatures of 300 K (red circles) and 400 K (blue squares)
S11
References [1] Z Wei J Yang W Chen K Bi D Li and Y Chen Phonon mean free path of graphite along the c-axis Appl Phys Lett 104 081903 (2014) [2] D W Brenner O A Shenderova J A Harrison S J Stuart B Ni and S B Sinnott A second-generation reactive empirical bond order (REBO) potential energy expression for hydrocarbons J Phys Condens Matter 14 783 (2002) [3] A Kınacı J B Haskins C Sevik and T Ccedilağın Thermal conductivity of BN-C nanostructures Phys Rev B 86 (2012) [4] W Ouyang I Azuri D Mandelli A Tkatchenko L Kronik M Urbakh and O Hod Mechanical and Tribological Properties of Layered Materials under High Pressure Assessing the Importance of Many-Body Dispersion Effects J Chem Theory Comput 16 666 (2020) [5] S Plimpton Fast Parallel Algorithms for Short-Range Molecular Dynamics J Comput Phys 117 1 (1995) [6] W Ouyang D Mandelli M Urbakh and O Hod Nanoserpents Graphene Nanoribbon Motion on Two-Dimensional Hexagonal Materials Nano Lett 18 6009 (2018) [7] W Shinoda M Shiga and M Mikami Rapid estimation of elastic constants by molecular dynamics simulation under constant stress Phys Rev B 69 134103 (2004) [8] G L Pollack Kapitza Resistance Rev Mod Phys 41 48 (1969) [9] H-S Yang G R Bai L J Thompson and J A Eastman Interfacial thermal resistance in nanocrystalline yttria-stabilized zirconia Acta Mater 50 2309 (2002) [10] Q Fu J Yang Y Chen D Li and D Xu Experimental evidence of very long intrinsic phonon mean free path along the c-axis of graphite Appl Phys Lett 106 031905 (2015) [11] J Chen J H Walther and P Koumoutsakos Strain Engineering of Kapitza Resistance in Few-Layer Graphene Nano Lett 14 819 (2014) [12] H Zhang X Chen Y-D Jho and A J Minnich Temperature-Dependent Mean Free Path Spectra of Thermal Phonons Along the c-Axis of Graphite Nano Lett 16 1643 (2016) [13] D P Sellan E S Landry J E Turney A J H McGaughey and C H Amon Size effects in molecular dynamics thermal conductivity predictions Phys Rev B 81 214305 (2010) [14] P Jiang X Qian R Yang and L Lindsay Anisotropic thermal transport in bulk hexagonal boron nitride Phys Rev Mater 2 064005 (2018) [15] J Zhang Y Hong and Y Yue Thermal transport across graphene and single layer hexagonal boron nitride J Appl Phys 117 134307 (2015) [16] B Liu F Meng C D Reddy J A Baimova N Srikanth S V Dmitriev and K Zhou Thermal transport in a graphenendashMoS2 bilayer heterostructure a molecular dynamics study RSC Advances 5 29193 (2015) [17] Z Ding Q-X Pei J-W Jiang W Huang and Y-W Zhang Interfacial thermal conductance in grapheneMoS2 heterostructures Carbon 96 888 (2016) [18] N N T Nam and M Koshino Lattice relaxation and energy band modulation in twisted bilayer graphene Phys Rev B 96 075311 (2017)
S12
[19] R Ramiacutearez and M C Boumlhm Simple geometric generation of special points in brillouin-zone integrations Two-dimensional bravais lattices Int J Quantum Chem 30 391 (1986) [20]S L Cunningham Special points in the two-dimensional Brillouin zone Phys Rev B 10 4988 (1974) [21] H J Monkhorst and J D Pack Special points for Brillouin-zone integrations Phys Rev B 13 5188 (1976) [22] D J Chadi Special points for Brillouin-zone integrations Phys Rev B 16 1746 (1977)
1
Fermirsquos golden rule for phonons
1 Basic theory for phonons 11 Basic notations
Let us consider a 3D crystal with a total of 119873119873 = 119873119873111987311987321198731198733 unit cells and periodic boundary conditions
To be specific let 119938119938119894119894 119894119894 = 123 be the lattice vectors that define the unit cell We index unit cells
with n = (n1n2n3) where each ni = 12 ∙∙∙ Ni and their locations are 119929119929119899119899 = sum 1198991198991198941198941199381199381198941198943119894119894=1 Assume that
there are r atoms in each unit cell which are indexed with 119904119904 = 1⋯ 119903119903 The mass and the equilibrium
distance of the sth atom are notated as 119872119872s and 119929119929s0 respectively Then the location of the sth atom in
the nth unit cell at time t can be expressed as
119955119955119899119899119899119899(119905119905) = 119929119929119899119899 + 119929119929s0 + 119958119958119899119899119899119899(119905119905) (11)
where 119958119958119899119899119899119899(119905119905) is its displacement from its equilibrium position
The Lagrangian for this classical problem can be written as
ℒ = sum sum 119872119872119904119904|119955119899119899119904119904|2(119905119905)2
119903119903119899119899=1
119873119873119899119899=1 minus 119881119881 (12)
where the second term is the sum of interactions between all pairs of atoms
Under the harmonic approximation ie expanding the total potential energy 119881119881 around the
equilibrium positions The Lagrangian can be simplified as
ℒ = sum sum 119872119872119904119904|119958119899119899119904119904|2(119905119905)2
119903119903119899119899=1
119873119873119899119899=1 minus 1
2sum sum sum 120601120601120572120572120572120572prime
119899119899119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime120572120572120572120572prime119899119899119899119899prime119899119899119899119899prime (13)
where 119906119906119899119899119899119899120572120572 120572120572 = 123 are the Cartesian coordinates of the displacement 119958119958119899119899119899119899(119905119905) and
120601120601120572120572120572120572prime 119899119899119899119899prime119904119904 119904119904prime = 1205971205972119881119881
120597120597119903119903119899119899119904119904119899119899119903119903119899119899prime119904119904prime119899119899primeeq
= 120601120601120572120572prime120572120572 119899119899prime119899119899119904119904prime 119904119904 = 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime (14)
Note that the first order term vanishes because we are expanding around the equilibrium positions
120601120601120572120572120572120572prime 119899119899119899119899prime119904119904 119904119904prime represents the component of the force acted on the sth atom in the nth unit cell along 120572120572
direction when the atom 119904119904prime in the unit cell 119899119899prime moves a unit displacement along 120572120572prime direction The
symmetries of 120601120601120572120572120572120572prime 119899119899 119899119899prime119904119904 119904119904prime appearing in Eq (14) arise from the intechangability of the second
derivative and the translational invariance of the interactions
12 Dynamical matrix
The equation of motion of the sth atom in the nth unit cell can be derived using the EulerndashLagrange
equation as follows
2
119872119872119899119899119906119899119899119899119899120572120572 = minussum 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119899119899prime119899119899prime120572120572prime 119906119906119899119899prime119899119899prime120572120572prime (15)
If we displace all atoms equally ie shifting 119906119906119899119899prime119899119899prime120572120572prime to 119906119906119899119899prime119899119899prime120572120572prime + 120575120575 the total force on the sth atom in
the nth unit cell does not change From the above equation we have
sum 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119899119899prime120572120572prime = 0 (16)
We are looking for normal modes (because any general solution can be written as a linear combination
of them) these are solutions where all atoms oscillate with the same frequency Moreover because
of the lattice structure we expect solutions to reflect this periodicity So we guess solutions of the
form
119906119906119899119899119899119899120572120572(119905119905) = 1119872119872119904119904
119890119890119899119899120572120572119890119890119894119894(119954119954∙119929119929119899119899minus120596120596119905119905) (17)
where 119890119890119899119899120572120572 are real-space solutions that will be determined later and 119954119954 is the wave-vector in
reciprocal space Substituting Eq (17) into Eq (15) we can get
1205961205962119890119890119899119899120572120572(119954119954) = sum 119863119863119899119899120572120572119899119899prime120572120572prime(119954119954)119890119890119899119899prime120572120572prime(119954119954)119899119899prime120572120572prime (18)
where
119863119863119899119899120572120572119899119899prime120572120572prime(119954119954) = sum 1
119872119872119904119904119872119872119904119904prime120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119890119890
minus119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime119899119899prime = sum 1
119872119872119904119904119872119872119904119904prime120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime 119890119890
minus119894119894119954119954∙119929119929119897119897119897119897 (19)
is called dynamical matrix (dimension 3119903119903 times 3119903119903) Note that we have defined the relative cell distance
vector 119929119929119897119897 equiv 119929119929119899119899 minus 119929119929119899119899prime and number index 119897119897 equiv 119899119899 minus 119899119899prime where the infinite sum over 119899119899prime can be
replaced by the sum over 119897119897 for any value of the index 119899119899 Note that dynamical matrix is Hermitian
symmetric (ie 119863119863119899119899120572120572119899119899prime120572120572prime(119954119954)
lowast= 119863119863119899119899prime120572120572prime
119899119899120572120572 (minus119954119954)) because ϕ is symmetric so all the eigenvalues of Eq (18)
(120596120596120582120582(119954119954) 120582120582 = 12⋯ 3119903119903) are real for each 119954119954 in the Brillouin zone (BZ) which is determined by
det1205961205961205821205822(119954119954)120575120575119899119899120572120572120575120575119899119899prime120572120572prime minus 119863119863119899119899120572120572119899119899prime120572120572prime(119954119954) = 0 (110)
Taking the conjugate of this equation we have
0 = det 1205961205961205821205822(119954119954)lowast120575120575119899119899120572120572120575120575119899119899prime120572120572prime minus 119863119863119899119899120572120572119899119899
prime120572120572prime(119954119954)lowast = det1205961205961205821205822(119954119954)120575120575119899119899120572120572120575120575119899119899prime120572120572prime minus 119863119863119899119899120572120572119899119899
prime120572120572prime(minus119954119954) (111)
while replacing 119954119954 by minus119954119954 in Eq (110) we have det1205961205961205821205822(minus119954119954)120575120575119899119899120572120572120575120575119899119899prime120572120572prime minus 119863119863119899119899120572120572119899119899prime120572120572prime(minus119954119954) = 0
Itrsquos clear that 1205961205961205821205822(119954119954) and 1205961205961205821205822(minus119954119954) obey the same equation thus we have
120596120596120582120582(minus119954119954) = 120596120596120582120582(119954119954) (112)
The corresponding eigenvectors are orthonormal
sum 119890119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582
prime119899119899120572120572 = 120575120575120582120582120582120582prime (113)
The complex conjugate of Eq (18) gives
1205961205962119890119890119899119899120572120572lowast (119954119954) = sum 119863119863119899119899120572120572119899119899prime120572120572prime(119954119954)
lowast119890119890119899119899prime120572120572primelowast (119954119954)119899119899prime120572120572prime = sum 119863119863119899119899prime120572120572prime
119899119899120572120572 (minus119954119954)119890119890119899119899prime120572120572primelowast (119954119954)119899119899prime120572120572prime (114)
3
While replacing 119954119954 by minus119954119954 in Eq (18) we get the following equation
1205961205962119890119890119899119899120572120572(minus119954119954) = sum 119863119863119899119899120572120572119899119899prime120572120572prime(minus119954119954)119890119890119899119899prime120572120572prime(minus119954119954)119899119899prime120572120572prime (115)
From Eq (114) and Eq (115) we see that eigenvectors 119890119890119899119899120572120572120582120582 (minus119954119954)lowast and 119890119890119899119899120572120572120582120582 (119954119954) obey the same
eigenvalue equation Since the eigenvectors are normalized we get the following property
119890119890119899119899120572120572120582120582 (minus119954119954)lowast
= 119890119890119899119899120572120572120582120582 (119954119954) (116)
2 Second quantization The general solution is a linear combination of all these normal modes thus we have
119906119906119899119899119899119899120572120572(119905119905) = sum 119862119862120582120582(119954119954)119873119873119872119872119904119904
119890119890119899119899120572120572120582120582 (119954119954)119890119890minus119894119894120596120596119954119954120582120582119905119905119890119890119894119894119954119954∙119929119929119899119899119954119954120582120582 = sum 119876119876120582120582(119954119954119905119905)119873119873119872119872119904119904
119890119890119899119899120572120572120582120582 (119954119954)119890119890119894119894119954119954∙119929119929119899119899119954119954120582120582 (21)
where the we define the normal coordinates as 119876119876120582120582(119954119954 119905119905) equiv 119862119862120582120582(119954119954)119890119890minus119894119894120596120596120582120582(119954119954)119905119905 in the eigenvectors
representation To ensure that the displacements 119906119906119899119899119899119899120572120572(119905119905) are real (namely 119906119906119899119899119899119899120572120572lowast (119905119905) = 119906119906119899119899119899119899120572120572(119905119905)) the
following relation on 119876119876120582120582(119954119954 119905119905) and 119890119890119899119899120572120572120582120582 is enforced
119876119876120582120582(119954119954 119905119905)119890119890119899119899120572120572120582120582 (119954119954)lowast
= 119876119876120582120582(minus119954119954 119905119905)119890119890119899119899120572120572120582120582 (minus119954119954) (22)
where we used the fact that the sum over 119954119954 runs symmetrically over both negative and positive
values Using Eq (116) we have
[119876119876120582120582(119954119954 119905119905)]lowast = 119876119876120582120582(minus119954119954 119905119905) (23)
21 Kinetic energy term
Using Eq (21) the kinetic energy of the system can be expressed as
119879119879 = 12sum 1198721198721198991198991199061198991198991198991198991205721205722 (119905119905)119899119899119899119899120572120572 = 1
2sum 119872119872119904119904
119873119873119872119872119904119904sum sum 119876120582120582(119954119954 119905119905)119876120582120582prime(119954119954prime 119905119905)119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582
prime(119954119954prime) sum 119890119890119894119894119954119954+119954119954prime∙119929119929119899119899119899119899119954119954prime120582120582prime119954119954120582120582119899119899120572120572 =
12sum sum 119876120582120582(119954119954 119905119905)119876120582120582prime(minus119954119954 119905119905)119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582
prime(minus119954119954)119954119954119954119954prime120582120582prime =119899119899120572120572
12sum 119876120582120582(119954119954 119905119905)119876120582120582prime
lowast (119954119954 119905119905) sum 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582prime(119954119954)
lowast119899119899120572120572 =119954119954120582120582120582120582prime
12sum 119876120582120582(119954119954 119905119905)119876120582120582lowast(119954119954 119905119905)119954119954120582120582 =
12sum 119876120582120582(119954119954 119905119905)119876120582120582(minus119954119954 119905119905)119954119954120582120582
ie
119879119879 = 12sum 119876120582120582(119954119954 119905119905)119876120582120582(minus119954119954 119905119905)119954119954120582120582 (24)
To derive Eq (24) we used Eqs (113) and (116) as well as the following equations
1119873119873sum 119890119890119894119894119954119954+119954119954prime∙119929119929119899119899119899119899 = 120575120575119954119954+119954119954prime120782120782 (25)
4
22 Potential energy term
Similarity the potential energy can be rewritten in terms of the normal coordinates as follows
119880119880 =12120601120601119899119899119899119899120572120572119899119899prime119899119899prime120572120572prime119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
120572120572120572120572prime119899119899119899119899prime119899119899119899119899prime=
12120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119899119899+119954119954prime∙119929119929119899119899prime
119873119873119872119872119899119899119872119872119899119899prime
[119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(119954119954prime 119905119905)] 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime120582120582prime (119954119954prime)
119954119954prime120582120582prime119954119954120582120582119899119899119899119899prime120572120572120572120572prime119899119899119899119899prime
=12120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894(119954119954+119954119954prime)∙119929119929119899119899119890119890minus119894119894119954119954prime∙119929119929119897119897
119873119873119872119872119899119899119872119872119899119899prime
[119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(119954119954prime 119905119905)] 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime120582120582prime (119954119954prime)
119954119954prime120582120582prime119954119954120582120582119899119899119897119897120572120572120572120572prime119899119899119899119899prime
=12120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890minus119894119894119954119954prime∙119929119929119897119897
119872119872119899119899119872119872119899119899prime[119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(119954119954prime 119905119905)] 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (119954119954prime)1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899
119899119899=120575120575119902119902+119902119902prime=0
119954119954prime120582120582prime119954119954120582120582119897119897120572120572120572120572prime119899119899119899119899prime
=12120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894119954119954119929119929119897119897
119872119872119899119899119872119872119899119899prime [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)] 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (minus119954119954)119954119954120582120582120582120582prime119897119897120572120572120572120572prime119899119899119899119899prime
=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]119890119890119899119899120572120572120582120582 (119954119954)
119899119899120572120572
119863119863119899119899120572120572119899119899prime120572120572prime(minus119954119954)119890119890119899119899prime120572120572prime
120582120582prime (minus119954119954)119899119899prime120572120572prime119954119954120582120582120582120582prime
=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]119890119890119899119899120572120572120582120582 (119954119954)
119899119899120572120572
1205961205961205821205822(minus119954119954)119890119890119899119899120572120572120582120582prime (minus119954119954)
119954119954120582120582120582120582prime
=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]1205961205961205821205822(119954119954)120575120575120582120582120582120582prime =119954119954120582120582120582120582prime
121205961205961205821205822(119954119954)119876119876120582120582(119954119954 119905119905)119876119876120582120582(minus119954119954 119905119905)
119954119954120582120582
ie
119880119880 = 12sum 1205961205961205821205822(119954119954)119876119876120582120582(119954119954 119905119905)119876119876120582120582(minus119954119954 119905119905)119954119954120582120582 (26)
where in above deriviation the orthogonality of its eigenvectores and the symmetry of its eigenvalues
are used Thus the Lagrangian reads as
ℒ = 119879119879 minus 119881119881 = 12sum 119876120582120582(minus119954119954 119905119905)119876120582120582(119954119954 119905119905) minus 120596120596120582120582
2(119954119954)119876119876120582120582(minus119954119954 119905119905)119876119876120582120582(119954119954 119905119905)119954119954120582120582 (27)
Using the relation 119875119875120582120582(119954119954 119905119905) = 120597120597ℒ part119876120582120582(119954119954 119905119905) the Hamiltonian of the system then can be written as
119867119867 = 119879119879 + 119881119881 = 12sum [119875119875120582120582(minus119954119954 119905119905)119875119875120582120582(119954119954 119905119905) + 120596120596120582120582
2(119954119954)119876119876120582120582(minus119954119954 119905119905)119876119876120582120582(119954119954 119905119905)]119954119954120582120582 (28)
Now we quantize H by asking the momenta and coordinates to be operators
119867119867 = 12sum 119875119875minus119954119954120582120582119875119875119954119954120582120582 + 120596120596119954119954120582120582
2 119876119876minus119954119954120582120582119876119876119954119954120582120582119954119954120582120582 (29)
here 119875119875119954119954120582120582 and 119876119876119954119954120582120582 obey the following commutation relations
119876119876119954119954120582120582119875119875119954119954prime120582120582prime = 119894119894ℏ120575120575119954119954119954119954prime120575120575120582120582120582120582prime
119876119876119954119954120582120582119876119876119954119954prime120582120582prime = 0 119875119875119954119954120582120582119875119875119954119954prime120582120582prime = 0 (210)
Similar to the case of ordinary quantum harmonic oscillators it is convenient to define ladder
operators for each mode as follows
119876119876119954119954120582120582 = ℏ
2120596120596119954119954120582120582119887119887119954119954120582120582 + 119887119887minus119954119954120582120582
dagger 119875119875119954119954120582120582 = 119894119894ℏ120596120596119954119954120582120582
2119887119887119954119954120582120582
dagger minus 119887119887minus119954119954120582120582 (211)
where 119887119887119954119954120582120582dagger and 119887119887119954119954120582120582 are Bosonic creation and annihilation operators for phonons with momentum
119954119954 branch index 120582120582 and frequency 120596120596119954119954120582120582 which obeys the Bosonic commutation relation
5
119887119887119954119954120582120582 119887119887119954119954prime120582120582prime
dagger = 120575120575119954119954119954119954prime120575120575120582120582120582120582prime
119887119887119954119954120582120582 119887119887119954119954prime120582120582prime = 0 119887119887119954119954120582120582dagger 119887119887119954119954prime120582120582prime
dagger = 0 (212)
Substituting Eqs (211) into (29) and using the properties Eq (210) and (212) we have
119867119867 =12119875119875minus119954119954120582120582119875119875119954119954120582120582 +120596120596119954119954120582120582
2 119876119876minus119954119954120582120582119876119876119954119954120582120582119954119954120582120582
=12minus
ℏ120596120596119954119954120582120582
2119887119887minus119954119954120582120582
dagger minus 119887119887119954119954120582120582119887119887119954119954120582120582dagger minus 119887119887minus119954119954120582120582+ 120596120596119954119954120582120582
2 ℏ2120596120596119954119954120582120582
119887119887minus119954119954120582120582 + 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 + 119887119887minus119954119954120582120582
dagger 119954119954120582120582
=ℏ4120596120596119954119954120582120582minus119887119887minus119954119954120582120582
dagger 119887119887119954119954120582120582dagger minus 119887119887minus119954119954120582120582
dagger 119887119887minus119954119954120582120582 minus 119887119887119954119954120582120582119887119887119954119954120582120582dagger + 119887119887119954119954120582120582119887119887minus119954119954120582120582
119954119954120582120582
+ 119887119887minus119954119954120582120582119887119887119954119954120582120582 + 119887119887minus119954119954120582120582119887119887minus119954119954120582120582dagger + 119887119887119954119954120582120582
dagger 119887119887119954119954120582120582 + 119887119887119954119954120582120582dagger 119887119887minus119954119954120582120582
dagger
=ℏ4120596120596119954119954120582120582119887119887minus119954119954120582120582
dagger 119887119887minus119954119954120582120582 + 119887119887minus119954119954120582120582119887119887minus119954119954120582120582dagger + 119887119887119954119954120582120582119887119887119954119954120582120582
dagger + 119887119887119954119954120582120582dagger 119887119887119954119954120582120582
119954119954120582120582
=ℏ41205961205961199541199541205821205822119887119887minus119954119954120582120582
dagger 119887119887minus119954119954120582120582 + 1+ 2119887119887119954119954120582120582dagger 119887119887119954119954120582120582 + 1
119954119954120582120582
= ℏ120596120596119954119954120582120582 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 +
12
119954119954120582120582
Therefore we finally get the quantized representation of non-interacting phonons
119867119867 = sum ℏ120596120596119954119954120582120582 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 + 1
2119954119954120582120582 (213)
The operator of atom displacements (Eq 114) is expressed in terms of the phonon operators by
119906119906119899119899119899119899120572120572 = sum ℏ
2119873119873119872119872119904119904120596120596119954119954120582120582119890119890119899119899120572120572120582120582 119890119890119894119894119954119954∙119929119929119899119899119887119887119954119954120582120582 + 119887119887minus119954119954120582120582
dagger 119954119954120582120582 (214)
These equations will be used below
3 Inter-phonon coupling within harmonic approximation 31 Hamiltonian with inter-phonon coupling
In this section we consider systems that consist of two (or more) covalently bonded units that are
weakly coupled between them Unlike previous studies that considered phonon-phonon couplings
resulting from anharmonicity effects [1] here all phonon mode considered are Harmonic and the
couplings arise from the division of the entire system into subunits The Hamiltonian of the whole
system can be written as
119867119867 = 1198671198671 + 1198671198672 + 11986711986712 (31)
To derive the expressions of 1198671198671 1198671198672 and 11986711986712 in Eq (31) we consider the Hamiltonian of the
whole system written as the function of the atomic displacements
119867119867 = 119879119879 + 119881119881 = sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)2119899119899119899119899120572120572 + 1
2sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime120572120572120572120572prime119899119899119899119899prime119899119899119899119899prime (32)
6
Letrsquos assume that subsystem I and II contain atoms with indeices ranging from 119904119904 = 12⋯ 1199031199032 and
119904119904 = 1199031199032
+ 1 1199031199032
+ 2⋯ 119903119903 respectively Then the kinetic energy term in Eq (32) can be rewritten as
119879119879 = sum sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)2
1199031199032119899119899=1 + sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)
2119903119903119899119899=1199031199032+1
119899119899120572120572 (33)
While the potential energy term is
119881119881 =12 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
1199031199032
119899119899119899119899prime=1
+ 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
119903119903
119899119899119899119899prime=1199031199032+1
+ 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime119899119899119899119899prime
+ 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
1199031199032
119899119899prime=1
119903119903
119899119899=1199031199032+1
= 11988111988111 + 11988111988122 + 11988111988112 + 11988111988121
Or equivalently
⎩⎪⎪⎨
⎪⎪⎧ 11988111988111 = 1
2sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
1199031199032119899119899119899119899prime=1120572120572120572120572prime119899119899119899119899prime
11988111988122 = 12sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
119903119903119899119899119899119899prime=1199031199032+1120572120572120572120572prime119899119899119899119899prime
11988111988112 = 12sum sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
119903119903119899119899prime=1199031199032+1
1199031199032119899119899=1120572120572120572120572prime119899119899119899119899prime
11988111988121 = 12sum sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
1199031199032119899119899prime=1
119903119903119899119899=1199031199032+1120572120572120572120572prime119899119899119899119899prime
(34)
32 Second quantization
We may now quantize this Hamiltonian and write it in the basis of the eigenstates of the coupled
subunits In second quantization the atomic displacement operators of the two subunits are given in
the following form
119906119906119899119899119899119899120572120572 =
⎩⎨
⎧ sum ℏ
2119873119873119872119872119904119904ω119954119954120582120582119890119890119899119899120572120572120582120582 119890119890119894119894119954119954∙119929119929119899119899119886119886119954119954120582120582 + 119886119886minus119954119954120582120582
dagger 119954119954120582120582 119904119904 isin 1 1199031199032
sum ℏ
2119873119873119872119872119904119904120596120596119954119954prime120582120582prime119890119899119899120572120572120582120582
prime119890119890119894119894119954119954prime∙119929119929119899119899 119886119886 119954119954prime120582120582prime + 119886119886minus119954119954prime120582120582primedagger 119954119954prime120582120582prime 119904119904 isin 119903119903
2+ 1 119903119903
(35)
Here we use different notations for the creation and annihilation operators (119886119886119954119954120582120582119886119886minus119954119954120582120582dagger and 119886119886119954119954120582120582119886119886minus119954119954120582120582
dagger )
eigenvalues (ω119954119954120582120582120596120596119954119954120582120582) and eigenvectors (119890119890119899119899120572120572120582120582 119890119899119899120572120572120582120582 ) for the two subunits Note that 119890119890119899119899120572120572120582120582 and 119890119899119899120572120572120582120582 are
of the dimensions of the whole system however their non-zero elements appear only on the relevant
subunits such that they obey the following relations sum 119890119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582
prime119899119899120572120572 = 120575120575120582120582120582120582prime sum 119890119899119899120572120572120582120582
lowast119890119899119899120572120572120582120582
prime119899119899120572120572 =
120575120575120582120582120582120582prime sum 119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582
prime119899119899120572120572 = sum 119890119890119899119899120572120572120582120582
lowast119890119899119899120572120572120582120582
prime119899119899120572120572 = 0 Defining the normal coordinates of the two subunits as
119876119876119954119954120582120582 equiv ℏ
2ω119954119954120582120582119886119886119954119954120582120582 + 119886119886minus119954119954120582120582
dagger 119876119876119954119954120582120582 equiv ℏ
2ω119954119954120582120582119886119886119954119954120582120582 + 119886119886minus119954119954120582120582
dagger (36)
and the corresponding momenta operators as
7
119875119875119954119954120582120582 = 119894119894ℏω119954119954120582120582
2119886119886119954119954120582120582
dagger minus 119886119886minus119954119954120582120582 119875119875119954119954120582120582 = 119894119894ℏω1199541199541205821205822
119886119886119954119954120582120582dagger minus 119886119886minus119954119954120582120582 (37)
Eq (35) reads
119906119906119899119899119899119899120572120572 = sum 119890119890119904119904119899119899120582120582 (119954119954)
119873119873119872119872119904119904119890119890119894119894119954119954∙119929119929119899119899119876119876119954119954120582120582119954119954120582120582 119904119904 isin 1 119903119903
2
sum 119890119904119904119899119899120582120582 (119954119954)119873119873119872119872119904119904
119890119890119894119894119954119954∙119929119929119899119899119876119876119954119954120582120582119954119954120582120582 119904119904 isin 1199031199032
+ 1 119903119903 (38)
The second quantized kinetic energy operator is then written as
119879119879 = 1198791198791 + 1198791198792 = 12sum 119875119875119954119954120582120582119875119875minus119954119954120582120582119954119954120582120582 + 1
2sum 119875119875119954119954prime120582120582prime119875119875minus119954119954prime120582120582prime119954119954prime120582120582prime (39)
Correspondingly the various potential energy terms are obtained by substituting Eq (38) in Eq (34)
11988111988111 =12 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119899119899119899119899prime
119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime120582120582prime (119954119954prime)
119872119872119899119899119872119872119899119899prime
1119873119873119890119890119894119894119954119954∙119929119929119899119899+119894119894119954119954prime∙119929119929119899119899prime119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954120582120582119954119954prime120582120582prime
1199031199032
119899119899119899119899prime=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954119954119954prime120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (119954119954prime)1119873119873 119890119890
119894119894(119954119954+119954119954prime)119929119929119899119899prime
119899119899prime120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime
119872119872119899119899119872119872119899119899prime119899119899
1199031199032
119899119899119899119899prime=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954119954119954prime120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (119954119954prime) 120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime
1199031199032
119899119899119899119899prime=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime
119954119954120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (minus119954119954)120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
1199031199032
119899119899119899119899prime=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime
119954119954120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954) 119863119863119899119899120572120572119899119899
prime120572120572prime(minus119954119954)119890119890119899119899prime120572120572prime120582120582prime (minus119954119954)
120572120572prime
1199031199032
119899119899prime=1
1199031199032
119899119899=1120572120572
=12 ω119954119954120582120582prime
2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582120582120582prime
119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582prime (119954119954)
lowast
1199031199032
119899119899=1
=120572120572
12ω119954119954120582120582
2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582
ie
11988111988111 = 12sum ω119954119954120582120582
2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582 (310)
Going from the second line to the third line we used the fact that the sum over 119899119899prime runs between plusmninfin
and the summand depends only on the difference between 119899119899 and 119899119899prime hence the sum is independent
of the value of the index 119899119899 Therefore we can replace the sum over 119899119899prime by a sum over 119897119897 equiv 119899119899 minus 119899119899prime
amd define 119929119929119897119897 equiv 119929119929119899119899 minus 119929119929119899119899prime Following the same procedure we can get the corresponding
expressions for the second diagonal term
11988111988122 = 12sum 1205961205961199541199541205821205822 119876119876119954119954120582120582119876119876minus119954119954120582120582119954119954120582120582 (311)
Similarly for the off-diagonal terms we get
8
11988111988112 =12 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119899119899119899119899prime
119903119903
119899119899prime=1199031199032+1
119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime
120582120582prime (119954119954prime)
119872119872119899119899119872119872119899119899prime
1119873119873119890119890119894119894119954119954∙119929119929119899119899+119894119894119954119954prime∙119929119929119899119899prime119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954120582120582119954119954prime120582120582prime
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954119954119954prime120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime
120582120582prime (119954119954prime)1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime
119872119872119899119899119872119872119899119899prime119899119899
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954119954119954prime120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime
120582120582prime (119954119954prime) 120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime
119954119954120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime
120582120582prime (minus119954119954)120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954
119890119899119899prime120572120572prime120582120582prime (119954119954)
lowast120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119890119890119899119899120572120572120582120582 (119954119954)119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954
Where we have defined
119881119881120582120582120582120582prime(119954119954) equiv sum sum sum 119890119899119899prime120572120572prime120582120582prime (119954119954)
lowast119881119881119899119899120572120572119899119899
prime120572120572prime(119954119954)119890119890119899119899120572120572120582120582 (119954119954)119903119903119899119899prime=1199031199032+1
1199031199032119899119899=1120572120572120572120572prime (312)
where
119881119881119899119899120572120572119899119899prime120572120572prime(119954119954) equiv sum 119890119890119894119894119954119954∙119929119929119897119897
119872119872119904119904119872119872119904119904prime119897119897 120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime (313)
Itrsquos easy to show that 119881119881120582120582120582120582prime(119954119954) has the following property
119881119881120582120582120582120582primelowast (119954119954) = 119881119881120582120582120582120582prime(minus119954119954) (314)
Following the same procedure we have
9
11988111988121 =12 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119899119899119899119899prime
119903119903
119899119899=1199031199032+1
119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (119954119954prime)
119872119872119899119899119872119872119899119899prime
1119873119873119890119890119894119894119954119954∙119929119929119899119899+119894119894119954119954prime∙119929119929119899119899prime119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954120582120582119954119954prime120582120582prime
1199031199032
119899119899prime=1120572120572120572120572prime
=12 120601120601120572120572prime120572120572
119899119899prime minus 119899119899119904119904prime 119904119904
119899119899prime119899119899
119903119903
119899119899prime=1199031199032+1
119890119899119899prime120572120572prime120582120582prime (119954119954prime)119890119890119899119899120572120572120582120582 (119954119954)
119872119872119899119899prime119872119872119899119899
1119873119873 119890119890
119894119894119954119954prime∙119929119929119899119899prime+119894119894119954119954∙119929119929119899119899119876119876119954119954prime120582120582prime119876119876119954119954120582120582119954119954prime120582120582prime119954119954120582120582
1199031199032
119899119899=1120572120572prime120572120572
=12 119876119876119954119954prime120582120582prime119876119876119954119954120582120582
119954119954119954119954prime120582120582120582120582prime119890119899119899prime120572120572prime120582120582prime (119954119954prime)119890119890119899119899120572120572120582120582 (119954119954)
1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime
119872119872119899119899119872119872119899119899prime119899119899
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954prime120582120582prime119876119876119954119954120582120582
119954119954119954119954prime120582120582120582120582prime119890119899119899prime120572120572prime120582120582prime (119954119954prime)119890119890119899119899120572120572120582120582 (119954119954) 120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954prime120582120582prime119876119876minus119954119954prime120582120582119890119899119899prime120572120572prime
120582120582prime (119954119954prime)119890119890119899119899120572120572120582120582 (minus119954119954prime)120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890minus119894119894119954119954prime∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897119954119954prime120582120582120582120582prime
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954prime120582120582prime119876119876119954119954prime120582120582
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954prime 119890119890119899119899120572120572120582120582 (119954119954prime)
lowast120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890minus119894119894119954119954prime∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119890119899119899prime120572120572prime120582120582prime (119954119954prime)
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582prime119876119876119954119954120582120582
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954
119890119890119899119899120572120572120582120582 (119954119954)lowast120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890minus119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119890119899119899prime120572120572prime120582120582prime (119954119954)
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger dagger
3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954
119890119899119899prime120572120572prime120582120582prime (119954119954)
lowast120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119890119890119899119899120572120572120582120582 (119954119954)119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
lowast
=
⎩⎨
⎧12 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954⎭⎬
⎫dagger
= 11988111988112dagger
Define
1198671198671 = 11987911987911 + 119881119881111198671198672 = 11987911987922 + 1198811198812211986711986712 = 11988111988112 + 11988111988121
(315)
we finally get the expressions of 1198671198671 1198671198672 and 11986711986712 in Eq (31) as follows
⎩⎪⎨
⎪⎧ 1198671198671 = sum sum ℏ120596120596119954119954120582120582 119886119886119954119954120582120582
dagger 119886119886119954119954120582120582 + 1231199031199032
120582120582=1119954119954
1198671198672 = sum sum ℏ120596120596119954119954120582120582prime 119886119886119954119954120582120582primedagger 119886119886119954119954120582120582prime + 1
23119903119903
120582120582prime=1+31199031199032119954119954
11986711986712 = 12sum sum sum 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903120582120582prime=31199031199032 +1
31199031199032120582120582=1 + h c 119954119954
(316)
Here hc means the Hermitian conjugate Since the indices of 119886119886119954119954120582120582 119886119886119954119954120582120582prime and 119876119876119954119954120582120582 119876119876119954119954120582120582prime belong to
the two system sections we can define an abbreviated notation 119886119886119954119954120582120582 and 119876119876119954119954120582120582 using the index 120582120582 to
identify which subsystem they belong to In this case the Hamiltonian of the coupled system can be
10
simplified as follows
119867119867 = 1198671198670 + 119867119867119862119862 (317)
where
1198671198670 = sum sum ℏ120596120596119954119954120582120582 119886119886119954119954120582120582
dagger 119886119886119954119954120582120582 + 123119903119903
120582120582=1119954119954
119867119867119862119862 = 12sum sum sum 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903120582120582prime=31199031199032 +1
31199031199032120582120582=1 + ℎ 119888119888 119954119954
(318)
4 Greenrsquos function 41 Dynamics of the ladder operators
In order to describe the dynamics of the ladder operators appearing in the Hamiltonian of Eq (317)
we express them in the Heisenberg picture as follows
119886119886119901119901(120591120591) = 119890119890119894119894ℏ119867119867119905119905119886119886119901119901119890119890
minus119894119894ℏ119867119867119905119905 = 119890119890
119867119867120591120591ℏ 119886119886119901119901119890119890
minus119867119867120591120591ℏ (41)
where we define the imaginary time 120591120591 equiv 119894119894119905119905 and introduce the notation 119953119953 equiv (119954119954 120582120582) and 119953119953 equiv (minus119954119954 120582120582)
The corresponding equation of motion for the ladder operators is given by
ℏ part119886119886119953119953(120591120591)
120597120597120591120591= 119867119867119886119886119953119953(120591120591) (42)
For the uncoupled system (119867119867119862119862 = 0) this gives
ℏpart119886119886119953119953(120591120591)120597120597120591120591
= 1198671198670119886119886119953119953(120591120591) = 1198671198670 1198901198901198671198670120591120591ℏ 119886119886119953119953119890119890
minus1198671198670120591120591ℏ = 1198671198670119890119890
1198671198670120591120591ℏ 119886119886119953119953119890119890
minus1198671198670120591120591ℏ minus 119890119890
1198671198670120591120591ℏ 119886119886119953119953119890119890
minus1198671198670120591120591ℏ 1198671198670
= 1198901198901198671198670120591120591ℏ 1198671198670119886119886119953119953 minus 1198861198861199531199531198671198670119890119890
minus1198671198670120591120591ℏ = 119890119890
1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890
minus1198671198670120591120591ℏ
ie
ℏ part119886119886119953119953(120591120591)
120597120597120591120591= 119890119890
1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890
minus1198671198670120591120591ℏ (43)
The commutator on the right-hand-side of Eq (43) can be evaluated as follows
1198671198670119886119886119953119953prime = sum ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953 + 1
2119953119953 119886119886119953119953prime = sum ℏ120596120596119953119953119886119886119953119953
dagger119886119886119953119953119886119886119953119953prime119953119953 = sum ℏ120596120596119953119953119886119886119953119953dagger119886119886119953119953119886119886119953119953prime minus 119886119886119953119953prime119886119886119953119953
dagger119886119886119953119953119953119953 =
sum ℏ120596120596119953119953119886119886119953119953dagger119886119886119953119953119886119886119953119953prime minus 120575120575119953119953prime119953119953 + 119886119886119953119953
dagger119886119886119953119953prime119886119886119953119953119953119953 = minusℏ120596120596119953119953prime119886119886119953119953prime + sum ℏ120596120596119953119953119886119886119953119953dagger119886119886119953119953119886119886119953119953prime minus 119886119886119953119953
dagger119886119886119953119953119886119886119953119953prime119953119953 = minusℏ120596120596119953119953prime119886119886119953119953prime
ie
1198671198670119886119886119953119953prime = minusℏ120596120596119953119953prime119886119886119953119953prime (44)
Therefore we have
ℏ part119886119886119953119953(120591120591)
120597120597120591120591= 119890119890
1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890
minus1198671198670120591120591ℏ = minusℏ120596120596119953119953119890119890
1198671198670120591120591ℏ 119886119886119953119953119890119890
minus1198671198670120591120591ℏ = minusℏ120596120596119953119953119886119886119953119953(120591120591) (45)
11
The solution of Eq (45) is 119886119886119953119953(120591120591) = 119886119886119953119953(0)119890119890minus120596120596119953119953120591120591 = 119886119886119953119953119890119890minus120596120596119953119953120591120591 For 119886119886119953119953dagger(120591120591) we have
1198671198670119886119886119953119953primedagger (0) = 1198671198670119886119886119953119953prime
dagger = ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953 +
12
119953119953
119886119886119953119953primedagger = ℏ120596120596119953119953119886119886119953119953
dagger119886119886119953119953119886119886119953119953primedagger
119953119953
= ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953119886119886119953119953prime
dagger minus 119886119886119953119953primedagger 119886119886119953119953
dagger119886119886119953119953119953119953
= ℏ120596120596119953119953 119886119886119953119953dagger 120575120575119953119953119953119953prime + 119886119886119953119953prime
dagger 119886119886119953119953 minus 119886119886119953119953primedagger 119886119886119953119953
dagger119886119886119953119953119953119953
= ℏ120596120596119953119953prime119886119886119953119953primedagger + ℏ120596120596119953119953 119886119886119953119953
dagger119886119886119953119953primedagger 119886119886119953119953 minus 119886119886119953119953prime
dagger 119886119886119953119953dagger119886119886119953119953
119953119953
= ℏ120596120596119953119953prime119886119886119953119953primedagger
In summary we have
119886119886119953119953(120591120591) = 119886119886119953119953(0)119890119890minus120596120596119953119953120591120591 = 119886119886119953119953119890119890minus120596120596119953119953120591120591
119886119886119953119953dagger(120591120591) = 119886119886119953119953
dagger(0)119890119890120596120596119953119953120591120591 = 119886119886119953119953dagger119890119890120596120596119953119953120591120591 (46)
42 Greenrsquos function
To describe thermal transport properties between different system sections we use the formalism of
thermal (or imaginary time) phonon Greenrsquos function [2] To this end we define the thermal Greenrsquos
function for phonons as
119866119866119953119953119953119953prime(120591120591 120591120591prime) = minuslang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime)rang = minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime) (47)
where 119879119879120591120591 is the time ordering operator and 120588120588119867119867 is the statistical operator for the grand canonical
ensemble (note that the chemical potential for phonons is zero)
120588120588119867119867 = 119890119890minus120573120573119867119867119885119885119867119867 (48)
where the partition function is given by 119885119885119867119867 equiv Tr119890119890minus120573120573119867119867 and 120573120573 = 1119896119896119861119861119879119879 with 119896119896119861119861 being
Boltzmannrsquos constant and 119879119879 the temperature For time independent Hamiltonians 119866119866119953119953119953119953prime(120591120591 120591120591prime)
depends only on 120591120591 minus 120591120591prime ie
119866119866119953119953119953119953prime(120591120591 120591120591prime) = 119866119866119953119953119953119953prime(120591120591 minus 120591120591prime 0) (49)
To show this we shall first assume that 120591120591 gt 120591120591prime such that
119866119866119953119953119953119953prime(120591120591 120591120591prime) = minuslang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime)rang = minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime) = minusTr 120588120588119867119867119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime)
= minusTr 120588120588119867119867119890119890119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591ℏ 119890119890
119867119867120591120591primeℏ 119886119886119953119953prime
dagger 119890119890minus119867119867120591120591primeℏ = minusTr 120588120588119867119867119890119890
minus119867119867120591120591primeℏ 119890119890
119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591minus120591120591prime
ℏ 119886119886119953119953primedagger
= minusTr 120588120588119867119867119890119890119867119867120591120591minus120591120591prime
ℏ 119886119886119953119953119890119890minus119867119867120591120591minus120591120591prime
ℏ 119886119886119953119953primedagger = minusTr 120588120588119867119867119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953prime
dagger (0)
= minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953primedagger (0) = minus lang119879119879120591120591119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953prime
dagger (0)rang = 119866119866119953119953119953119953prime(120591120591 minus 120591120591prime 0)
Where we have used the commutativity of 120588120588119867119867 and 119890119890plusmn119867119867120591120591prime
ℏ and the invariance of the trace operation
12
towards cyclic permutations Similarly for 120591120591 lt 120591120591prime we have
119866119866119953119953119953119953prime(120591120591 120591120591prime) = minuslang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime)rang = minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime) = minusTr 120588120588119867119867119886119886119953119953primedagger (120591120591prime)119886119886119953119953(120591120591)
= minusTr 120588120588119867119867119890119890119867119867120591120591primeℏ 119886119886119953119953prime
dagger 119890119890minus119867119867120591120591primeℏ 119890119890
119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591ℏ = minusTr 120588120588119867119867119886119886119953119953prime
dagger 119890119890119867119867120591120591minus120591120591prime
ℏ 119886119886119953119953119890119890minus119867119867120591120591ℏ 119890119890
119867119867120591120591primeℏ
= minusTr 120588120588119867119867119886119886119953119953primedagger 119890119890
119867119867120591120591minus120591120591primeℏ 119886119886119953119953119890119890
minus119867119867120591120591minus120591120591prime
ℏ = minusTr 120588120588119867119867119886119886119953119953primedagger (0)119886119886119953119953(120591120591 minus 120591120591prime)
= minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953primedagger (0)119886119886119953119953(120591120591 minus 120591120591prime) = minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953prime
dagger (0)
= minuslang119879119879120591120591119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953primedagger (0)rang = 119866119866119953119953119953119953prime(120591120591 minus 120591120591prime 0)
For simplicity we will introduce the notation 119866119866119953119953119953119953prime(120591120591 0) equiv 119866119866119953119953119953119953prime(120591120591) We further note that when using
imaginary time 119866119866119953119953119953119953prime(120591120591) is a periodic functions in the domain [minus120573120573ℏ120573120573ℏ] with a period of 120573120573ℏ (see
Page 236 of Ref [2])
119866119866119953119953119953119953prime(120591120591 + 120573120573ℏ 0) = 119866119866119953119953119953119953prime(120591120591 0) 120591120591 lt 0119866119866119953119953119953119953prime(120591120591 minus 120573120573ℏ 0) = 119866119866119953119953119953119953prime(120591120591 0) 120591120591 gt 0 (410)
To show this we shall again assume first that minus120573120573ℏ lt 120591120591 lt 0 to write that is
119866119866119953119953119953119953prime(120591120591 + 120573120573ℏ 0) = minus lang119879119879120591120591119886119886119953119953(120591120591 + 120573120573ℏ)119886119886119953119953primedagger (0)rang = minuslang119886119886119953119953(120591120591 + 120573120573ℏ)119886119886119953119953prime
dagger (0)rang
= minusTr 120588120588119867119867119890119890119867119867(120591120591+120573120573ℏ)
ℏ 119886119886119953119953119890119890minus119867119867(120591120591+120573120573ℏ)
ℏ 119886119886119953119953prime = minusTr 120588120588119867119867119890119890120573120573119867119867119890119890
119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591ℏ 119890119890minus120573120573119867119867119886119886119953119953prime
= minusTr120588120588119867119867119890119890120573120573119867119867119886119886119953119953(120591120591)119890119890minus120573120573119867119867119886119886119953119953prime(0) = minus
1119885119885119867119867
Tr119890119890minus120573120573119867119867119890119890120573120573119867119867119886119886119953119953(120591120591)119890119890minus120573120573119867119867119886119886119953119953prime(0)
= minus1119885119885119867119867
Tr119886119886119953119953(120591120591)119890119890minus120573120573119867119867119886119886119953119953prime(0) = minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119886119886119953119953prime(0)119886119886119953119953(120591120591)
= minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime(0) = minusTr120588120588119867119867119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime = 119866119866119953119953119953119953prime(120591120591 0)
Similarly for 120573120573ℏ gt 120591120591 gt 0 we have
119866119866119953119953119953119953prime(120591120591 minus 120573120573ℏ 0) = minus lang119879119879120591120591119886119886119953119953(120591120591 minus 120573120573ℏ)119886119886119953119953primedagger (0)rang = minuslang119886119886119953119953prime
dagger (0)119886119886119953119953(120591120591 minus 120573120573ℏ)rang
= minusTr 120588120588119867119867119886119886119953119953prime119890119890119867119867(120591120591minus120573120573ℏ)
ℏ 119886119886119953119953119890119890minus119867119867(120591120591minus120573120573ℏ)
ℏ = minusTr 120588120588119867119867119886119886119953119953prime119890119890minus120573120573119867119867119890119890
119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591ℏ 119890119890120573120573119867119867
= minusTr120588120588119867119867119886119886119953119953prime(0)119890119890minus120573120573119867119867119886119886119953119953(120591120591)119890119890120573120573119867119867 = minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119886119886119953119953prime(0)119890119890minus120573120573119867119867119886119886119953119953(120591120591)119890119890120573120573119867119867
= minus1119885119885119867119867
Tr119886119886119953119953prime(0)119890119890minus120573120573119867119867119886119886119953119953(120591120591) = minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119886119886119953119953(120591120591)119886119886119953119953prime(0)
= minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime(0) = minusTr120588120588119867119867119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime = 119866119866119953119953119953119953prime(120591120591 0)
13
Therefore 119866119866119953119953119953119953prime(120591120591) can be expanded as a Fourier series in the domain [0120573120573ℏ] as follows
119866119866119953119953119953119953prime(120591120591) = 1120573120573ℏsum 119890119890minus119894119894120596120596119899119899120591120591119866119866119953119953119953119953prime(119894119894120596120596119899119899)infinminusinfin (411)
where 120596120596119899119899 = 2120587120587119899119899120573120573ℏ
and the associated Fourier coefficient is given by
119866119866119953119953119953119953prime(119894119894120596120596119899119899) = int d120591120591119890119890119894119894120596120596119899119899120591120591119866119866119953119953119953119953prime(120591120591)120573120573ℏ0 120596120596119899119899 = 2119899119899120587120587
120573120573ℏ (412)
Having proven the translational time invariance and the periodicity of the Greenrsquos functios we can
now calculate it for the uncoupled system
119866119866119953119953119953119953prime0 (120591120591 120591120591prime) = minuslang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime)rang0 == minus lang119886119886119953119953(0)119890119890minus120596120596119953119953120591120591119886119886119953119953prime
dagger (0)119890119890120596120596119953119953prime120591120591primerang 120591120591 minus 120591120591prime gt 0
minus lang119886119886119953119953primedagger (0)119890119890120596120596119953119953prime120591120591
prime119886119886119953119953(0)119890119890minus120596120596119953119953120591120591rang 120591120591 minus 120591120591prime lt 0
= minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591
primelang119886119886119953119953(0)119886119886119953119953prime
dagger (0)rang 120591120591 minus 120591120591prime gt 0
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang 120591120591 minus 120591120591prime lt 0
= minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591
primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0) + 119886119886119953119953(0)119886119886119953119953primedagger (0)rang 120591120591 minus 120591120591prime gt 0
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang 120591120591 minus 120591120591prime lt 0
= minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591
primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang + 120575120575119953119953119953119953prime 120591120591 minus 120591120591prime gt 0
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang 120591120591 minus 120591120591prime lt 0
ie
119866119866119953119953119953119953prime0 (120591120591 120591120591prime) =
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang + 120575120575119953119953119953119953prime 120591120591 minus 120591120591prime gt 0
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang 120591120591 minus 120591120591prime lt 0 (413)
where we have used Eq (46) for 119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime) and Commutation relation 119886119886119953119953(0)119886119886119953119953prime
dagger (0) = 120575120575119953119953119953119953prime
To calculate lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang we first prove following equation
119890119890119860119860119861119861119890119890minus119860119860 = sum 1119899119899119860(119899119899)119861119861infin
119899119899=0 (414)
where 119860(119896119896)119861119861 equiv 119860 119860(119896119896minus1)119861119861 To prove this identity we define the operator 119891119891(119905119905) = 119890119890119905119905A119861119861119890119890minus119905119905119860119860
and taylor expand it around 119905119905 = 0
119891119891(119905119905) = 119891119891(0) + 119905119905119891119891prime(0) + 1199051199052
2119891119891primeprime(0) + ⋯ = sum 119905119905119899119899
119899119899119889119889119899119899119891119889119889119905119905119899119899
119905119905=0
infin119899119899=0 (415)
The corresponding derivatives are given by
⎩⎪⎨
⎪⎧ 119891119891prime(119905119905) = 119890119890119905119905119860119860119860119861119861119890119890minus119905119905119860119860 minus 119890119890119905119905119860119860119861119861119860119890119890minus119905119905119860119860 = 119890119890119905119905119860119860119860119861119861119890119890minus119905119905119860119860
119891119891primeprime(119905119905) = 119890119890119905119905119860119860119860119860119861119861 minus 119860119861119861119860119890119890minus119905119905119860119860 = 119890119890119905119905119860119860119860(2)119861119861119890119890minus119905119905119860119860
⋮119891119891(119899119899)(119905119905) = 119890119890119905119905119860119860119860(119899119899)119861119861119890119890minus119905119905119860119860
(416)
14
Substituting Eq (416) into Eq (415) we have
119890119890119905119905A119861119861119890119890minus119905119905119860119860 = sum 119905119905119899119899
119899119899119860(119899119899)119861119861infin
119899119899=0 (417)
Itrsquos clear that Eq (413) is a spectial case of Eq (416) with 119905119905 = 1 With this we can proceed as
follows
lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang =
11198851198851198671198670
Tr119890119890minus1205731205731198671198670119886119886119953119953primedagger (0)119886119886119953119953(0) =
11198851198851198671198670
Tr119890119890minus1205731205731198671198670119886119886119953119953primedagger (0)1198901198901205731205731198671198670119890119890minus1205731205731198671198670119886119886119953119953(0)
=11198851198851198671198670
Tr (minus120573120573)119899119899
1198991198991198671198670
(119899119899)119886119886119953119953primedagger (0)
infin
119899119899=0
119890119890minus1205731205731198671198670119886119886119953119953(0)
Note that 1198671198670119886119886119953119953primedagger (0) = ℏ120596120596119953119953prime119886119886119953119953prime
dagger (0) we have
1198671198670(119899119899)119886119886119953119953prime
dagger (0) = ℏ120596120596119953119953prime119899119899119886119886119953119953primedagger (0) (418)
such that
lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang =
11198851198851198671198670
Tr (minus120573120573)119899119899
1198991198991198671198670
(119899119899)119886119886119953119953primedagger (0)
infin
119899119899=0
119890119890minus1205731205731198671198670119886119886119953119953(0)
=11198851198851198671198670
Tr minus120573120573ℏ120596120596119953119953prime
119899119899
119899119899119886119886119953119953primedagger (0)
infin
119899119899=0
119890119890minus1205731205731198671198670119886119886119953119953(0)
= 119890119890minus120573120573ℏ120596120596119953119953prime11198851198851198671198670
Tr119890119890minus1205731205731198671198670119886119886119953119953(0)119886119886119953119953primedagger (0) = 119890119890minus120573120573ℏ120596120596119953119953primelang119886119886119953119953(0)119886119886119953119953prime
dagger (0)rang
= 119890119890minus120573120573ℏ120596120596119953119953primelang119886119886119953119953primedagger (0)119886119886119953119953(0)rang + 120575120575119953119953119953119953prime
therefore we obtain lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang 119890119890120573120573ℏ120596120596119953119953prime minus 1 = 120575120575119953119953119953119953prime For 119953119953prime ne 119953119953 and general 119890119890120573120573ℏ120596120596119953119953prime we have
lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang = 0 For 119953119953prime = 119953119953 we obtain lang119886119886119953119953
dagger(0)119886119886119953119953(0)rang = 119890119890120573120573ℏ120596120596119953119953 minus 1minus1
Hence we have
lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang = 120575120575119953119953prime119953119953
1119890119890120573120573ℏ120596120596119953119953minus1
equiv 120575120575119953119953prime119953119953119899119899119861119861120596120596119953119953 (419)
where 119899119899119861119861120596120596119953119953 is the Bose-Einstein distribution for phonons Substituting Eq (418) in Eq (413)
we obtain
119866119866119953119953119953119953prime0 (120591120591 120591120591prime) =
minus120575120575119953119953119953119953prime1 + 119899119899119861119861120596120596119953119953119890119890minus120596120596119953119953120591120591minus120591120591prime 120591120591 minus 120591120591prime gt 0
minus120575120575119953119953119953119953prime119899119899119861119861120596120596119953119953119890119890minus120596120596119953119953120591120591minus120591120591prime 120591120591 minus 120591120591prime lt 0 (420)
Note that only those Greenrsquos functions of the form 119866119866119953119953119953119953prime0 (120591120591 120591120591prime) = 1198661198661199531199530(120591120591 minus 120591120591prime)120575120575119953119953119953119953prime are non-zero due
to the orthogonality of normal modes Looking at the non-vanishing terms and setting 120591120591prime = 0
without loss of generality we can calculate the Fourier transform of 1198661198661199531199530(120591120591) from Eq (412)
1198661198661199531199530(119894119894120596120596119899119899) = int d1205911205911198901198901198941198941205961205961198991198991205911205911198661198661199531199530(120591120591)120573120573ℏ0 = minusint d120591120591119890119890119894119894120596120596119899119899120591120591120573120573ℏ
0 1 + 119899119899119861119861120596120596119953119953119890119890minus120596120596119953119953120591120591 = minus1+119899119899119861119861120596120596119953119953119894119894120596120596119899119899minus120596120596119953119953
119890119890119894119894120596120596119899119899minus120596120596119953119953120573120573ℏ minus
15
1 = minus1+ 1
119890119890120573120573ℏ120596120596119953119953minus1
119894119894120596120596119899119899minus120596120596119953119953119890119890119894119894
2120587120587119899119899120573120573ℏ 120573120573ℏ119890119890minus120573120573ℏ120596120596119953119953 minus 1 = minus 119890119890120573120573ℏ120596120596119953119953
119894119894120596120596119899119899minus120596120596119953119953
119890119890minus120573120573ℏ120596120596119953119953minus1
119890119890120573120573ℏ120596120596119953119953minus1= minus 1
119894119894120596120596119899119899minus120596120596119953119953
1minus119890119890120573120573ℏ120596120596119953119953
119890119890120573120573ℏ120596120596119953119953minus1= 1
119894119894120596120596119899119899minus120596120596119953119953
ie
1198661198661199531199530(119894119894120596120596119899119899) == 1119894119894120596120596119899119899minus120596120596119953119953
(421)
5 Fermirsquos golden rule 51 Interaction picture
Next we can proceed with calculating the Greenrsquos function of the coupled system To this end we
define the coupling Hamiltonian operator term in the interaction picture as
119867119867119862119862119868119868 (120591120591) = 1198901198901198671198670120591120591ℏ 119867119867119862119862119890119890
minus1198671198670120591120591ℏ (51)
The time evolution of 119867119867119862119862(120591120591) is then given by
ℏ part119867119867119862119862119868119868 (120591120591)120597120597120591120591
= 1198671198670119867119867119862119862119868119868 (120591120591) (52)
Note that the Greenrsquos function defined above was given in the Heisenberg picture To proceed we
need to transform it to the interaction picture
119866119866119953119953119953119953prime(120591120591 0) = minus lang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953primedagger (0)rang = minusTr 120588120588119867119867119879119879120591120591 119890119890
119867119867120591120591ℏ 119886119886119953119953(0)119890119890minus
119867119867120591120591ℏ 119886119886119953119953prime
dagger (0)
= minusTr 120588120588119867119867119879119879120591120591 119890119890119867119867120591120591ℏ 119890119890minus
1198671198670120591120591ℏ 119890119890
1198671198670120591120591ℏ 119886119886119953119953(0)119890119890minus
1198671198670120591120591ℏ 119890119890
1198671198670120591120591ℏ 119890119890minus
119867119867120591120591ℏ 119886119886119953119953prime
dagger (0)
= minusTr 120588120588119867119867119879119879120591120591 119880119880(0 120591120591)119886119886119953119953119868119868 (120591120591)119880119880(120591120591 0)119886119886119953119953primedagger (0)
where
119886119886119953119953119868119868 (120591120591) equiv 1198901198901198671198670120591120591ℏ 119886119886119953119953(0)119890119890minus
1198671198670120591120591ℏ (53)
is the operator in interaction picture and the operator 119880119880 is defined by
119880119880(1205911205911 1205911205912) equiv 11989011989011986711986701205911205911ℏ 119890119890minus
119867119867 (1205911205911minus1205911205912)ℏ 119890119890minus
11986711986701205911205912ℏ (54)
Note that while 119880119880 is not unitary it satisfies the following group property
119880119880(1205911205911 1205911205912)119880119880(1205911205912 1205911205913) = 119880119880(1205911205911 1205911205913) (55)
and the boundary condition 119880119880(1205911205911 1205911205911) = 1 In addition the 120591120591 derivative of 119880119880 is simply
ℏpart119880119880(120591120591 120591120591prime)
120597120597120591120591= 1198671198670119890119890
1198671198670120591120591ℏ 119890119890minus
119867119867120591120591minus120591120591primeℏ minus 119890119890
1198671198670120591120591ℏ 119867119867119890119890minus
119867119867120591120591minus120591120591primeℏ 119890119890minus
1198671198670120591120591primeℏ
= 1198901198901198671198670120591120591ℏ 1198671198670 minus 119867119867119890119890minus
1198671198670120591120591ℏ 119890119890
1198671198670120591120591ℏ 119890119890minus
119867119867120591120591minus120591120591primeℏ 119890119890minus
1198671198670120591120591primeℏ = 119890119890
1198671198670120591120591ℏ minus119867119867119862119862119890119890
minus1198671198670120591120591ℏ 119880119880(120591120591 120591120591prime)
= minus119867119867119862119862119868119868 (120591120591)119880119880(120591120591 120591120591prime)
16
ie
ℏ part119880119880120591120591120591120591prime120597120597120591120591
= minus119867119867119862119862119868119868 (120591120591)119880119880(120591120591 120591120591prime) (56)
The solution of Eq (56) is (see page 235 of Ref [2])
119880119880(120591120591 120591120591prime) = summinus1ℏ
119899119899
119899119899 int 1198891198891205911205911120591120591120591120591prime ⋯int 119889119889120591120591119899119899
120591120591120591120591prime 119879119879120591120591119867119867119862119862119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119899119899)infin
119899119899=0 (57)
The exact thermal Greens function now may be rewritten in the interaction picture as
119866119866119953119953119953119953prime(120591120591 0) = minus1119885119885119867119867
Tr 119890119890minus120573120573119867119867119879119879120591120591119880119880(0 120591120591)119886119886119953119953119868119868 (120591120591)119880119880(120591120591 0)119886119886119953119953primedagger (0)
= minusTr 119890119890minus120573120573119867119867119879119879120591120591 119880119880(0 120591120591)119886119886119953119953119868119868 (120591120591)119880119880(120591120591 0)119886119886119953119953prime
dagger (0)
Tr119890119890minus120573120573119867119867
= minusTr 119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)119879119879120591120591 119886119886119953119953119868119868 (120591120591)119880119880(0 120591120591)119880119880(120591120591 0)119886119886119953119953prime
dagger (0)
Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)
= minusTr 119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)119879119879120591120591 119886119886119953119953119868119868 (120591120591)119880119880(00)119886119886119953119953prime
dagger (0)
Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)=
= minusTr 119890119890minus1205731205731198671198670119879119879120591120591 119880119880(120573120573ℏ 0)119886119886119953119953119868119868 (120591120591)119886119886119953119953prime
dagger (0)
Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)
Where we used the fact that we are free to change the order of the operators within the time ordering
operation (see pages 241-242 of Ref [2])
52 Wickrsquos theorem
Thus the Greenrsquos function can be expanded as [2]
119866119866119953119953119953119953prime(120591120591 0) = minussum 1
119898119898minus1ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)119886119886119953119953119868119868 (120591120591)119886119886
119953119953primedagger (0)rang0infin
119898119898=0
sum 1119898119898minus
1ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)rang0infin
119898119898=0 (58)
where lang⋯ rang0 represents the ensemble average with respect to the non-interacting basis
Tr119890119890minus1205731205731198671198670(⋯ ) Or explicitly
119866119866119953119953119953119953prime(120591120591 0) = minus119866119866119953119953119953119953prime0 (120591120591)minus1ℏint 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953primedagger (0)rang0+
12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0+⋯
1minus1ℏint 1198891198891205911205911120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)rang0+12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0+⋯
(59)
For consiceness we introduce the following notation
119863119863119898119898 equiv 1119898119898minus 1
ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)rang0 (510)
To simplify the calculation in Eq (58) we adopt Wickrsquos theorem (see pages 237-241 of Ref
[2])which can be expressed as follows
lang119879119879120591120591119860119861119861119862119863119863 ⋯119884119884119885rang0 = lang119879119879120591120591119860119861119861rang0lang119879119879120591120591119862119863119863rang0⋯ lang119879119879120591120591119884119884119885rang0 + lang119879119879120591120591119860119862rang0lang119879119879120591120591119861119861119863119863rang0⋯ lang119879119879120591120591119883119883119885rang0 + ⋯ (511)
17
Here 119860119861119861 hellip 119884119884 119885 represent 119886119886119953119953(120591120591) or 119886119886119953119953dagger(120591120591) In Eq (511) each term corresponds to a particular
pairing of the operators 119860119861119861119862119863119863 ⋯119884119884119885 and all possible pairings are taken into account Here the only
non-vanishing propagators will have the form [see Eq (420)]
lang119879119879120591120591 119886119886119953119953primedagger (120591120591prime)119886119886119953119953(120591120591)rang0 = minuslang119879119879120591120591 119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime)rang0 = 119866119866119953119953119953119953prime0 (120591120591 120591120591prime) = 1198661198661199531199530(120591120591 minus 120591120591prime)120575120575119953119953119953119953prime (512)
Therefore the second term in the numerator of Eq (59) can be calculated as
1198681198682 = minus1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0
= minus1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 lang119879119879120591120591119886119886119953119953prime
dagger (0)119886119886119953119953(120591120591)rang0 minus1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
= 119866119866119953119953119953119953prime0 (120591120591 0) minus
1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 minus
1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
The first term in above equation is called disconnected part since the pairing is performed separately
on 119867119867119862119862(1205911205911) and 119886119886119953119953(120591120591)119886119886119953119953primedagger (0) All other terms have pairs that mix creation and annihilation operators
of the Hamiltonian with 119886119886119953119953(120591120591) or 119886119886119953119953primedagger (0) and are said to have connected party For simplicity we
include all these terms in the notation lang119879119879120591120591119867119867119862119862(1205911205911)119886119886119953119953(120591120591)119886119886119953119953primedagger (0)rang0119888119888 Furthermore we define 119866119866119953119953119953119953prime
(1) (120591120591) equiv
minus1ℏ int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862(1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 Then we have
1198681198683 = minus1ℏint 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 = 119866119866119953119953119953119953prime0 (120591120591 0)1198631198631 + 119866119866119953119953119953119953prime
(1) (120591120591) (513)
Using this method the third term in the numerator of Eq (59) can be calculated as
1198681198683 = 12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 = 12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0 lang119879119879120591120591119886119886119953119953prime
dagger (0)119886119886119953119953(120591120591)rang0 +
12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 + 12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 +
12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 = 119866119866119953119953119953119953prime0 (120591120591 0) 1
2ℏ2 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0 +
12ℏ2 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 + 12ℏ2 int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 +
12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
Here the last term is denoted as 119866119866119953119953119953119953prime(2) (120591120591)
Using Eq (513) the second and third terms on the right hand above equation can be simplified as
follows
18
12ℏ2
1198891198891205911205912120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 +1
2ℏ2 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
=12 minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0
minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
+12 minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0
minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
=12 minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0
minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
+12 minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0
minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
= minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0
minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 = 1198631198631119866119866119953119953119953119953prime(1) (120591120591)
Where we performed the following integration variables interchange 1205911205911 ⟷ 1205911205912 Thus we have
1198681198683 = 119866119866119953119953119953119953prime(2) (120591120591) + 119866119866119953119953119953119953prime
(1) (120591120591)1198631198631 + 119866119866119953119953119953119953prime0 (120591120591 0)1198631198632 (514)
Similarity we can calculate the fourth term of Eq (59) as
1198681198684 = minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 =
minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0 lang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 + 3 times
minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 + 3 times
minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 +
minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 =
119866119866119953119953119953119953prime0 (120591120591 0) minus1
6ℏ3 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0+
12ℏ2 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0
minus1ℏ int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888+
12ℏ2 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 minus1ℏ int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0+ 119866119866119953119953119953119953prime
(3) (120591120591) = 119866119866119953119953119953119953prime(3) (120591120591) +
119866119866119953119953119953119953prime(2) (120591120591)1198631198631 + 119866119866119953119953119953119953prime
(1) (120591120591)1198631198632 + 119866119866119953119953119953119953prime0 (120591120591 0)1198631198633
Higher order terms can be treated in the same manner (see pages 95-96 of Ref [2]) Then when
substituting 1198681198682 1198681198683 1198681198684 and all higher order terms into Eq (59) we can simplify the numerator as
follows
119866119866119953119953119953119953prime0 (120591120591) minus
1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0
+1
2ℏ2 1198891198891205911205911120573120573ℏ
0 1198891198891205911205912120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 + ⋯
= 119866119866119953119953119953119953prime0 (120591120591) + 119866119866119953119953119953119953prime
(1) (120591120591) + 119866119866119953119953119953119953prime(2) (120591120591) + ⋯ (1 + 1198631198631 + 1198631198632 + ⋯ )
Noting that the expression (1 + 1198631198631 + 1198631198632 + ⋯ ) is exactly canceled with the denominator the
Greenrsquos function can be simplified as
19
119866119866119953119953119953119953prime(120591120591) = 119866119866119953119953119953119953prime0 (120591120591) + 119866119866119953119953119953119953prime
(1) (120591120591) + 119866119866119953119953119953119953prime(2) (120591120591) + ⋯ (515)
where
119866119866119953119953119953119953prime
(1) (120591120591) = minus1ℏ int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888
119866119866119953119953119953119953prime(2) (120591120591) = 1
2ℏ2 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888 (516)
53 Calculation of Greenrsquos function
531 First order appriximation In Eq (516) we go back to the full notation 119886119886119954119954120582120582 to distinguish phonons of different branches
then 119866119866119953119953119953119953prime(1) (120591120591) is calculated as
119866119866119953119953119953119953prime(1) (120591120591) = minus
1ℏ
12 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591 119881119881119895119895119895119895prime(119948119948)119876119876119948119948119895119895(1205911205911)119876119876119948119948119895119895prime
dagger (1205911205911)3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ 119881119881119895119895119895119895primelowast (119948119948)119876119876119948119948119895119895prime(1205911205911)119876119876119948119948119895119895
dagger (1205911205911)3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= minus12ℏ
1198891198891205911205911120573120573ℏ
0lang119879119879120591120591
ℏ119881119881119895119895119895119895prime(119948119948)2120596120596119948119948119895119895120596120596119948119948119895119895prime
119886119886119948119948119895119895(1205911205911) + 119886119886minus119948119948119895119895dagger (1205911205911) 119886119886minus119948119948119895119895prime(1205911205911) + 119886119886119948119948119895119895prime
dagger (1205911205911)3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ ℏ119881119881119895119895119895119895prime
lowast (119948119948)
2120596120596119948119948119895119895120596120596119948119948119895119895prime119886119886119948119948119895119895prime(1205911205911) + 119886119886minus119948119948119895119895prime
dagger (1205911205911) 119886119886minus119948119948119895119895(1205911205911) + 119886119886119948119948119895119895dagger (1205911205911)
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
According to Wickrsquos theorem [2] the terms that contain 119886119886119948119948119895119895119886119886minus119948119948119895119895prime and 119886119886minus119948119948119895119895dagger 119886119886119948119948119895119895prime
dagger are equal to zero
thus the first term of above equation are calculated as
11989211989211 = minus14
119881119881119895119895119895119895prime(119948119948)
120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119886119886minus119948119948119895119895
dagger (1205911205911)119886119886minus119948119948119895119895prime(1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ lang119879119879120591120591119886119886119948119948119895119895(1205911205911)119886119886119948119948119895119895primedagger (1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888
= minus14
119881119881119895119895119895119895prime(119948119948)
120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119886119886minus119948119948119895119895
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus119948119948119895119895prime(1205911205911)rang0
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ lang119879119879120591120591119886119886119948119948119895119895primedagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)119886119886119948119948119895119895(1205911205911)rang0
= minus14
119881119881119895119895119895119895prime(119948119948)
120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
01198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954minus119948119948120575120575120582120582119895119895119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119954119954primeminus119948119948120575120575120582120582prime119895119895prime3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ 1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119948119948120575120575120582120582119895119895prime119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119954119954prime119948119948120575120575120582120582prime119895119895
20
Simplification of above equation gives
11989211989211 =
⎩⎪⎨
⎪⎧minus119881119881120582120582120582120582prime(minus119954119954)
4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954120582120582
0 (120591120591 1205911205911)119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903
2lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582(119954119954)
4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954120582120582
0 (120591120591 1205911205911)119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582prime le 3119903119903
2lt 120582120582 le 3119903119903
0 else
(517)
Similarity the second term of 119866119866119953119953119953119953prime(1) (120591120591) is calculated as
11989211989212 = minus14
minus119881119881119895119895119895119895primelowast (119948119948)
4120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119886119886119948119948119895119895prime(1205911205911)119886119886119948119948119895119895
dagger (1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ lang119879119879120591120591119886119886minus119948119948119895119895primedagger (1205911205911)119886119886minus119948119948119895119895(1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888
= minus14
minus119881119881119895119895119895119895primelowast (119948119948)
4120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)119886119886119948119948119895119895prime(1205911205911)rang0 lang119879119879120591120591119886119886119948119948119895119895dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus119948119948119895119895(1205911205911)rang0 lang119879119879120591120591119886119886minus119948119948119895119895prime
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
ie
11989211989212 =
⎩⎪⎨
⎪⎧
minus119881119881120582120582120582120582primelowast (119954119954)
4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954prime120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903
2lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582lowast (minus119954119954)
4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954prime120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119954119954prime 1 le 120582120582prime le 3119903119903
2lt 120582120582 le 3119903119903
0 else
(518)
Note that since 119881119881120582120582120582120582primelowast (119954119954) = 119881119881120582120582120582120582prime(minus119954119954) [Eq (314)] the first order approximation of Greenrsquos function
can be written as
119866119866119954119954120582120582119954119954prime120582120582prime(1) (120591120591) =
⎩⎪⎨
⎪⎧minus119881119881
120582120582120582120582prime(minus119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ
0 1198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582 le3119903119903
2lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582
(119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ
0 1198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582prime le3119903119903
2lt 120582120582 le 3119903119903
0 else
(519)
To derive the Greenrsquos function in frequency domain we use the expression for Fourier series then
we have
1198891198891205911205911120573120573ℏ
01198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0) = 1198891198891205911205911120573120573ℏ
0
1120573120573ℏ
119890119890minus119894119894120596120596119899119899(120591120591minus1205911205911)1198661198661199541199541205821205820 (119894119894120596120596119899119899)
119899119899
1120573120573ℏ
119890119890minus119894119894120596120596119899119899prime1205911205911119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899prime)
119899119899prime
=1
(120573120573ℏ)2119890119890minus119894119894120596120596119899119899120591120591 1198891198891205911205911120573120573ℏ
0119890119890119894119894120596120596119899119899minus120596120596119899119899prime1205911205911 119866119866119954119954120582120582
0 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899prime)
119899119899119899119899prime=
=1
(120573120573ℏ)2119890119890minus119894119894120596120596119899119899120591120591[120573120573ℏ120575120575119899119899119899119899prime]1198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899prime)119899119899119899119899prime
=1120573120573ℏ
119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899)119899119899
According to the definition of Fourier series [see Eq (412)] we get the Fourier coefficient of
21
119866119866119954119954120582120582119954119954prime120582120582prime(1) (120591120591)
119866119866119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) =
⎩⎪⎨
⎪⎧minus119881119881120582120582120582120582prime(minus119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582prime1198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime 1 le 120582120582 le 31199031199032
lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582(119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582prime1198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime 1 le 120582120582prime le 31199031199032
lt 120582120582 le 3119903119903
0 else
(520)
where 119866119866119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) = 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) times Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) times 119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899) and Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) is defined as
Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) =
⎩⎪⎨
⎪⎧minus119881119881120582120582120582120582prime(minus119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582prime120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903
2lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582(119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582prime120575120575119954119954119954119954prime 1 le 120582120582prime le 3119903119903
2lt 120582120582 le 3119903119903
0 else
(521)
532 second order approximation The second order approximation of the Greenrsquos function in Eq (516) 119866119866119954119954120582120582119954119954prime120582120582prime
(2) (120591120591) is calculated as
119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) =
18ℏ2 1198891198891205911205911
120573120573ℏ
0 1198891198891205911205912120573120573ℏ
0lang119879119879120591120591
⎣⎢⎢⎡ 11988111988111989511989511198951198951prime(119948119948120783120783)1198761198761199481199481207831207831198951198951(1205911205911)1198761198761199481199481207831207831198951198951prime
dagger (1205911205911)3119903119903
1198951198951prime=31199031199032 +1
31199031199032
1198951198951=1119948119948120783120783
+ 11988111988111989411989411198941198941primelowast (119948119948120783120783)1198761198761199481199481207831207831198941198941prime(1205911205911)1198761198761199481199481207831207831198941198941
dagger (1205911205911)3119903119903
1198941198941prime=31199031199032 +1
31199031199032
1198941198941=1119948119948120783120783 ⎦⎥⎥⎤
⎣⎢⎢⎡ 11988111988111989511989521198951198952prime(119948119948120784120784)1198761198761199481199481207841207841198951198952(1205911205912)1198761198761199481199481207841207841198951198952prime
dagger (1205911205912)3119903119903
1198951198952prime=31199031199032 +1
31199031199032
1198951198952=1119948119948120784120784
+ 11988111988111989411989421198941198942primelowast (119948119948120784120784)1198761198761199481199481207841207841198941198942prime(1205911205912)1198761198761199481199481207841207841198941198942
dagger (1205911205912)3119903119903
1198941198942prime=31199031199032 +1
31199031199032
1198941198942=1119948119948120784120784 ⎦⎥⎥⎤119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888 = 1198921198921 + 1198921198922 + 1198921198923 + 1198921198924
where
1198921198921 = 18ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0
sum sum 11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989511989521198951198952prime(119948119948120784120784)1le1198951198952le
31199031199032 lt1198951198952
primele311990311990311994811994812078412078411989511989521198951198952prime
1le1198951198951le31199031199032 lt1198951198951
primele311990311990311994811994812078312078311989511989511198951198951prime
times lang1198791198791205911205911198761198761199481199481207831207831198951198951(1205911205911)1198761198761199481199481207831207831198951198951primedagger (1205911205911)1198761198761199481199481207841207841198951198952(1205911205912)1198761198761199481199481207841207841198951198952prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(522)
1198921198922 = 18ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0
sum sum 11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942primelowast (119948119948120784120784)
1le1198941198942le31199031199032 lt1198941198942
primele311990311990311994811994812078412078411989411989421198941198942prime
1le1198951198951le31199031199032 lt1198951198951
primele311990311990311994811994812078312078311989511989511198951198951prime
times lang1198791198791205911205911198761198761199481199481207831207831198951198951(1205911205911)1198761198761199481199481207831207831198951198951primedagger (1205911205911)1198761198761199481199481207841207841198941198942prime (1205911205912)1198761198761199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(523)
1198921198923 = 18ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0
sum sum 11988111988111989411989411198941198941primelowast (119948119948120783120783)11988111988111989511989521198951198952prime(119948119948120784120784)
1le1198951198952le31199031199032 lt1198951198952
primele311990311990311994811994812078412078411989511989521198951198952prime
1le1198941198941le31199031199032 lt1198941198941
primele311990311990311994811994812078312078311989411989411198941198941prime
times lang1198791198791205911205911198761198761199481199481207831207831198941198941prime (1205911205911)1198761198761199481199481207831207831198941198941dagger (1205911205911)1198761198761199481199481207841207841198951198952(1205911205912)1198761198761199481199481207841207841198951198952prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(524)
22
1198921198924 = 18ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0
sum sum 11988111988111989411989411198941198941primelowast (119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)1le1198941198942le
31199031199032 lt1198941198942
primele311990311990311994811994812078412078411989411989421198941198942prime
1le1198941198941le31199031199032 lt1198941198941
primele311990311990311994811994812078312078311989411989411198941198941prime
times lang1198791198791205911205911198761198761199481199481207831207831198941198941prime (1205911205911)1198761198761199481199481207831207831198941198941dagger (1205911205911)1198761198761199481199481207841207841198941198942prime (1205911205912)1198761198761199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(525)
In what follows we will expand the term 1198921198922 The expansion of all other terms follows the same lines
and eventually results in the same expression Expanding the product 11987611987611994811994812078312078311989511989511198761198761199481199481207831207831198951198951primedagger 1198761198761199481199481207841207841198941198942prime1198761198761199481199481207841207841198941198942
dagger we can
rewrite Eq (523) as follows
1198921198922 = 132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)119881119881
11989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032 lt1198941198942
primele311990311990311994811994812078412078411989411989421198941198942prime
1le1198951198951le31199031199032 lt1198951198951
primele311990311990311994811994812078312078311989511989511198951198951prime
times 119868119868 (526)
Where
119868119868 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911) + 119886119886minus1199481199481207831207831198951198951dagger (1205911205911) 119886119886minus1199481199481207831207831198951198951prime(1205911205911) + 1198861198861199481199481207831207831198951198951prime
dagger (1205911205911) 1198861198861199481199481207841207841198941198942prime (1205911205912) + 119886119886minus1199481199481207841207841198941198942primedagger (1205911205912) 119886119886minus1199481199481207841207841198941198942(1205911205912) + 1198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0c
= lang119879119879120591120591 1198861198861199481199481207831207831198951198951119886119886minus1199481199481207831207831198951198951prime + 11988611988611994811994812078312078311989511989511198861198861199481199481207831207831198951198951primedagger + 119886119886minus1199481199481207831207831198951198951
dagger 119886119886minus1199481199481207831207831198951198951prime + 119886119886minus1199481199481207831207831198951198951dagger 1198861198861199481199481207831207831198951198951prime
dagger 12059112059111198861198861199481199481207841207841198941198942prime119886119886minus1199481199481207841207841198941198942 + 1198861198861199481199481207841207841198941198942prime1198861198861199481199481207841207841198941198942
dagger + 119886119886minus1199481199481207841207841198941198942primedagger 119886119886minus1199481199481207841207841198941198942
+ 119886119886minus1199481199481207841207841198941198942primedagger 1198861198861199481199481207841207841198941198942
dagger 1205911205912119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0c
= lang119879119879120591120591 11988611988611994811994812078312078311989511989511198861198861199481199481207831207831198951198951primedagger + 119886119886minus1199481199481207831207831198951198951
dagger 119886119886minus1199481199481207831207831198951198951prime12059112059111198861198861199481199481207841207841198941198942prime1198861198861199481199481207841207841198941198942
dagger + 119886119886minus1199481199481207841207841198941198942primedagger 119886119886minus11994811994812078412078411989411989421205911205912
119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
Where the symbol [hellip ]1205911205911 signifies that the operators in the brackets are given in the interaction
picture The last equality in above equation comes from the fact that the contractions of the product
of the operators are equal to zero when the number of creation and annihilation operators are not the
same (see Wickrsquos theorem in Ref [2]) Using Wickrsquos theorem [2] we are able to calculate the above
equation term by term as follows
⎩⎪⎨
⎪⎧ 1198681198681 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951prime
dagger (1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)1198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888
1198681198682 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
1198681198683 = lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)1198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
1198681198684 = lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(527)
We shall now calculate them term by term
23
1198681198681 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951primedagger (1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)1198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942
dagger (1205911205912)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
+ lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951prime
dagger (1205911205911)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0
+ lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)1198861198861199481199481207841207841198941198942prime(1205911205912)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0119888119888
= 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199481199481207841207841198951198952
0 (1205911205912 1205911205912)12057512057511989411989421198941198942prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582
+ 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 1198661198661199481199481207831207831198951198951
0 (1205911205911 1205911205911)12057512057511989511989511198951198951prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582
+ 1198661198661199481199481207831207831198951198951prime0 (1205911205912 1205911205911)1205751205751199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951prime ∙ 119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582
= 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582
+ 1198661198661199481199481207831207831198951198951prime0 (1205911205912 1205911205911)1205751205751199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951prime ∙ 119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582
The last equality results from the fact that 1198951198951 and 1198951198951prime are indices of different ranges (belonging to
different subsystems) and the same holds for 1198941198942 and 1198941198942prime Similarity
1198681198684 = lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0
= 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 119866119866minus11994811994812078412078411989411989420 (1205911205912 1205911205912)12057512057511989411989421198941198942prime
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 119866119866minus1199481199481207831207831198951198951prime 0 (1205911205911 1205911205912)1205751205751199481199481207831207831199481199481207841207841205751205751198951198951prime1198941198942prime
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205911 1205911205911)12057512057511989511989511198951198951prime
+ 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205912 1205911205911)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942
= 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 119866119866minus1199481199481207831207831198951198951prime 0 (1205911205911 1205911205912)1205751205751199481199481207831207831199481199481207841207841205751205751198951198951prime1198941198942prime
+ 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205912 1205911205911)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942
Where again the last equality results from the fact that 1198951198951 and 1198951198951prime are indices of different ranges
(belonging to different subsystems) and the same holds for 1198941198942 and 1198941198942prime Similarly we obtain 1198681198682 =
1198681198683 = 0 for the same reason as shown below
24
1198681198682 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951prime
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0
+ lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)1198861198861199481199481207831207831198951198951(1205911205911)rang0
+ lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0
= 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582 ∙ 119866119866minus11994811994812078412078411989411989420 (1205911205912 1205911205912)12057512057511989411989421198941198942prime
+ 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)120575120575minus11994811994812078412078411994811994812078312078312057512057511989511989511198941198942prime ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205911)12057512057511989511989511198951198951prime
+ 1198661198661199481199481207831207831198951198951prime0 (1205911205912 1205911205911)120575120575minus11994811994812078412078411994811994812078312078312057512057511989411989421198951198951prime ∙ 119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 = 0
1198681198683 = lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)1198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0
+ lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0
+ lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0
= 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 11986611986611994811994812078412078411989411989420 (1205911205912 1205911205912)12057512057511989411989421198941198942prime
+ 11986611986611994811994812078412078411989411989420 (1205911205911 1205911205912)120575120575minus1199481199481207831207831199481199481207841207841205751205751198951198951prime1198941198942 ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205911 1205911205911)12057512057511989511989511198951198951prime
+ 1198661198661199481199481207841207841198941198942prime0 (1205911205912 1205911205911)120575120575minus1199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951 ∙ 119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 = 0
The two non-vanishing terms (1198681198681 and 1198681198684) produce the following contributions to 1198921198922 of Eq (526)
11989211989221 = 132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime1198681198681 =
132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime1198661198661199481199481207831207831198951198951
0 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙
1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582 + 1
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime1198661198661199481199481207831207831198951198951prime
0 (1205911205912 1205911205911)1205751205751199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951prime ∙
119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 =
⎩⎪⎨
⎪⎧
120575120575119954119954119954119954prime
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast (119954119954prime)
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (1205911205911 1205911205912) ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
120575120575119954119954119954119954prime
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032+1 1198661198661199541199541198951198951prime
0 (1205911205912 1205911205911) ∙ 119866119866119954119954prime120582120582prime0 (1205911205911 0)119866119866119954119954120582120582
0 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032
0 else
and
25
11989211989224 = 132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime1198681198684 =
132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙
119866119866minus1199481199481207831207831198951198951prime 0 (1205911205911 1205911205912)1205751205751199481199481207831207831199481199481207841207841205751205751198951198951prime1198941198942prime + 1
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙
1198661198661199541199541205821205820 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 119866119866minus1199481199481207831207831198951198951
0 (1205911205912 1205911205911)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 =
⎩⎪⎨
⎪⎧120575120575119954119954119954119954prime
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032+1 1198661198661199541199541198951198951prime
0 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime0 (1205911205912 0)119866119866119954119954120582120582
0 (120591120591 1205911205911) 120582120582 120582120582prime isin 1 31199031199032
120575120575119954119954119954119954prime
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast (119954119954prime)
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
0 else
Here we used the following properties 120596120596minus119954119954120582120582 = 120596120596119954119954120582120582 and 1198811198811205821205821198951198951prime(minus119954119954) = 1198811198811205821205821198951198951primelowast (119954119954) [see Eqs (112) and
(314)]
The equivalence of 11989211989221 and 11989211989224 can be seen by changing the integration variables 1205911205912 harr 1205911205911
Substituting 11989211989221 and 11989211989224 into Eq (526) 1198921198922 is given by
1198921198922 =
⎩⎪⎨
⎪⎧
120575120575119954119954119954119954prime
16 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast (119954119954prime)
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime
0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
120575120575119954119954119954119954prime
16 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032+1 1198661198661199541199541198951198951prime
0 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime0 (1205911205911 0)119866119866119954119954120582120582
0 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032
0 else
(528)
By changing the indices and integration variables itrsquos straightforward to show that the other three
terms (119892119892111989211989231198921198924 ) are identical to 1198921198922 thus the final expression of 119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) [Eq (516)) can be
calculated as
119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) = 41198921198922 =
⎩⎪⎨
⎪⎧120575120575119954119954119954119954prime
4 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032
120575120575119954119954119954119954prime
4 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast 119954119954prime
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime
0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 31199031199032
+ 13119903119903
0 else
(529)
To calculate 119866119866119954119954120582120582119954119954prime120582120582prime(2) in frequency space we expand 119866119866119954119954120582120582119954119954prime120582120582prime
(2) (120591120591) in a Fourier series
119866119866119954119954120582120582119954119954prime120582120582prime
(2) (120591120591) = 1120573120573ℏsum 119890119890minus119894119894120596120596119899119899120591120591119866119866119954119954120582120582119954119954prime120582120582prime
(2) (119894119894120596120596119899119899)119899119899
119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = int d120591120591119890119890119894119894120596120596119899119899120591120591119866119866119954119954120582120582119954119954prime120582120582prime
(2) (120591120591)120573120573ℏ0
(530)
Using Eq (421) we get for 120582120582 120582120582prime isin 1 31199031199032
26
119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) =
120575120575119954119954119954119954prime4
1198891198891205911205911120573120573ℏ
0 1198891198891205911205912120573120573ℏ
0
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1198661198661199541199541198951198951prime0 (1205911205912 1205911205911)119866119866119954119954120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912)
=120575120575119954119954119954119954prime
4
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1198891198891205911205911120573120573ℏ
0 1198891198891205911205912120573120573ℏ
0
1120573120573ℏ
119890119890minus1198941198941205961205961198991198992(1205911205912minus1205911205911)1198661198661199541199541198951198951prime0 1198941198941205961205961198991198992
1198991198992
∙1120573120573ℏ
119890119890minus11989411989412059612059611989911989911205911205911119866119866119954119954120582120582prime0 1198941198941205961205961198991198991
1198991198991
∙1120573120573ℏ
119890119890minus119894119894120596120596119899119899(120591120591minus1205911205912)1198661198661199541199541205821205820 (119894119894120596120596119899119899)
119899119899
=120575120575119954119954119954119954prime
4
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1120573120573ℏ
119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954120582120582prime
0 11989411989412059612059611989911989911198661198661199541199541198951198951prime0 1198941198941205961205961198991198992
11989911989911989911989911198991198992⎩⎪⎨
⎪⎧ 1120573120573ℏ
1198891198891205911205911120573120573ℏ
0119890119890minus1198941198941205961205961198991198991minus12059612059611989911989921205911205911
times1120573120573ℏ
1198891198891205911205912120573120573ℏ
0119890119890minus1198941198941205961205961198991198992minus1205961205961198991198991205911205912
⎭⎪⎬
⎪⎫
=120575120575119954119954119954119954prime
4
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1120573120573ℏ
119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)
11989911989911989911989911198991198992
119866119866119954119954120582120582prime0 11989411989412059612059611989911989911198661198661199541199541198951198951prime
0 1198941198941205961205961198991198992120575120575119899119899111989911989921205751205751198991198991198991198992
=120575120575119954119954119954119954prime120573120573ℏ
119890119890minus119894119894120596120596119899119899120591120591
119899119899
1198661198661199541199541205821205820 (119894119894120596120596119899119899)
14
119881119881120582120582prime1198951198951prime(119954119954
prime)1198811198811205821205821198951198951primelowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899)119866119866119954119954120582120582prime
0 (119894119894120596120596119899119899)
Comparing above expression with Eq (530) the Fourier transform of 119866119866119954119954120582120582119954119954120582120582prime(2) (120591120591) reads as
119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) 120575120575119954119954119954119954prime
4sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 119866119866119954119954120582120582prime
0 (119894119894120596120596119899119899) (531)
where we have introduced the notation
Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) equiv
120575120575119954119954119954119954prime
4sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) (532)
and 119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954120582120582prime(2) (119894119894120596120596119899119899) ∙ 119866119866119954119954120582120582prime
0 (119894119894120596120596119899119899) Similarly for 120582120582 120582120582prime isin 31199031199032
+ 13119903119903 we
have
Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) =
120575120575119954119954119954119954prime
4sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast 119954119954prime
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (119894119894120596120596119899119899) (533)
and all together we have for Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899)
Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) =
⎩⎪⎪⎨
⎪⎪⎧120575120575119954119954119954119954prime
4sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 1 3119903119903
2
120575120575119954119954119954119954prime
4sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast 119954119954prime
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
0 else
(534)
27
533 Dysonrsquos equation Substituting Eqs (521) and (534) into Eq (516) and performing a Fourier transform we have
119866119866119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) = 119866119866119954119954120582120582119954119954prime120582120582prime0 (119894119894120596120596119899119899) + 119866119866119954119954120582120582119954119954prime120582120582prime
(1) (119894119894120596120596119899119899) + 119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) + ⋯
= 1198661198661199541199541205821205820 (119894119894120596120596119899119899)120575120575119954119954119954119954prime120575120575120582120582120582120582prime + 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) ∙ 119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899) + 1198661198661199541199541205821205820 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime
(2) (119894119894120596120596119899119899)
∙ 119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899) + ⋯ = 119866119866119954119954120582120582
0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime120575120575120582120582120582120582prime + 1198661198661199541199541205821205820 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) ∙ 119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899)
or
119866119866119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) = 1198661198661199541199541205821205820 (119894119894120596120596119899119899)120575120575119954119954119954119954prime120575120575120582120582120582120582prime + 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) ∙ 119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899) (535)
Eq (535) is the so-called Dysonrsquos equation (see Ref [2]) where
Σ119954119954120582120582119954119954120582120582prime(119894119894120596120596119899119899) = Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) + Σ119954119954120582120582119954119954prime120582120582prime
(2) (119894119894120596120596119899119899) + ⋯
is called self-energy and Σ119954119954120582120582119954119954120582120582prime(119899119899) (119894119894120596120596119899119899)119899119899 = 12⋯ is the self-energy of nth order approximation Up
to the second order approximation the self-energy is written as
Σ119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) =
⎩⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎧ minus
120575120575119954119954119954119954prime
2
119881119881120582120582120582120582primelowast (119954119954)
120596120596119954119954120582120582120596120596119954119954120582120582prime 1 le 120582120582 le 3119903119903
2lt 120582120582prime le 3119903119903
minus120575120575119954119954119954119954prime
2
119881119881120582120582prime120582120582(119954119954)
120596120596119954119954120582120582120596120596119954119954120582120582prime 1 le 120582120582prime le 3119903119903
2lt 120582120582 le 3119903119903
120575120575119954119954119954119954prime
4sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 1 3119903119903
2
120575120575119954119954119954119954prime
4sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast 119954119954prime
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
(536)
54 Fermirsquos golden Rule
To get the Fermirsquos golden Rule we need to calculate the retarded Greenrsquos function which can be
calculated from 119866119866119954119954120582120582119954119954120582120582prime(119894119894120596120596119899119899) by analytic continuation to the real axis via 119894119894120596120596119899119899 rarr 120596120596 + 119894119894119894119894 with an
infinitesimal positive 119894119894 [12]
119866119866119954119954120582120582119954119954120582120582prime119903119903 (120596120596) equiv 119866119866119954119954120582120582119954119954120582120582prime(119894119894120596120596119899119899 rarr 120596120596 + 119894119894119894119894) 119894119894 rarr 0 (537)
Similarity the retarded self-energy is defined as
Σ119954119954120582120582119954119954120582120582prime119903119903 (120596120596) equiv Σ119954119954120582120582119954119954120582120582prime(119894119894120596120596119899119899 rarr 120596120596 + 119894119894119894119894) 119894119894 rarr 0 (538)
The transition rate of phonons of branch 120582120582 at 119954119954 Γ119954119954120582120582(120596120596) is related to the retarded self energy
Σ119954119954120582120582119903119903 (120596120596) via [13]
Γ119954119954120582120582(120596120596) = minus2ImΣ119954119954120582120582119954119954120582120582119903119903 (120596120596) (539)
Using the expression for 1198661198661199541199541205821205820 (119894119894120596120596119899119899) [Eq (421)] and focused on the second order approximation [13]
we have
28
Σ119954119954120582120582119954119954120582120582119903119903 (119894119894120596120596119899119899) =
⎩⎪⎨
⎪⎧14sum
1198811198811205821205821198951198951prime(119954119954)
2
1205961205961199541199541198951198951prime120596120596119954119954120582120582
31199031199031198951198951prime=
31199031199032 +1
1
120596120596minus1205961205961199541199541198951198951prime +119894119894119894119894
120582120582 isin 1 31199031199032
14sum 1198811198811198951198951120582120582(119954119954)2
1205961205961199541199541198951198951120596120596119954119954120582120582
311990311990321198951198951=1
1120596120596minus1205961205961199541199541198951198951+119894119894119894119894
120582120582 isin 31199031199032
+ 13119903119903
(540)
Using the relation
Im 1
120596120596minus1205961205961199541199541198951198951prime +119894119894119894119894
= minus120587120587120575120575120596120596 minus 1205961205961199541199541198951198951prime (541)
The transition rate reads as
Γ119954119954120582120582(120596120596) =
⎩⎪⎨
⎪⎧1205871205872sum
1198811198811205821205821198951198951prime(119954119954)
2
1205961205961199541199541198951198951prime120596120596119954119954120582120582
31199031199031198951198951prime=
31199031199032 +1
120575120575120596120596 minus 1205961205961199541199541198951198951prime 120582120582 isin 1 31199031199032
1205871205872sum 1198811198811198951198951120582120582(119954119954)2
1205961205961199541199541198951198951120596120596119954119954120582120582
311990311990321198951198951=1
120575120575120596120596 minus 1205961205961199541199541198951198951 120582120582 isin 31199031199032
+ 13119903119903
(542)
Or equivalently
Γ119954119954120582120582(119864119864 = ℏ120596120596) =
⎩⎪⎨
⎪⎧120587120587ℏ3
2sum
1198811198811205821205821198951198951prime(119954119954)
2
1198641198641199541199541198951198951prime119864119864119954119954120582120582
31199031199031198951198951prime=
31199031199032 +1
120575120575119864119864 minus 1198641198641199541199541198951198951prime 120582120582 isin 1 31199031199032
120587120587ℏ3
2sum 1198811198811198951198951120582120582(119954119954)2
1198641198641199541199541198951198951119864119864119954119954120582120582
311990311990321198951198951=1
120575120575119864119864 minus 1198641198641199541199541198951198951 120582120582 isin 31199031199032
+ 13119903119903
(543)
Γ119954119954120582120582(119864119864) represents the probability per unit time of a transition with energy 119864119864 from phonon branch 120582120582
at wave number 119954119954 in subsystem I (II) to a set of phonon branches in subsystem II (I) with the same
wave number Here state 119954119954120582120582 in one subsystem is coupled to state 1199541199541198951198951prime in the other subsystem via
1198811198811205821205821198951198951prime(119954119954) Since the two expressions in Eq (543) are completely equivalent just representing
transitions of opposite directions we consider only the first one in what follows Assuming that
subsystem II sufficiently large such that its density of states is nearly continuous the sum over its
states in the above expression can be replaced by an integral over the energy sum (⋯ )119895119895 rarr
int(⋯ )120588120588119864119864119954119954d119864119864119954119954 giving
Γ119954119954120582120582(119864119864) = 120587120587ℏ3
2 intd119864119864119954119954prime120588120588119864119864119954119954prime 119881119881120582120582119895119895prime(119954119954)
2119864119864119954119954119895119895prime=119864119864119954119954prime
119864119864119954119954prime119864119864119954119954120582120582120575120575119864119864 minus 119864119864119954119954prime = 120587120587ℏ3
2120588120588(119864119864)
119881119881120582120582119895119895prime(119954119954)2119864119864119954119954119895119895prime=119864119864
119864119864119864119864119954119954120582120582 (544)
Eq (544) represents the transition rate of the phonons with energy 119864119864 from branch 120582120582 and wave
number 119954119954 in subsystem I to the continuous manifold of states in subsystem II The total transition
rate between the two subsystems is then given by the sum of transition rates from all states in
subsystem I weighted by their phonon populations to the manifold of states in subsystem II
Assuming that the two subsystems are weakly coupled such that the width of state 119954119954120582120582 in subsystem
I is small and the probability to leave it at an energy other than 119864119864119954119954120582120582 is negligible we can now write
29
total transition rate as
Γtot = 1119885119885119890119890minus120573120573119864119864119954119954120582120582Γ119954119954120582120582119864119864119954119954120582120582
119954119954120582120582
=120587120587ℏ3
2
119890119890minus120573120573119864119864119954119954120582120582
119885119885
120588120588119864119864119954119954120582120582 119881119881120582120582119895119895prime(119954119954)2119864119864119954119954119895119895prime=119864119864119954119954120582120582
1198641198641199541199541205821205822119954119954120582120582
=120587120587ℏ3
2
119890119890minus120573120573119864119864119954119954120582120582
119885119885
120588120588119864119864119954119954120582120582 119881119881120582120582120582120582+31199031199032(119954119954)
2
1198641198641199541199541205821205822119954119954120582120582
Finally we get
Γtot = Γ119954119954120582120582(119864119864) =120587120587ℏ3
2sum 119890119890minus120573120573119864119864119954119954120582120582
119885119885
120588120588119864119864119954119954120582120582119881119881120582120582120582120582+31199031199032(119954119954)
2
1198641198641199541199541205821205822119954119954120582120582 (545)
where 119885119885 = sum 119890119890minus120573120573119864119864119954119954120582120582119954119954120582120582 is the partition function Here we choose the index of phonon branches 120582120582
such that the phonon branch with lower energy has a smaller index ie 1198641198641199541199541205821205821 lt 1198641198641199541199541205821205822 for index 1205821205821 lt
1205821205822 Note that phonon branches 120582120582 and 1198951198951prime belong to the subsystem I (with index from 1 to 31199031199032) and
subsystem II (with index from 1 to 1 + 31199031199032 to 3119903119903) respectively we choose 119895119895prime = 120582120582 + 31199031199032
to make
sure that 119864119864119954119954119895119895prime = 119864119864119954119954120582120582 since phonon energy of two uncoupled system is identical Eq (545) is the
Fermirsquos golden rule for inter-phonon coupling
References
[1] Y Xu J-S Wang W Duan B-L Gu and B Li Nonequilibrium Greens function method for phonon-phonon interactions and ballistic-diffusive thermal transport Phys Rev B 78 224303 (2008) [2] A L Fetter and J D Walecka Quantum theory of many-particle systems (Courier Corporation 2012) [3] G E Stedman Fermis Golden RulemdashAn Exercise in Quantum Field Theory Am J Phys 39 205 (1971)
8
ITR weakly depend on the average temperature over the entire thickness range considered This
conclusion is consistent with recent experimental findings [11] We may attribute this to the fact that
the thickness of the graphite and h-BN stack model considered is much smaller than their phonon
mean-free path (~200 nm for graphite [1011133536] and ~100 nm for bulk h-BN [15]) such that
the phonon transport is dominated by phonon-boundary scattering which weakly depends on
temperature in the range considered Therefore temperature dependent Umklapp processes have only
marginal contribution to our results [11]
Finally we note that previous calculations of the heat conductivity of twisted graphitic interfaces
relied on Lennard-Jones (LJ) potentials describing the interlayer interactions [8-1013] To
demonstrate the importance of using registry-dependent interlayer potentials we have repeated our
calculations of the heat conductivity of graphitic slabs with the REBO intralayer potential augmented
by LJ interlayer interactions [37] (120576120576 = 284 meV120590120590 = 34 Å ) We find that the calculated heat
conductivities obtained using the LJ interlayer potential are consistently higher than those obtained
by our ILP and that the difference between them grows with the model system thickness Notably the
heat conductivity obtained using the LJ potential for a graphitic slab of thickness ~34 nm is 154
W m sdot Kfrasl overestimating the experimental value by more than a factor of 2
The excellent agreement of our ILP calculations with experimental data of nanoscale graphitic stacks
therefore demonstrates the reliability of our predictions for the strong interfacial misfit angle
dependence of cross-layer thermal conductivity in graphite and h-BN The observed sharp
conductivity decrease of twisted graphitic interfaces at misfit angles lt 5deg opens the way to control
the thermal evacuation rate and thermal isolation of active layers in graphene-based electronic and
mechanical devices The revealed underlying mechanism suggests that design rules can be obtained
by carefully tailoring the phonon-phonon couplings across the twisted interface While the misfit
angle dependence of h-BN is found to be weaker than that of graphite the overall thermal
conductivity of the former is found to be higher This may be utilized to achieve higher conductivity
and controllability in twisted heterogeneous junctions of layered materials
ASSOCIATED CONTENT
Supporting Information
The supporting Information section includes a description of the methodology thermal conductivity
of twisted bilayer graphene theory for calculating the phonon coupling of twisted bilayer graphene
9
derivation of Fermirsquos golden rule and temperature dependence of the cross-plane thermal
conductivity
AUTHOR INFORMATION
Corresponding Author
E-mail urbakhtauextauacil
Notes
The authors declare no competing financial interest
ACKNOWLEDGMENTS
The authors would like to thank Prof Abraham Nitzan Dr Guy Cohen and Dr Yiming Pan for
helpful discussions WO acknowledges the financial support from a fellowship program for
outstanding postdoctoral researchers from China and India in Israeli Universities and the support from
the National Natural Science Foundation of China (Nos 11890673 and 11890674) HQ
acknowledges the financial support from the National Natural Science Foundation of China (No
11890674) MU acknowledges the financial support of the Israel Science Foundation Grant
No 114118 and the ISF-NSFC joint grant 319119 OH is grateful for the generous financial
support of the Israel Science Foundation under grant no 158617 and the Naomi Foundation for
generous financial support via the 2017 Kadar Award This work is supported in part by COST Action
MP1303
10
References
[1] A A Balandin Thermal properties of graphene and nanostructured carbon materials Nat Mater 10 569 (2011) [2] S Ghosh I Calizo D Teweldebrhan E P Pokatilov D L Nika A A Balandin W Bao F Miao and C N Lau Extremely high thermal conductivity of graphene Prospects for thermal management applications in nanoelectronic circuits Appl Phys Lett 92 151911 (2008) [3] A A Balandin S Ghosh W Bao I Calizo D Teweldebrhan F Miao and C N Lau Superior thermal conductivity of single-layer graphene Nano Lett 8 902 (2008) [4] J H Seol et al Two-Dimensional Phonon Transport in Supported Graphene Science 328 213 (2010) [5] R Taylor The thermal conductivity of pyrolytic graphite J Philosophical Magazine 13 157 (1966) [6] S Ghosh W Bao D L Nika S Subrina E P Pokatilov C N Lau and A A Balandin Dimensional crossover of thermal transport in few-layer graphene Nat Mater 9 555 (2010) [7] Z Wang R Xie C T Bui D Liu X Ni B Li and J T L Thong Thermal Transport in Suspended and Supported Few-Layer Graphene Nano Lett 11 113 (2011) [8] Z Wei Z Ni K Bi M Chen and Y Chen Interfacial thermal resistance in multilayer graphene structures Phys Lett A 375 1195 (2011) [9] Y Ni Y Chalopin and S Volz Significant thickness dependence of the thermal resistance between few-layer graphenes Appl Phys Lett 103 061906 (2013) [10] J Chen J H Walther and P Koumoutsakos Strain Engineering of Kapitza Resistance in Few-Layer Graphene Nano Lett 14 819 (2014) [11] Q Fu J Yang Y Chen D Li and D Xu Experimental evidence of very long intrinsic phonon mean free path along the c-axis of graphite Appl Phys Lett 106 031905 (2015) [12] G Fugallo A Cepellotti L Paulatto M Lazzeri N Marzari and F Mauri Thermal Conductivity of Graphene and Graphite Collective Excitations and Mean Free Paths Nano Lett 14 6109 (2014) [13] Z Wei J Yang W Chen K Bi D Li and Y Chen Phonon mean free path of graphite along the c-axis Appl Phys Lett 104 081903 (2014) [14] E K Sichel R E Miller M S Abrahams and C J Buiocchi Heat capacity and thermal conductivity of hexagonal pyrolytic boron nitride Phys Rev B 13 4607 (1976) [15] P Jiang X Qian R Yang and L Lindsay Anisotropic thermal transport in bulk hexagonal boron nitride Phys Rev Mater 2 064005 (2018) [16] M-H Wang Y-E Xie and Y-P Chen Thermal transport in twisted few-layer graphene Chinese Physics B 26 116503 (2017) [17] X Nie L Zhao S Deng Y Zhang and Z Du How interlayer twist angles affect in-plane and cross-plane thermal conduction of multilayer graphene A non-equilibrium molecular dynamics study Int J Heat Mass Tran 137 161 (2019) [18] A N Kolmogorov and V H Crespi Registry-dependent interlayer potential for graphitic systems Phys Rev B 71 235415 (2005) [19] M Reguzzoni A Fasolino E Molinari and M C Righi Potential energy surface for graphene on graphene Ab initio derivation analytical description and microscopic interpretation Phys Rev B 86 (2012) [20] M M van Wijk A Schuring M I Katsnelson and A Fasolino Moire Patterns as a Probe of Interplanar Interactions for Graphene on h-BN Phys Rev Lett 113 135504 (2014) [21] D Mandelli W Ouyang M Urbakh and O Hod The Princess and the Nanoscale Pea Long-Range Penetration of Surface Distortions into Layered Materials Stacks ACS Nano 13 7603 (2019) [22] E Koren E Loumlrtscher C Rawlings A W Knoll and U Duerig Adhesion and friction in mesoscopic graphite contacts Science 348 679 (2015) [23] E Koren I Leven E Loumlrtscher A Knoll O Hod and U Duerig Coherent commensurate electronic states at the interface between misoriented graphene layers Nat Nanotechnol 11 752 (2016)
11
[24] W Ouyang I Azuri D Mandelli A Tkatchenko L Kronik M Urbakh and O Hod Mechanical and Tribological Properties of Layered Materials under High Pressure Assessing the Importance of Many-Body Dispersion Effects J Chem Theory Comput 16 666 (2020) [25] W Ouyang D Mandelli M Urbakh and O Hod Nanoserpents Graphene Nanoribbon Motion on Two-Dimensional Hexagonal Materials Nano Lett 18 6009 (2018) [26] I Leven T Maaravi I Azuri L Kronik and O Hod Interlayer Potential for Grapheneh-BN Heterostructures J Chem Theory Comput 12 2896 (2016) [27] T Maaravi I Leven I Azuri L Kronik and O Hod Interlayer Potential for Homogeneous Graphene and Hexagonal Boron Nitride Systems Reparametrization for Many-Body Dispersion Effects J Phys Chem C 121 22826 (2017) [28] L H Liang and B Li Size-dependent thermal conductivity of nanoscale semiconducting systems Phys Rev B 73 153303 (2006) [29] P K Schelling S R Phillpot and P Keblinski Comparison of atomic-level simulation methods for computing thermal conductivity Phys Rev B 65 144306 (2002) [30] J Shiomi Nonequilirium molecular dynamics methods for lattice heat conduction calculations Annual Review of Heat Transfer 17 (2014) [31] G E Stedman Fermis Golden RulemdashAn Exercise in Quantum Field Theory Am J Phys 39 205 (1971) [32] L T Kong G Bartels C Campantildeaacute C Denniston and M H Muumlser Implementation of Greens function molecular dynamics An extension to LAMMPS Comput Phys Commun 180 1004 (2009) [33] S Plimpton Fast Parallel Algorithms for Short-Range Molecular Dynamics J Comput Phys 117 1 (1995) [34] M Harb et al The c-axis thermal conductivity of graphite film of nanometer thickness measured by time resolved X-ray diffraction Appl Phys Lett 101 233108 (2012) [35] H Zhang X Chen Y-D Jho and A J Minnich Temperature-Dependent Mean Free Path Spectra of Thermal Phonons Along the c-Axis of Graphite Nano Lett 16 1643 (2016) [36] D P Sellan E S Landry J E Turney A J H McGaughey and C H Amon Size effects in molecular dynamics thermal conductivity predictions Phys Rev B 81 214305 (2010) [37] S J Stuart A B Tutein and J A Harrison A reactive potential for hydrocarbons with intermolecular interactions J Chem Phys 112 6472 (2000)
S1
Supporting information for ldquoControllable Thermal Conductivity of
Twisted Graphene and Hexagonal Boron Nitride Interfacesrdquo
Wengen Ouyang1 Huasong Qin2 Michael Urbakh1 and Oded Hod1
1Department of Physical Chemistry School of Chemistry The Raymond and Beverly
Sackler Faculty of Exact Sciences and The Sackler Center for Computational
Molecular and Materials Science Tel Aviv University Tel Aviv 6997801 Israel
2State Key Laboratory for Strength and Vibration of Mechanical Structures School of
Aerospace Xirsquoan Jiaotong University Xirsquoan 710049 China
The Supporting information includes following sections
1 Methodology
2 Thermal conductivity of twisted bilayer graphene
3 Theory for calculating the phonon coupling of twisted bilayer graphene
4 Derivation of Fermirsquos golden rule
5 Temperature dependence of cross-plane thermal conductivity
S2
1 Methodology
11 Model system
The initial interlayer distance across the layered stack was set equal to 34 Aring and 33 Aring for graphite
and bulk h-BN respectively Periodic boundary conditions were applied in all directions It should be
noted that the lattice structure is rigorously periodic only at some specific twist angles the values of
which are listed in Table S1 in section 3 below While the cross-sectional area for each misfit angle
θ is different all systems considered have a contact area exceeding 12 nm2 which was shown to
provide converged results with respect to unit-cell dimensions [1] The intralayer interactions within
each graphene and h-BN layer were modeled via the second generation REBO potential [2] and
Tersoff potential [3] respectively The interlayer interactions between the layers of graphite and bulk
h-BN were described via our dedicated interlayer potential (ILP) [4] which is implemented in the
LAMMPS [5] suite of codes [6]
Fig S1 Schematic representation of the simulation setup (a) and steady-state temperature profile (b) respectively In panel (a) two identical AB-stacked graphite slabs (gray and orange respectively) are twisted with respect to each other to create a stacking fault of misfit angle θ A thermal bias is induced by applying Langevin thermostats to the two layers marked by dashed red (Thot) and green (Tcold) rectangles The arrows indicate the direction of the vertical heat flux Since periodic boundary conditions are applied also in the vertical direction two twisted interfaces are shown across which heat flows in opposite directions The steady-state temperature profiles are illustrated in panel (b) where N is the total number of layers in the model system and RAB dAB and Rθ and dθ mark the interfacial Kapitza resistance [89] and interlayer distance for contacting graphene layers with AB-stacking and misfit angle θ respectively The red lines in panel (b) mark the temperature variation across the twisted interface where the vertical axis corresponds to the position of the various layers along the stack and the horizontal axis marks the temperature of the various layers
S3
12 Simulation Protocol
All MD simulations were performed with the LAMMPS simulation package [5] The velocity-Verlet
algorithm with a time-step of 05 fs was used to propagate the equations of motion A Noseacute-Hoover
thermostat with a time constant of 025 ps was used for constant temperature simulations To maintain
a specified hydrostatic pressure the three translational vectors of the simulation cell were adjusted
independently by a Noseacute-Hoover barostat with a time constant of 10 ps [7] To relax the box we first
equilibrated the systems in the NPT ensemble at a temperature of T = 300 K and zero pressure for
250 ps (see Fig S2) After equilibration Langevin thermostats with damping coefficients 10 ps-1
were applied to the bottom and middle layer of the graphene stack (see Fig S1) with target
temperature Thot = 375 K (hot reservoir) and Tcold = 225 K (cold reservoir) respectively Then the
system was allowed to reach steady-state over a subsequent simulation period of 750 ps (see Fig S2)
during which the dynamics of all non-thermostated layers followed the NVE ensemble For the larger
model systems the length of the NPT and Langevin stages was doubled (for the 32 and 48 layers
systems) or tripled (for the 104 layers graphitic system) to ensure convergence of the obtained steady-
state Once steady-state was obtained the last 500 ps were used to calculate the thermal conductivity
of the twisted graphite and bulk h-BN The statistical errors were estimated using ten different data
sets each calculated over a time interval of 50 ps
Fig S2 Time evolution of the temperature of thermostated layers for 16 layers twisted graphite with misfit angle (a) θ = 0deg (b) θ = 509deg (c) θ = 1518deg and (d) θ = 3016deg Note that the thermal fluctuations increase with increasing the misfit angle due to the growing interfacial thermal resistance that enhances phonon back scattering at the twisted junction
S4
13 Calculation of the interfacial thermal resistance
According to Fourierrsquos law the cross-plane thermal conductivity (120581120581CP) of a twisted graphitic interface
of misfit angle θ can be calculated as
120581120581CP = 119876119860119860Δ119879119879Δ119911119911
(S1)
where 119876 is the heat flux A is the cross-section area and Δ119879119879Δ119911119911 is the temperature gradient along
the direction of heat flux (perpendicular to the basal plane in our case) Fig S1(b) shows a schematic
temperature profile along the z-direction where the vertical axis corresponds to the position of the
various layers along the stack and the horizontal axis marks the temperature of the various layers The
actual temperature profiles extracted from the NEMD simulations for twisted graphite with different
number of layers can be found in Fig S3 For Bernal-stacked graphite (ie θ = 0deg red circles in Fig
S3) only the linear region of the temperature profile was used to calculate 120581120581CP and the points
corresponding to the layers where the thermostats were applied were omitted (marked with green
triangle in Fig S3) The 120581120581CP of the system was calculated using Eq (S1) by averaging over the two
linear regions of the temperature profiles For the twisted case (θ ne 0deg) we found a sudden
temperature decrease Δ119879119879120579120579 at the position of the twisted interface (see black squares in Fig S3) 120581120581CP
in this case was calculated using the temperature gradient calculated for the same layer range as that
for θ = 0deg To characterize the thermal properties of the twisted interface the concept of interfacial
thermal resistance (ITR) ie Kapitza resistance [89] was introduced According the definition of
the Kapitza resistance [8] 119877119877 = 119860119860Δ119879119879119876 and noticing that Δ119879119879tot = (1198731198732 minus 3)Δ119879119879AB + Δ119879119879120579120579 and
Δ119911119911 = (1198731198732 minus 3)119889119889AB + 119889119889120579120579 Eq (1) can be rewritten considering two-resistors in series as [see Fig
S1(b)]
(1198731198732 minus 3)119877119877AB + 119877119877120579120579 = [(1198731198732 minus 3)119889119889AB + 119889119889120579120579] 120581120581CPfrasl (S2)
Here 119877119877AB 119889119889AB Δ119879119879AB and 119877119877120579120579 119889119889120579120579 Δ119879119879120579120579 are the ITR interlayer distance and temperature
difference for adjacent AB-stacked and twisted graphene layers respectively For the aligned contact
(120579120579 = 0deg) the ITR can be simply calculated as 119877119877AB = 119889119889AB120581120581AB Once 119877119877AB is known 119877119877120579120579 can be
calculated from Eq (S2) We note that the sharp temperature drop at the twisted interface indicates
that 119877119877120579120579 should be much larger than 119877119877AB
S5
Fig S3 Temperature profiles for graphitic stacks consisting of (a) 8 and (b) 16 layers The red circles and black squares represent the temperature profiles for the aligned (120579120579 = 0deg) and twisted (120579120579 =3016deg) junctions respectively Green triangles represent data points that were omitted in the 120581120581CP calculation
2 Thermal conductivity of twisted bilayer graphene
As comparison we also calculated the interfacial thermal conductivity (ITC) and ITR of twisted
bilayer graphene (tBLG) with the transient MD simulation approach [15-17] since the NEMD
simulation protocol used in the main text becomes invalid in this case In this protocol the system
was first equilibrated within the NPT ensemble at T = 200 K and zero pressure for 100 ps which was
followed by a 100 ps NVT ensemble equilibration stage and a 100 ps of NVE ensemble equilibration
stage After the system reached steady-state an ultrafast heat impulse was imposed on the top layer
of the t-BLG for 50 fs to increase the temperature of the top layer from 200 K to 400 K while that of
bottom layer of tBLG remained unchanged After the external heat source was removed thermal
energy flowed from the top layer to the bottom layer due to the temperature difference and the
temperature of both layers approached 300 K when quasi-steady-state was reached During the
thermal relaxation time interval (500 ps) the temperature and energy of the system sections were
recorded The ITR could then be extracted using the following equation [15-17]
part119864119864t120597120597120597120597
= 119860119860119877119877119879119879bot(119905119905) minus 119879119879top(119905119905) (S3)
where 119864119864t is the total energy of the top graphene layer R is the ITR of the tBLG A is the interfacial
cross-section area and 119879119879bot and 119879119879top are the instantaneous temperatures measured for the bottom
and top layers of the tBLG respectively Note that in Eq S3 we assume a linear dependence of the
heat flux on the temperature difference between the layers The ITC of the tBLG is simply defined as
ITC equiv 119889119889ITR where d is the average interlayer distance
S6
The ITC and ITR of tBLG as functions of misfit angle calculated with the transient MD simulation
protocol are illustrated in Fig S4 demonstrating similar misfit-angle dependence as that for the
NEMD protocol with Langevin thermostat exercised to obtain the results presented in the main text
This further validates the reliability of the simulation protocol adopted in the main text which is more
suitable to treat thick slabs and allows to obtain a true stead-state
Fig S4 Misfit-angle dependence of (a) ITC and (b) ITR for a twisted bilayer graphene obtained using the transient MD simulation approach
3 Theory for calculating the phonon-phonon coupling of twisted bilayer graphene
31 Brillouin Zone of supercell in tBLG
For tBLG the lattice structure is rigorously periodic only at some specific misfit angles θ where the
lattice vector 1199231199231 = 11989911989911199381199381 + 11989911989921199381199382 in the bottom layer equals the vector 1199231199232 = 11989811989811199381199381 + 11989811989821199381199382 in the
top layer with certain integers m1 m2 and n1 n2 Here 1199381199381 = 119886119886(10) and 1199381199382 = 11988611988612radic32 are
the primitive lattice vectors of the bottom layer and a is the lattice constant of monolayer graphene
Thus the exact superlattice period is then given by [18]
119871119871 = |11989911989911199381199381 + 11989911989921199381199382| = 11988611988611989911989912 + 11989911989922 + 11989911989911198991198992 = |1198991198991minus1198991198992|1198861198862 sin(1205791205792) (S4)
where θ is the angle between two lattice vectors 1199231199231 and 1199231199232 In the simulations below we always
rotated the supercell such that its lattice vector is 1199231199231 = 119871119871(10) and 1199231199232 = 11987111987112radic32 In this case
the corresponding reciprocal lattice vector of the moireacute superlattice satisfies the relation 119918119918119894119894 ∙ 119923119923119895119895 =
2120587120587120575120575119894119894119895119895 such that
1199181199181 = 4120587120587radic3119871119871
radic32
minus12 1199181199182 = 4120587120587
radic3119871119871(01) (S5)
S7
Both the lattice vectors and the corresponding reciprocal lattice vectors of the superlattice of the tBLG
are presented in Fig S5 Table S1 reports the parameters used to construct rhombus periodic
supercells of different misfit angles that can be duplicated to construct a rectangular periodic supercell
Table S1 The parameters used to construct periodic supercells of various misfit angles
120579120579 (deg) 119860119860 (nm2) 1198991198991 1198991198992 1198981198981 1198981198982
0 60383688 24 0 24 0
0696407 709613169 48 47 47 48
1121311 273718420 30 29 29 30
2000628 85648391 17 16 16 17
3006558 152322047 23 21 21 23
4048894 189013524 26 23 23 26
5085849 53255058 14 12 12 14
7926470 49376245 14 11 11 14
9998709 86277388 19 14 14 19
15178179 31554670 15 4 4 15
19932013 42876612 15 8 8 15
25039660 13942761 9 4 4 9
30158276 18974735 11 4 4 11
32204228 21805221 12 4 4 12
32 Special points for Brillouin Zone integration
The calculation of the sum over wave vector q in Eq (4) in the main text can be transformed to an
integral using the relation sum (⋯ )119954119954 = 1119881119881119887119887int (⋯ )BZ d119954119954 where 119881119881119887119887 = (2120587120587)3119881119881 is the volume of the
Brillouin Zone (BZ) and V is volume of the real-space unit-cell The calculation of integral is usually
inefficient since it requires calculating the value of the function over a large set of k points in the first
BZ To calculate such integrations more efficiently simple k-point meshes can be replaced by a
carefully selected set of special points in the BZ 119954119954119894119894 [19-22] over which the function is evaluated
The integral can then be estimated via
119868119868 = 1119881119881119887119887int 119891119891(119954119954)BZ d119954119954 asymp 1
119873119873sum 119908119908119894119894119891119891(119954119954119894119894)119894119894 (S6)
S8
where 119881119881119887119887 is the volume of the BZ 119908119908119894119894 is the weight of the ith data point and N normalizes the
weighting factors to unity 119873119873 = sum 119908119908119894119894119894119894 The set of selected 119954119954119894119894 forms a grid in the irreducible
Brillouin zone (IBZ) as is illustrated by the red points in Fig S5(b) The coordinates of these points
for a hexagonal lattice are presented in Eq (S7)
Fig S5 (a) Twisted bilayer graphene of misfit angle θ = 509deg L1 and L2 are the superlattice vectors (b) The corresponding first Brillouin Zone of (a) G1 and G2 are the reciprocal lattice vectors of the superlattice The triangle ΔΓMK represents the irreducible Brillouin zone Red circles mark the position of the special points used to evaluate the integral over the first Brillouin Zone
⎩⎪⎨
⎪⎧ 1199541199541 = 1
9 1205971205973 1199541199542 = 2
9 1205971205973 1199541199543 = 1
18 512059712059718
1199541199544 = 518
512059712059718 1199541199545 = 1
9 21205971205979 1199541199546 = 2
9 21205971205979
1199541199547 = 118
312059712059718 1199541199548 = 1
9 1205971205979 1199541199549 = 1
18 12059712059718
(S7)
Here 119905119905 = radic3 and the unit of the coordinates is 2120587120587119871119871 The weighting factors 119908119908119894119894 are [19]
119908119908124689 = 112
119908119908357 = 16 (S8)
Using Eqs (S6-S8) Eq (4) in the main text can be evaluated as follows
Γtot = 120587120587ℏ3
2sum 119908119908119896119896 sum
119890119890minus120573120573119864119864119954119954119948119948120582120582
119885119885
120588120588119864119864119954119954119948119948120582120582119881119881120582120582120582120582+31199031199032(119954119954119948119948)
2
1198641198641199541199541199481199481205821205822120582120582
9119896119896=1 (S9)
This equation was used to calculate the transition rate presented in Fig 3 in the main text
4 Derivation of Fermirsquos golden rule
The derivation of Fermirsquos golden rule is provided in a separate file (Fermirsquos Golden Rule for
phononspdf) due to its extent
S9
5 Temperature dependence of interfacial thermal conductivity
In the main text the target temperatures of the Langevin thermostats for the bottom and middle layers
of graphene and h-BN were set to 225 K and 375 K respectively After reaching the steady-state the
average temperature of the system was found to be ~300 K To check the effect of average temperature
on our results we calculated the cross-plane thermal conductivity (120581120581CP ) and the corresponding
interfacial thermal resistance (ITR) at a different temperature gradient (325 K ndash 475 K) resulting the
average steady-state temperature of ~400K The protocol described in Section 1 above was used to
perform these calculations as well Both ILP and Lennard-Jones (LJ) potential were tested for
graphite whereas for the bulk h-BN simulations only the ILP was used The results for graphite and
bulk h-BN are illustrated in Fig S6 and Fig S7 respectively For the ILP we find that the overall
values of 120581120581CP (ITR) decrease (increase) slightly with increasing average temperature which is
consistent with a recent experiment [10] The LJ potential calculations as well exhibit very week
dependence on average temperature within the range studied Altogether the layer dependences of
both quantities remain mostly insensitive to the average temperature The reason is that the thickness
of the graphite and h-BN model systems was chosen to be much smaller than their phonon mean free
path (~200 nm for graphite [110-13] and ~100 nm for bulk h-BN [14]) such that phonon transport is
dominated by phonon-boundary scattering and the Umklapp process only makes marginal
contribution [10]
S10
Fig S6 Layer dependence of 120581120581CP (a c) and ITR (b d) for Bernal-stacked graphite at average steady-state temperatures of 300 K (red circles) and 400 K (black squares) The left and right columns correspond to the 120581120581CP and ITR calculated with ILP and LJ potential respectively
Fig S7 Layer dependence of 120581120581CP (a) and ITR (b) for AArsquo-stacked h-BN at average temperatures of 300 K (red circles) and 400 K (blue squares)
S11
References [1] Z Wei J Yang W Chen K Bi D Li and Y Chen Phonon mean free path of graphite along the c-axis Appl Phys Lett 104 081903 (2014) [2] D W Brenner O A Shenderova J A Harrison S J Stuart B Ni and S B Sinnott A second-generation reactive empirical bond order (REBO) potential energy expression for hydrocarbons J Phys Condens Matter 14 783 (2002) [3] A Kınacı J B Haskins C Sevik and T Ccedilağın Thermal conductivity of BN-C nanostructures Phys Rev B 86 (2012) [4] W Ouyang I Azuri D Mandelli A Tkatchenko L Kronik M Urbakh and O Hod Mechanical and Tribological Properties of Layered Materials under High Pressure Assessing the Importance of Many-Body Dispersion Effects J Chem Theory Comput 16 666 (2020) [5] S Plimpton Fast Parallel Algorithms for Short-Range Molecular Dynamics J Comput Phys 117 1 (1995) [6] W Ouyang D Mandelli M Urbakh and O Hod Nanoserpents Graphene Nanoribbon Motion on Two-Dimensional Hexagonal Materials Nano Lett 18 6009 (2018) [7] W Shinoda M Shiga and M Mikami Rapid estimation of elastic constants by molecular dynamics simulation under constant stress Phys Rev B 69 134103 (2004) [8] G L Pollack Kapitza Resistance Rev Mod Phys 41 48 (1969) [9] H-S Yang G R Bai L J Thompson and J A Eastman Interfacial thermal resistance in nanocrystalline yttria-stabilized zirconia Acta Mater 50 2309 (2002) [10] Q Fu J Yang Y Chen D Li and D Xu Experimental evidence of very long intrinsic phonon mean free path along the c-axis of graphite Appl Phys Lett 106 031905 (2015) [11] J Chen J H Walther and P Koumoutsakos Strain Engineering of Kapitza Resistance in Few-Layer Graphene Nano Lett 14 819 (2014) [12] H Zhang X Chen Y-D Jho and A J Minnich Temperature-Dependent Mean Free Path Spectra of Thermal Phonons Along the c-Axis of Graphite Nano Lett 16 1643 (2016) [13] D P Sellan E S Landry J E Turney A J H McGaughey and C H Amon Size effects in molecular dynamics thermal conductivity predictions Phys Rev B 81 214305 (2010) [14] P Jiang X Qian R Yang and L Lindsay Anisotropic thermal transport in bulk hexagonal boron nitride Phys Rev Mater 2 064005 (2018) [15] J Zhang Y Hong and Y Yue Thermal transport across graphene and single layer hexagonal boron nitride J Appl Phys 117 134307 (2015) [16] B Liu F Meng C D Reddy J A Baimova N Srikanth S V Dmitriev and K Zhou Thermal transport in a graphenendashMoS2 bilayer heterostructure a molecular dynamics study RSC Advances 5 29193 (2015) [17] Z Ding Q-X Pei J-W Jiang W Huang and Y-W Zhang Interfacial thermal conductance in grapheneMoS2 heterostructures Carbon 96 888 (2016) [18] N N T Nam and M Koshino Lattice relaxation and energy band modulation in twisted bilayer graphene Phys Rev B 96 075311 (2017)
S12
[19] R Ramiacutearez and M C Boumlhm Simple geometric generation of special points in brillouin-zone integrations Two-dimensional bravais lattices Int J Quantum Chem 30 391 (1986) [20]S L Cunningham Special points in the two-dimensional Brillouin zone Phys Rev B 10 4988 (1974) [21] H J Monkhorst and J D Pack Special points for Brillouin-zone integrations Phys Rev B 13 5188 (1976) [22] D J Chadi Special points for Brillouin-zone integrations Phys Rev B 16 1746 (1977)
1
Fermirsquos golden rule for phonons
1 Basic theory for phonons 11 Basic notations
Let us consider a 3D crystal with a total of 119873119873 = 119873119873111987311987321198731198733 unit cells and periodic boundary conditions
To be specific let 119938119938119894119894 119894119894 = 123 be the lattice vectors that define the unit cell We index unit cells
with n = (n1n2n3) where each ni = 12 ∙∙∙ Ni and their locations are 119929119929119899119899 = sum 1198991198991198941198941199381199381198941198943119894119894=1 Assume that
there are r atoms in each unit cell which are indexed with 119904119904 = 1⋯ 119903119903 The mass and the equilibrium
distance of the sth atom are notated as 119872119872s and 119929119929s0 respectively Then the location of the sth atom in
the nth unit cell at time t can be expressed as
119955119955119899119899119899119899(119905119905) = 119929119929119899119899 + 119929119929s0 + 119958119958119899119899119899119899(119905119905) (11)
where 119958119958119899119899119899119899(119905119905) is its displacement from its equilibrium position
The Lagrangian for this classical problem can be written as
ℒ = sum sum 119872119872119904119904|119955119899119899119904119904|2(119905119905)2
119903119903119899119899=1
119873119873119899119899=1 minus 119881119881 (12)
where the second term is the sum of interactions between all pairs of atoms
Under the harmonic approximation ie expanding the total potential energy 119881119881 around the
equilibrium positions The Lagrangian can be simplified as
ℒ = sum sum 119872119872119904119904|119958119899119899119904119904|2(119905119905)2
119903119903119899119899=1
119873119873119899119899=1 minus 1
2sum sum sum 120601120601120572120572120572120572prime
119899119899119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime120572120572120572120572prime119899119899119899119899prime119899119899119899119899prime (13)
where 119906119906119899119899119899119899120572120572 120572120572 = 123 are the Cartesian coordinates of the displacement 119958119958119899119899119899119899(119905119905) and
120601120601120572120572120572120572prime 119899119899119899119899prime119904119904 119904119904prime = 1205971205972119881119881
120597120597119903119903119899119899119904119904119899119899119903119903119899119899prime119904119904prime119899119899primeeq
= 120601120601120572120572prime120572120572 119899119899prime119899119899119904119904prime 119904119904 = 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime (14)
Note that the first order term vanishes because we are expanding around the equilibrium positions
120601120601120572120572120572120572prime 119899119899119899119899prime119904119904 119904119904prime represents the component of the force acted on the sth atom in the nth unit cell along 120572120572
direction when the atom 119904119904prime in the unit cell 119899119899prime moves a unit displacement along 120572120572prime direction The
symmetries of 120601120601120572120572120572120572prime 119899119899 119899119899prime119904119904 119904119904prime appearing in Eq (14) arise from the intechangability of the second
derivative and the translational invariance of the interactions
12 Dynamical matrix
The equation of motion of the sth atom in the nth unit cell can be derived using the EulerndashLagrange
equation as follows
2
119872119872119899119899119906119899119899119899119899120572120572 = minussum 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119899119899prime119899119899prime120572120572prime 119906119906119899119899prime119899119899prime120572120572prime (15)
If we displace all atoms equally ie shifting 119906119906119899119899prime119899119899prime120572120572prime to 119906119906119899119899prime119899119899prime120572120572prime + 120575120575 the total force on the sth atom in
the nth unit cell does not change From the above equation we have
sum 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119899119899prime120572120572prime = 0 (16)
We are looking for normal modes (because any general solution can be written as a linear combination
of them) these are solutions where all atoms oscillate with the same frequency Moreover because
of the lattice structure we expect solutions to reflect this periodicity So we guess solutions of the
form
119906119906119899119899119899119899120572120572(119905119905) = 1119872119872119904119904
119890119890119899119899120572120572119890119890119894119894(119954119954∙119929119929119899119899minus120596120596119905119905) (17)
where 119890119890119899119899120572120572 are real-space solutions that will be determined later and 119954119954 is the wave-vector in
reciprocal space Substituting Eq (17) into Eq (15) we can get
1205961205962119890119890119899119899120572120572(119954119954) = sum 119863119863119899119899120572120572119899119899prime120572120572prime(119954119954)119890119890119899119899prime120572120572prime(119954119954)119899119899prime120572120572prime (18)
where
119863119863119899119899120572120572119899119899prime120572120572prime(119954119954) = sum 1
119872119872119904119904119872119872119904119904prime120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119890119890
minus119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime119899119899prime = sum 1
119872119872119904119904119872119872119904119904prime120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime 119890119890
minus119894119894119954119954∙119929119929119897119897119897119897 (19)
is called dynamical matrix (dimension 3119903119903 times 3119903119903) Note that we have defined the relative cell distance
vector 119929119929119897119897 equiv 119929119929119899119899 minus 119929119929119899119899prime and number index 119897119897 equiv 119899119899 minus 119899119899prime where the infinite sum over 119899119899prime can be
replaced by the sum over 119897119897 for any value of the index 119899119899 Note that dynamical matrix is Hermitian
symmetric (ie 119863119863119899119899120572120572119899119899prime120572120572prime(119954119954)
lowast= 119863119863119899119899prime120572120572prime
119899119899120572120572 (minus119954119954)) because ϕ is symmetric so all the eigenvalues of Eq (18)
(120596120596120582120582(119954119954) 120582120582 = 12⋯ 3119903119903) are real for each 119954119954 in the Brillouin zone (BZ) which is determined by
det1205961205961205821205822(119954119954)120575120575119899119899120572120572120575120575119899119899prime120572120572prime minus 119863119863119899119899120572120572119899119899prime120572120572prime(119954119954) = 0 (110)
Taking the conjugate of this equation we have
0 = det 1205961205961205821205822(119954119954)lowast120575120575119899119899120572120572120575120575119899119899prime120572120572prime minus 119863119863119899119899120572120572119899119899
prime120572120572prime(119954119954)lowast = det1205961205961205821205822(119954119954)120575120575119899119899120572120572120575120575119899119899prime120572120572prime minus 119863119863119899119899120572120572119899119899
prime120572120572prime(minus119954119954) (111)
while replacing 119954119954 by minus119954119954 in Eq (110) we have det1205961205961205821205822(minus119954119954)120575120575119899119899120572120572120575120575119899119899prime120572120572prime minus 119863119863119899119899120572120572119899119899prime120572120572prime(minus119954119954) = 0
Itrsquos clear that 1205961205961205821205822(119954119954) and 1205961205961205821205822(minus119954119954) obey the same equation thus we have
120596120596120582120582(minus119954119954) = 120596120596120582120582(119954119954) (112)
The corresponding eigenvectors are orthonormal
sum 119890119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582
prime119899119899120572120572 = 120575120575120582120582120582120582prime (113)
The complex conjugate of Eq (18) gives
1205961205962119890119890119899119899120572120572lowast (119954119954) = sum 119863119863119899119899120572120572119899119899prime120572120572prime(119954119954)
lowast119890119890119899119899prime120572120572primelowast (119954119954)119899119899prime120572120572prime = sum 119863119863119899119899prime120572120572prime
119899119899120572120572 (minus119954119954)119890119890119899119899prime120572120572primelowast (119954119954)119899119899prime120572120572prime (114)
3
While replacing 119954119954 by minus119954119954 in Eq (18) we get the following equation
1205961205962119890119890119899119899120572120572(minus119954119954) = sum 119863119863119899119899120572120572119899119899prime120572120572prime(minus119954119954)119890119890119899119899prime120572120572prime(minus119954119954)119899119899prime120572120572prime (115)
From Eq (114) and Eq (115) we see that eigenvectors 119890119890119899119899120572120572120582120582 (minus119954119954)lowast and 119890119890119899119899120572120572120582120582 (119954119954) obey the same
eigenvalue equation Since the eigenvectors are normalized we get the following property
119890119890119899119899120572120572120582120582 (minus119954119954)lowast
= 119890119890119899119899120572120572120582120582 (119954119954) (116)
2 Second quantization The general solution is a linear combination of all these normal modes thus we have
119906119906119899119899119899119899120572120572(119905119905) = sum 119862119862120582120582(119954119954)119873119873119872119872119904119904
119890119890119899119899120572120572120582120582 (119954119954)119890119890minus119894119894120596120596119954119954120582120582119905119905119890119890119894119894119954119954∙119929119929119899119899119954119954120582120582 = sum 119876119876120582120582(119954119954119905119905)119873119873119872119872119904119904
119890119890119899119899120572120572120582120582 (119954119954)119890119890119894119894119954119954∙119929119929119899119899119954119954120582120582 (21)
where the we define the normal coordinates as 119876119876120582120582(119954119954 119905119905) equiv 119862119862120582120582(119954119954)119890119890minus119894119894120596120596120582120582(119954119954)119905119905 in the eigenvectors
representation To ensure that the displacements 119906119906119899119899119899119899120572120572(119905119905) are real (namely 119906119906119899119899119899119899120572120572lowast (119905119905) = 119906119906119899119899119899119899120572120572(119905119905)) the
following relation on 119876119876120582120582(119954119954 119905119905) and 119890119890119899119899120572120572120582120582 is enforced
119876119876120582120582(119954119954 119905119905)119890119890119899119899120572120572120582120582 (119954119954)lowast
= 119876119876120582120582(minus119954119954 119905119905)119890119890119899119899120572120572120582120582 (minus119954119954) (22)
where we used the fact that the sum over 119954119954 runs symmetrically over both negative and positive
values Using Eq (116) we have
[119876119876120582120582(119954119954 119905119905)]lowast = 119876119876120582120582(minus119954119954 119905119905) (23)
21 Kinetic energy term
Using Eq (21) the kinetic energy of the system can be expressed as
119879119879 = 12sum 1198721198721198991198991199061198991198991198991198991205721205722 (119905119905)119899119899119899119899120572120572 = 1
2sum 119872119872119904119904
119873119873119872119872119904119904sum sum 119876120582120582(119954119954 119905119905)119876120582120582prime(119954119954prime 119905119905)119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582
prime(119954119954prime) sum 119890119890119894119894119954119954+119954119954prime∙119929119929119899119899119899119899119954119954prime120582120582prime119954119954120582120582119899119899120572120572 =
12sum sum 119876120582120582(119954119954 119905119905)119876120582120582prime(minus119954119954 119905119905)119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582
prime(minus119954119954)119954119954119954119954prime120582120582prime =119899119899120572120572
12sum 119876120582120582(119954119954 119905119905)119876120582120582prime
lowast (119954119954 119905119905) sum 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582prime(119954119954)
lowast119899119899120572120572 =119954119954120582120582120582120582prime
12sum 119876120582120582(119954119954 119905119905)119876120582120582lowast(119954119954 119905119905)119954119954120582120582 =
12sum 119876120582120582(119954119954 119905119905)119876120582120582(minus119954119954 119905119905)119954119954120582120582
ie
119879119879 = 12sum 119876120582120582(119954119954 119905119905)119876120582120582(minus119954119954 119905119905)119954119954120582120582 (24)
To derive Eq (24) we used Eqs (113) and (116) as well as the following equations
1119873119873sum 119890119890119894119894119954119954+119954119954prime∙119929119929119899119899119899119899 = 120575120575119954119954+119954119954prime120782120782 (25)
4
22 Potential energy term
Similarity the potential energy can be rewritten in terms of the normal coordinates as follows
119880119880 =12120601120601119899119899119899119899120572120572119899119899prime119899119899prime120572120572prime119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
120572120572120572120572prime119899119899119899119899prime119899119899119899119899prime=
12120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119899119899+119954119954prime∙119929119929119899119899prime
119873119873119872119872119899119899119872119872119899119899prime
[119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(119954119954prime 119905119905)] 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime120582120582prime (119954119954prime)
119954119954prime120582120582prime119954119954120582120582119899119899119899119899prime120572120572120572120572prime119899119899119899119899prime
=12120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894(119954119954+119954119954prime)∙119929119929119899119899119890119890minus119894119894119954119954prime∙119929119929119897119897
119873119873119872119872119899119899119872119872119899119899prime
[119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(119954119954prime 119905119905)] 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime120582120582prime (119954119954prime)
119954119954prime120582120582prime119954119954120582120582119899119899119897119897120572120572120572120572prime119899119899119899119899prime
=12120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890minus119894119894119954119954prime∙119929119929119897119897
119872119872119899119899119872119872119899119899prime[119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(119954119954prime 119905119905)] 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (119954119954prime)1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899
119899119899=120575120575119902119902+119902119902prime=0
119954119954prime120582120582prime119954119954120582120582119897119897120572120572120572120572prime119899119899119899119899prime
=12120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894119954119954119929119929119897119897
119872119872119899119899119872119872119899119899prime [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)] 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (minus119954119954)119954119954120582120582120582120582prime119897119897120572120572120572120572prime119899119899119899119899prime
=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]119890119890119899119899120572120572120582120582 (119954119954)
119899119899120572120572
119863119863119899119899120572120572119899119899prime120572120572prime(minus119954119954)119890119890119899119899prime120572120572prime
120582120582prime (minus119954119954)119899119899prime120572120572prime119954119954120582120582120582120582prime
=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]119890119890119899119899120572120572120582120582 (119954119954)
119899119899120572120572
1205961205961205821205822(minus119954119954)119890119890119899119899120572120572120582120582prime (minus119954119954)
119954119954120582120582120582120582prime
=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]1205961205961205821205822(119954119954)120575120575120582120582120582120582prime =119954119954120582120582120582120582prime
121205961205961205821205822(119954119954)119876119876120582120582(119954119954 119905119905)119876119876120582120582(minus119954119954 119905119905)
119954119954120582120582
ie
119880119880 = 12sum 1205961205961205821205822(119954119954)119876119876120582120582(119954119954 119905119905)119876119876120582120582(minus119954119954 119905119905)119954119954120582120582 (26)
where in above deriviation the orthogonality of its eigenvectores and the symmetry of its eigenvalues
are used Thus the Lagrangian reads as
ℒ = 119879119879 minus 119881119881 = 12sum 119876120582120582(minus119954119954 119905119905)119876120582120582(119954119954 119905119905) minus 120596120596120582120582
2(119954119954)119876119876120582120582(minus119954119954 119905119905)119876119876120582120582(119954119954 119905119905)119954119954120582120582 (27)
Using the relation 119875119875120582120582(119954119954 119905119905) = 120597120597ℒ part119876120582120582(119954119954 119905119905) the Hamiltonian of the system then can be written as
119867119867 = 119879119879 + 119881119881 = 12sum [119875119875120582120582(minus119954119954 119905119905)119875119875120582120582(119954119954 119905119905) + 120596120596120582120582
2(119954119954)119876119876120582120582(minus119954119954 119905119905)119876119876120582120582(119954119954 119905119905)]119954119954120582120582 (28)
Now we quantize H by asking the momenta and coordinates to be operators
119867119867 = 12sum 119875119875minus119954119954120582120582119875119875119954119954120582120582 + 120596120596119954119954120582120582
2 119876119876minus119954119954120582120582119876119876119954119954120582120582119954119954120582120582 (29)
here 119875119875119954119954120582120582 and 119876119876119954119954120582120582 obey the following commutation relations
119876119876119954119954120582120582119875119875119954119954prime120582120582prime = 119894119894ℏ120575120575119954119954119954119954prime120575120575120582120582120582120582prime
119876119876119954119954120582120582119876119876119954119954prime120582120582prime = 0 119875119875119954119954120582120582119875119875119954119954prime120582120582prime = 0 (210)
Similar to the case of ordinary quantum harmonic oscillators it is convenient to define ladder
operators for each mode as follows
119876119876119954119954120582120582 = ℏ
2120596120596119954119954120582120582119887119887119954119954120582120582 + 119887119887minus119954119954120582120582
dagger 119875119875119954119954120582120582 = 119894119894ℏ120596120596119954119954120582120582
2119887119887119954119954120582120582
dagger minus 119887119887minus119954119954120582120582 (211)
where 119887119887119954119954120582120582dagger and 119887119887119954119954120582120582 are Bosonic creation and annihilation operators for phonons with momentum
119954119954 branch index 120582120582 and frequency 120596120596119954119954120582120582 which obeys the Bosonic commutation relation
5
119887119887119954119954120582120582 119887119887119954119954prime120582120582prime
dagger = 120575120575119954119954119954119954prime120575120575120582120582120582120582prime
119887119887119954119954120582120582 119887119887119954119954prime120582120582prime = 0 119887119887119954119954120582120582dagger 119887119887119954119954prime120582120582prime
dagger = 0 (212)
Substituting Eqs (211) into (29) and using the properties Eq (210) and (212) we have
119867119867 =12119875119875minus119954119954120582120582119875119875119954119954120582120582 +120596120596119954119954120582120582
2 119876119876minus119954119954120582120582119876119876119954119954120582120582119954119954120582120582
=12minus
ℏ120596120596119954119954120582120582
2119887119887minus119954119954120582120582
dagger minus 119887119887119954119954120582120582119887119887119954119954120582120582dagger minus 119887119887minus119954119954120582120582+ 120596120596119954119954120582120582
2 ℏ2120596120596119954119954120582120582
119887119887minus119954119954120582120582 + 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 + 119887119887minus119954119954120582120582
dagger 119954119954120582120582
=ℏ4120596120596119954119954120582120582minus119887119887minus119954119954120582120582
dagger 119887119887119954119954120582120582dagger minus 119887119887minus119954119954120582120582
dagger 119887119887minus119954119954120582120582 minus 119887119887119954119954120582120582119887119887119954119954120582120582dagger + 119887119887119954119954120582120582119887119887minus119954119954120582120582
119954119954120582120582
+ 119887119887minus119954119954120582120582119887119887119954119954120582120582 + 119887119887minus119954119954120582120582119887119887minus119954119954120582120582dagger + 119887119887119954119954120582120582
dagger 119887119887119954119954120582120582 + 119887119887119954119954120582120582dagger 119887119887minus119954119954120582120582
dagger
=ℏ4120596120596119954119954120582120582119887119887minus119954119954120582120582
dagger 119887119887minus119954119954120582120582 + 119887119887minus119954119954120582120582119887119887minus119954119954120582120582dagger + 119887119887119954119954120582120582119887119887119954119954120582120582
dagger + 119887119887119954119954120582120582dagger 119887119887119954119954120582120582
119954119954120582120582
=ℏ41205961205961199541199541205821205822119887119887minus119954119954120582120582
dagger 119887119887minus119954119954120582120582 + 1+ 2119887119887119954119954120582120582dagger 119887119887119954119954120582120582 + 1
119954119954120582120582
= ℏ120596120596119954119954120582120582 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 +
12
119954119954120582120582
Therefore we finally get the quantized representation of non-interacting phonons
119867119867 = sum ℏ120596120596119954119954120582120582 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 + 1
2119954119954120582120582 (213)
The operator of atom displacements (Eq 114) is expressed in terms of the phonon operators by
119906119906119899119899119899119899120572120572 = sum ℏ
2119873119873119872119872119904119904120596120596119954119954120582120582119890119890119899119899120572120572120582120582 119890119890119894119894119954119954∙119929119929119899119899119887119887119954119954120582120582 + 119887119887minus119954119954120582120582
dagger 119954119954120582120582 (214)
These equations will be used below
3 Inter-phonon coupling within harmonic approximation 31 Hamiltonian with inter-phonon coupling
In this section we consider systems that consist of two (or more) covalently bonded units that are
weakly coupled between them Unlike previous studies that considered phonon-phonon couplings
resulting from anharmonicity effects [1] here all phonon mode considered are Harmonic and the
couplings arise from the division of the entire system into subunits The Hamiltonian of the whole
system can be written as
119867119867 = 1198671198671 + 1198671198672 + 11986711986712 (31)
To derive the expressions of 1198671198671 1198671198672 and 11986711986712 in Eq (31) we consider the Hamiltonian of the
whole system written as the function of the atomic displacements
119867119867 = 119879119879 + 119881119881 = sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)2119899119899119899119899120572120572 + 1
2sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime120572120572120572120572prime119899119899119899119899prime119899119899119899119899prime (32)
6
Letrsquos assume that subsystem I and II contain atoms with indeices ranging from 119904119904 = 12⋯ 1199031199032 and
119904119904 = 1199031199032
+ 1 1199031199032
+ 2⋯ 119903119903 respectively Then the kinetic energy term in Eq (32) can be rewritten as
119879119879 = sum sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)2
1199031199032119899119899=1 + sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)
2119903119903119899119899=1199031199032+1
119899119899120572120572 (33)
While the potential energy term is
119881119881 =12 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
1199031199032
119899119899119899119899prime=1
+ 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
119903119903
119899119899119899119899prime=1199031199032+1
+ 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime119899119899119899119899prime
+ 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
1199031199032
119899119899prime=1
119903119903
119899119899=1199031199032+1
= 11988111988111 + 11988111988122 + 11988111988112 + 11988111988121
Or equivalently
⎩⎪⎪⎨
⎪⎪⎧ 11988111988111 = 1
2sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
1199031199032119899119899119899119899prime=1120572120572120572120572prime119899119899119899119899prime
11988111988122 = 12sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
119903119903119899119899119899119899prime=1199031199032+1120572120572120572120572prime119899119899119899119899prime
11988111988112 = 12sum sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
119903119903119899119899prime=1199031199032+1
1199031199032119899119899=1120572120572120572120572prime119899119899119899119899prime
11988111988121 = 12sum sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
1199031199032119899119899prime=1
119903119903119899119899=1199031199032+1120572120572120572120572prime119899119899119899119899prime
(34)
32 Second quantization
We may now quantize this Hamiltonian and write it in the basis of the eigenstates of the coupled
subunits In second quantization the atomic displacement operators of the two subunits are given in
the following form
119906119906119899119899119899119899120572120572 =
⎩⎨
⎧ sum ℏ
2119873119873119872119872119904119904ω119954119954120582120582119890119890119899119899120572120572120582120582 119890119890119894119894119954119954∙119929119929119899119899119886119886119954119954120582120582 + 119886119886minus119954119954120582120582
dagger 119954119954120582120582 119904119904 isin 1 1199031199032
sum ℏ
2119873119873119872119872119904119904120596120596119954119954prime120582120582prime119890119899119899120572120572120582120582
prime119890119890119894119894119954119954prime∙119929119929119899119899 119886119886 119954119954prime120582120582prime + 119886119886minus119954119954prime120582120582primedagger 119954119954prime120582120582prime 119904119904 isin 119903119903
2+ 1 119903119903
(35)
Here we use different notations for the creation and annihilation operators (119886119886119954119954120582120582119886119886minus119954119954120582120582dagger and 119886119886119954119954120582120582119886119886minus119954119954120582120582
dagger )
eigenvalues (ω119954119954120582120582120596120596119954119954120582120582) and eigenvectors (119890119890119899119899120572120572120582120582 119890119899119899120572120572120582120582 ) for the two subunits Note that 119890119890119899119899120572120572120582120582 and 119890119899119899120572120572120582120582 are
of the dimensions of the whole system however their non-zero elements appear only on the relevant
subunits such that they obey the following relations sum 119890119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582
prime119899119899120572120572 = 120575120575120582120582120582120582prime sum 119890119899119899120572120572120582120582
lowast119890119899119899120572120572120582120582
prime119899119899120572120572 =
120575120575120582120582120582120582prime sum 119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582
prime119899119899120572120572 = sum 119890119890119899119899120572120572120582120582
lowast119890119899119899120572120572120582120582
prime119899119899120572120572 = 0 Defining the normal coordinates of the two subunits as
119876119876119954119954120582120582 equiv ℏ
2ω119954119954120582120582119886119886119954119954120582120582 + 119886119886minus119954119954120582120582
dagger 119876119876119954119954120582120582 equiv ℏ
2ω119954119954120582120582119886119886119954119954120582120582 + 119886119886minus119954119954120582120582
dagger (36)
and the corresponding momenta operators as
7
119875119875119954119954120582120582 = 119894119894ℏω119954119954120582120582
2119886119886119954119954120582120582
dagger minus 119886119886minus119954119954120582120582 119875119875119954119954120582120582 = 119894119894ℏω1199541199541205821205822
119886119886119954119954120582120582dagger minus 119886119886minus119954119954120582120582 (37)
Eq (35) reads
119906119906119899119899119899119899120572120572 = sum 119890119890119904119904119899119899120582120582 (119954119954)
119873119873119872119872119904119904119890119890119894119894119954119954∙119929119929119899119899119876119876119954119954120582120582119954119954120582120582 119904119904 isin 1 119903119903
2
sum 119890119904119904119899119899120582120582 (119954119954)119873119873119872119872119904119904
119890119890119894119894119954119954∙119929119929119899119899119876119876119954119954120582120582119954119954120582120582 119904119904 isin 1199031199032
+ 1 119903119903 (38)
The second quantized kinetic energy operator is then written as
119879119879 = 1198791198791 + 1198791198792 = 12sum 119875119875119954119954120582120582119875119875minus119954119954120582120582119954119954120582120582 + 1
2sum 119875119875119954119954prime120582120582prime119875119875minus119954119954prime120582120582prime119954119954prime120582120582prime (39)
Correspondingly the various potential energy terms are obtained by substituting Eq (38) in Eq (34)
11988111988111 =12 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119899119899119899119899prime
119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime120582120582prime (119954119954prime)
119872119872119899119899119872119872119899119899prime
1119873119873119890119890119894119894119954119954∙119929119929119899119899+119894119894119954119954prime∙119929119929119899119899prime119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954120582120582119954119954prime120582120582prime
1199031199032
119899119899119899119899prime=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954119954119954prime120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (119954119954prime)1119873119873 119890119890
119894119894(119954119954+119954119954prime)119929119929119899119899prime
119899119899prime120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime
119872119872119899119899119872119872119899119899prime119899119899
1199031199032
119899119899119899119899prime=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954119954119954prime120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (119954119954prime) 120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime
1199031199032
119899119899119899119899prime=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime
119954119954120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (minus119954119954)120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
1199031199032
119899119899119899119899prime=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime
119954119954120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954) 119863119863119899119899120572120572119899119899
prime120572120572prime(minus119954119954)119890119890119899119899prime120572120572prime120582120582prime (minus119954119954)
120572120572prime
1199031199032
119899119899prime=1
1199031199032
119899119899=1120572120572
=12 ω119954119954120582120582prime
2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582120582120582prime
119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582prime (119954119954)
lowast
1199031199032
119899119899=1
=120572120572
12ω119954119954120582120582
2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582
ie
11988111988111 = 12sum ω119954119954120582120582
2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582 (310)
Going from the second line to the third line we used the fact that the sum over 119899119899prime runs between plusmninfin
and the summand depends only on the difference between 119899119899 and 119899119899prime hence the sum is independent
of the value of the index 119899119899 Therefore we can replace the sum over 119899119899prime by a sum over 119897119897 equiv 119899119899 minus 119899119899prime
amd define 119929119929119897119897 equiv 119929119929119899119899 minus 119929119929119899119899prime Following the same procedure we can get the corresponding
expressions for the second diagonal term
11988111988122 = 12sum 1205961205961199541199541205821205822 119876119876119954119954120582120582119876119876minus119954119954120582120582119954119954120582120582 (311)
Similarly for the off-diagonal terms we get
8
11988111988112 =12 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119899119899119899119899prime
119903119903
119899119899prime=1199031199032+1
119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime
120582120582prime (119954119954prime)
119872119872119899119899119872119872119899119899prime
1119873119873119890119890119894119894119954119954∙119929119929119899119899+119894119894119954119954prime∙119929119929119899119899prime119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954120582120582119954119954prime120582120582prime
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954119954119954prime120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime
120582120582prime (119954119954prime)1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime
119872119872119899119899119872119872119899119899prime119899119899
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954119954119954prime120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime
120582120582prime (119954119954prime) 120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime
119954119954120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime
120582120582prime (minus119954119954)120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954
119890119899119899prime120572120572prime120582120582prime (119954119954)
lowast120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119890119890119899119899120572120572120582120582 (119954119954)119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954
Where we have defined
119881119881120582120582120582120582prime(119954119954) equiv sum sum sum 119890119899119899prime120572120572prime120582120582prime (119954119954)
lowast119881119881119899119899120572120572119899119899
prime120572120572prime(119954119954)119890119890119899119899120572120572120582120582 (119954119954)119903119903119899119899prime=1199031199032+1
1199031199032119899119899=1120572120572120572120572prime (312)
where
119881119881119899119899120572120572119899119899prime120572120572prime(119954119954) equiv sum 119890119890119894119894119954119954∙119929119929119897119897
119872119872119904119904119872119872119904119904prime119897119897 120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime (313)
Itrsquos easy to show that 119881119881120582120582120582120582prime(119954119954) has the following property
119881119881120582120582120582120582primelowast (119954119954) = 119881119881120582120582120582120582prime(minus119954119954) (314)
Following the same procedure we have
9
11988111988121 =12 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119899119899119899119899prime
119903119903
119899119899=1199031199032+1
119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (119954119954prime)
119872119872119899119899119872119872119899119899prime
1119873119873119890119890119894119894119954119954∙119929119929119899119899+119894119894119954119954prime∙119929119929119899119899prime119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954120582120582119954119954prime120582120582prime
1199031199032
119899119899prime=1120572120572120572120572prime
=12 120601120601120572120572prime120572120572
119899119899prime minus 119899119899119904119904prime 119904119904
119899119899prime119899119899
119903119903
119899119899prime=1199031199032+1
119890119899119899prime120572120572prime120582120582prime (119954119954prime)119890119890119899119899120572120572120582120582 (119954119954)
119872119872119899119899prime119872119872119899119899
1119873119873 119890119890
119894119894119954119954prime∙119929119929119899119899prime+119894119894119954119954∙119929119929119899119899119876119876119954119954prime120582120582prime119876119876119954119954120582120582119954119954prime120582120582prime119954119954120582120582
1199031199032
119899119899=1120572120572prime120572120572
=12 119876119876119954119954prime120582120582prime119876119876119954119954120582120582
119954119954119954119954prime120582120582120582120582prime119890119899119899prime120572120572prime120582120582prime (119954119954prime)119890119890119899119899120572120572120582120582 (119954119954)
1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime
119872119872119899119899119872119872119899119899prime119899119899
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954prime120582120582prime119876119876119954119954120582120582
119954119954119954119954prime120582120582120582120582prime119890119899119899prime120572120572prime120582120582prime (119954119954prime)119890119890119899119899120572120572120582120582 (119954119954) 120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954prime120582120582prime119876119876minus119954119954prime120582120582119890119899119899prime120572120572prime
120582120582prime (119954119954prime)119890119890119899119899120572120572120582120582 (minus119954119954prime)120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890minus119894119894119954119954prime∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897119954119954prime120582120582120582120582prime
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954prime120582120582prime119876119876119954119954prime120582120582
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954prime 119890119890119899119899120572120572120582120582 (119954119954prime)
lowast120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890minus119894119894119954119954prime∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119890119899119899prime120572120572prime120582120582prime (119954119954prime)
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582prime119876119876119954119954120582120582
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954
119890119890119899119899120572120572120582120582 (119954119954)lowast120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890minus119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119890119899119899prime120572120572prime120582120582prime (119954119954)
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger dagger
3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954
119890119899119899prime120572120572prime120582120582prime (119954119954)
lowast120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119890119890119899119899120572120572120582120582 (119954119954)119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
lowast
=
⎩⎨
⎧12 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954⎭⎬
⎫dagger
= 11988111988112dagger
Define
1198671198671 = 11987911987911 + 119881119881111198671198672 = 11987911987922 + 1198811198812211986711986712 = 11988111988112 + 11988111988121
(315)
we finally get the expressions of 1198671198671 1198671198672 and 11986711986712 in Eq (31) as follows
⎩⎪⎨
⎪⎧ 1198671198671 = sum sum ℏ120596120596119954119954120582120582 119886119886119954119954120582120582
dagger 119886119886119954119954120582120582 + 1231199031199032
120582120582=1119954119954
1198671198672 = sum sum ℏ120596120596119954119954120582120582prime 119886119886119954119954120582120582primedagger 119886119886119954119954120582120582prime + 1
23119903119903
120582120582prime=1+31199031199032119954119954
11986711986712 = 12sum sum sum 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903120582120582prime=31199031199032 +1
31199031199032120582120582=1 + h c 119954119954
(316)
Here hc means the Hermitian conjugate Since the indices of 119886119886119954119954120582120582 119886119886119954119954120582120582prime and 119876119876119954119954120582120582 119876119876119954119954120582120582prime belong to
the two system sections we can define an abbreviated notation 119886119886119954119954120582120582 and 119876119876119954119954120582120582 using the index 120582120582 to
identify which subsystem they belong to In this case the Hamiltonian of the coupled system can be
10
simplified as follows
119867119867 = 1198671198670 + 119867119867119862119862 (317)
where
1198671198670 = sum sum ℏ120596120596119954119954120582120582 119886119886119954119954120582120582
dagger 119886119886119954119954120582120582 + 123119903119903
120582120582=1119954119954
119867119867119862119862 = 12sum sum sum 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903120582120582prime=31199031199032 +1
31199031199032120582120582=1 + ℎ 119888119888 119954119954
(318)
4 Greenrsquos function 41 Dynamics of the ladder operators
In order to describe the dynamics of the ladder operators appearing in the Hamiltonian of Eq (317)
we express them in the Heisenberg picture as follows
119886119886119901119901(120591120591) = 119890119890119894119894ℏ119867119867119905119905119886119886119901119901119890119890
minus119894119894ℏ119867119867119905119905 = 119890119890
119867119867120591120591ℏ 119886119886119901119901119890119890
minus119867119867120591120591ℏ (41)
where we define the imaginary time 120591120591 equiv 119894119894119905119905 and introduce the notation 119953119953 equiv (119954119954 120582120582) and 119953119953 equiv (minus119954119954 120582120582)
The corresponding equation of motion for the ladder operators is given by
ℏ part119886119886119953119953(120591120591)
120597120597120591120591= 119867119867119886119886119953119953(120591120591) (42)
For the uncoupled system (119867119867119862119862 = 0) this gives
ℏpart119886119886119953119953(120591120591)120597120597120591120591
= 1198671198670119886119886119953119953(120591120591) = 1198671198670 1198901198901198671198670120591120591ℏ 119886119886119953119953119890119890
minus1198671198670120591120591ℏ = 1198671198670119890119890
1198671198670120591120591ℏ 119886119886119953119953119890119890
minus1198671198670120591120591ℏ minus 119890119890
1198671198670120591120591ℏ 119886119886119953119953119890119890
minus1198671198670120591120591ℏ 1198671198670
= 1198901198901198671198670120591120591ℏ 1198671198670119886119886119953119953 minus 1198861198861199531199531198671198670119890119890
minus1198671198670120591120591ℏ = 119890119890
1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890
minus1198671198670120591120591ℏ
ie
ℏ part119886119886119953119953(120591120591)
120597120597120591120591= 119890119890
1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890
minus1198671198670120591120591ℏ (43)
The commutator on the right-hand-side of Eq (43) can be evaluated as follows
1198671198670119886119886119953119953prime = sum ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953 + 1
2119953119953 119886119886119953119953prime = sum ℏ120596120596119953119953119886119886119953119953
dagger119886119886119953119953119886119886119953119953prime119953119953 = sum ℏ120596120596119953119953119886119886119953119953dagger119886119886119953119953119886119886119953119953prime minus 119886119886119953119953prime119886119886119953119953
dagger119886119886119953119953119953119953 =
sum ℏ120596120596119953119953119886119886119953119953dagger119886119886119953119953119886119886119953119953prime minus 120575120575119953119953prime119953119953 + 119886119886119953119953
dagger119886119886119953119953prime119886119886119953119953119953119953 = minusℏ120596120596119953119953prime119886119886119953119953prime + sum ℏ120596120596119953119953119886119886119953119953dagger119886119886119953119953119886119886119953119953prime minus 119886119886119953119953
dagger119886119886119953119953119886119886119953119953prime119953119953 = minusℏ120596120596119953119953prime119886119886119953119953prime
ie
1198671198670119886119886119953119953prime = minusℏ120596120596119953119953prime119886119886119953119953prime (44)
Therefore we have
ℏ part119886119886119953119953(120591120591)
120597120597120591120591= 119890119890
1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890
minus1198671198670120591120591ℏ = minusℏ120596120596119953119953119890119890
1198671198670120591120591ℏ 119886119886119953119953119890119890
minus1198671198670120591120591ℏ = minusℏ120596120596119953119953119886119886119953119953(120591120591) (45)
11
The solution of Eq (45) is 119886119886119953119953(120591120591) = 119886119886119953119953(0)119890119890minus120596120596119953119953120591120591 = 119886119886119953119953119890119890minus120596120596119953119953120591120591 For 119886119886119953119953dagger(120591120591) we have
1198671198670119886119886119953119953primedagger (0) = 1198671198670119886119886119953119953prime
dagger = ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953 +
12
119953119953
119886119886119953119953primedagger = ℏ120596120596119953119953119886119886119953119953
dagger119886119886119953119953119886119886119953119953primedagger
119953119953
= ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953119886119886119953119953prime
dagger minus 119886119886119953119953primedagger 119886119886119953119953
dagger119886119886119953119953119953119953
= ℏ120596120596119953119953 119886119886119953119953dagger 120575120575119953119953119953119953prime + 119886119886119953119953prime
dagger 119886119886119953119953 minus 119886119886119953119953primedagger 119886119886119953119953
dagger119886119886119953119953119953119953
= ℏ120596120596119953119953prime119886119886119953119953primedagger + ℏ120596120596119953119953 119886119886119953119953
dagger119886119886119953119953primedagger 119886119886119953119953 minus 119886119886119953119953prime
dagger 119886119886119953119953dagger119886119886119953119953
119953119953
= ℏ120596120596119953119953prime119886119886119953119953primedagger
In summary we have
119886119886119953119953(120591120591) = 119886119886119953119953(0)119890119890minus120596120596119953119953120591120591 = 119886119886119953119953119890119890minus120596120596119953119953120591120591
119886119886119953119953dagger(120591120591) = 119886119886119953119953
dagger(0)119890119890120596120596119953119953120591120591 = 119886119886119953119953dagger119890119890120596120596119953119953120591120591 (46)
42 Greenrsquos function
To describe thermal transport properties between different system sections we use the formalism of
thermal (or imaginary time) phonon Greenrsquos function [2] To this end we define the thermal Greenrsquos
function for phonons as
119866119866119953119953119953119953prime(120591120591 120591120591prime) = minuslang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime)rang = minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime) (47)
where 119879119879120591120591 is the time ordering operator and 120588120588119867119867 is the statistical operator for the grand canonical
ensemble (note that the chemical potential for phonons is zero)
120588120588119867119867 = 119890119890minus120573120573119867119867119885119885119867119867 (48)
where the partition function is given by 119885119885119867119867 equiv Tr119890119890minus120573120573119867119867 and 120573120573 = 1119896119896119861119861119879119879 with 119896119896119861119861 being
Boltzmannrsquos constant and 119879119879 the temperature For time independent Hamiltonians 119866119866119953119953119953119953prime(120591120591 120591120591prime)
depends only on 120591120591 minus 120591120591prime ie
119866119866119953119953119953119953prime(120591120591 120591120591prime) = 119866119866119953119953119953119953prime(120591120591 minus 120591120591prime 0) (49)
To show this we shall first assume that 120591120591 gt 120591120591prime such that
119866119866119953119953119953119953prime(120591120591 120591120591prime) = minuslang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime)rang = minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime) = minusTr 120588120588119867119867119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime)
= minusTr 120588120588119867119867119890119890119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591ℏ 119890119890
119867119867120591120591primeℏ 119886119886119953119953prime
dagger 119890119890minus119867119867120591120591primeℏ = minusTr 120588120588119867119867119890119890
minus119867119867120591120591primeℏ 119890119890
119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591minus120591120591prime
ℏ 119886119886119953119953primedagger
= minusTr 120588120588119867119867119890119890119867119867120591120591minus120591120591prime
ℏ 119886119886119953119953119890119890minus119867119867120591120591minus120591120591prime
ℏ 119886119886119953119953primedagger = minusTr 120588120588119867119867119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953prime
dagger (0)
= minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953primedagger (0) = minus lang119879119879120591120591119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953prime
dagger (0)rang = 119866119866119953119953119953119953prime(120591120591 minus 120591120591prime 0)
Where we have used the commutativity of 120588120588119867119867 and 119890119890plusmn119867119867120591120591prime
ℏ and the invariance of the trace operation
12
towards cyclic permutations Similarly for 120591120591 lt 120591120591prime we have
119866119866119953119953119953119953prime(120591120591 120591120591prime) = minuslang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime)rang = minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime) = minusTr 120588120588119867119867119886119886119953119953primedagger (120591120591prime)119886119886119953119953(120591120591)
= minusTr 120588120588119867119867119890119890119867119867120591120591primeℏ 119886119886119953119953prime
dagger 119890119890minus119867119867120591120591primeℏ 119890119890
119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591ℏ = minusTr 120588120588119867119867119886119886119953119953prime
dagger 119890119890119867119867120591120591minus120591120591prime
ℏ 119886119886119953119953119890119890minus119867119867120591120591ℏ 119890119890
119867119867120591120591primeℏ
= minusTr 120588120588119867119867119886119886119953119953primedagger 119890119890
119867119867120591120591minus120591120591primeℏ 119886119886119953119953119890119890
minus119867119867120591120591minus120591120591prime
ℏ = minusTr 120588120588119867119867119886119886119953119953primedagger (0)119886119886119953119953(120591120591 minus 120591120591prime)
= minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953primedagger (0)119886119886119953119953(120591120591 minus 120591120591prime) = minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953prime
dagger (0)
= minuslang119879119879120591120591119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953primedagger (0)rang = 119866119866119953119953119953119953prime(120591120591 minus 120591120591prime 0)
For simplicity we will introduce the notation 119866119866119953119953119953119953prime(120591120591 0) equiv 119866119866119953119953119953119953prime(120591120591) We further note that when using
imaginary time 119866119866119953119953119953119953prime(120591120591) is a periodic functions in the domain [minus120573120573ℏ120573120573ℏ] with a period of 120573120573ℏ (see
Page 236 of Ref [2])
119866119866119953119953119953119953prime(120591120591 + 120573120573ℏ 0) = 119866119866119953119953119953119953prime(120591120591 0) 120591120591 lt 0119866119866119953119953119953119953prime(120591120591 minus 120573120573ℏ 0) = 119866119866119953119953119953119953prime(120591120591 0) 120591120591 gt 0 (410)
To show this we shall again assume first that minus120573120573ℏ lt 120591120591 lt 0 to write that is
119866119866119953119953119953119953prime(120591120591 + 120573120573ℏ 0) = minus lang119879119879120591120591119886119886119953119953(120591120591 + 120573120573ℏ)119886119886119953119953primedagger (0)rang = minuslang119886119886119953119953(120591120591 + 120573120573ℏ)119886119886119953119953prime
dagger (0)rang
= minusTr 120588120588119867119867119890119890119867119867(120591120591+120573120573ℏ)
ℏ 119886119886119953119953119890119890minus119867119867(120591120591+120573120573ℏ)
ℏ 119886119886119953119953prime = minusTr 120588120588119867119867119890119890120573120573119867119867119890119890
119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591ℏ 119890119890minus120573120573119867119867119886119886119953119953prime
= minusTr120588120588119867119867119890119890120573120573119867119867119886119886119953119953(120591120591)119890119890minus120573120573119867119867119886119886119953119953prime(0) = minus
1119885119885119867119867
Tr119890119890minus120573120573119867119867119890119890120573120573119867119867119886119886119953119953(120591120591)119890119890minus120573120573119867119867119886119886119953119953prime(0)
= minus1119885119885119867119867
Tr119886119886119953119953(120591120591)119890119890minus120573120573119867119867119886119886119953119953prime(0) = minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119886119886119953119953prime(0)119886119886119953119953(120591120591)
= minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime(0) = minusTr120588120588119867119867119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime = 119866119866119953119953119953119953prime(120591120591 0)
Similarly for 120573120573ℏ gt 120591120591 gt 0 we have
119866119866119953119953119953119953prime(120591120591 minus 120573120573ℏ 0) = minus lang119879119879120591120591119886119886119953119953(120591120591 minus 120573120573ℏ)119886119886119953119953primedagger (0)rang = minuslang119886119886119953119953prime
dagger (0)119886119886119953119953(120591120591 minus 120573120573ℏ)rang
= minusTr 120588120588119867119867119886119886119953119953prime119890119890119867119867(120591120591minus120573120573ℏ)
ℏ 119886119886119953119953119890119890minus119867119867(120591120591minus120573120573ℏ)
ℏ = minusTr 120588120588119867119867119886119886119953119953prime119890119890minus120573120573119867119867119890119890
119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591ℏ 119890119890120573120573119867119867
= minusTr120588120588119867119867119886119886119953119953prime(0)119890119890minus120573120573119867119867119886119886119953119953(120591120591)119890119890120573120573119867119867 = minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119886119886119953119953prime(0)119890119890minus120573120573119867119867119886119886119953119953(120591120591)119890119890120573120573119867119867
= minus1119885119885119867119867
Tr119886119886119953119953prime(0)119890119890minus120573120573119867119867119886119886119953119953(120591120591) = minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119886119886119953119953(120591120591)119886119886119953119953prime(0)
= minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime(0) = minusTr120588120588119867119867119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime = 119866119866119953119953119953119953prime(120591120591 0)
13
Therefore 119866119866119953119953119953119953prime(120591120591) can be expanded as a Fourier series in the domain [0120573120573ℏ] as follows
119866119866119953119953119953119953prime(120591120591) = 1120573120573ℏsum 119890119890minus119894119894120596120596119899119899120591120591119866119866119953119953119953119953prime(119894119894120596120596119899119899)infinminusinfin (411)
where 120596120596119899119899 = 2120587120587119899119899120573120573ℏ
and the associated Fourier coefficient is given by
119866119866119953119953119953119953prime(119894119894120596120596119899119899) = int d120591120591119890119890119894119894120596120596119899119899120591120591119866119866119953119953119953119953prime(120591120591)120573120573ℏ0 120596120596119899119899 = 2119899119899120587120587
120573120573ℏ (412)
Having proven the translational time invariance and the periodicity of the Greenrsquos functios we can
now calculate it for the uncoupled system
119866119866119953119953119953119953prime0 (120591120591 120591120591prime) = minuslang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime)rang0 == minus lang119886119886119953119953(0)119890119890minus120596120596119953119953120591120591119886119886119953119953prime
dagger (0)119890119890120596120596119953119953prime120591120591primerang 120591120591 minus 120591120591prime gt 0
minus lang119886119886119953119953primedagger (0)119890119890120596120596119953119953prime120591120591
prime119886119886119953119953(0)119890119890minus120596120596119953119953120591120591rang 120591120591 minus 120591120591prime lt 0
= minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591
primelang119886119886119953119953(0)119886119886119953119953prime
dagger (0)rang 120591120591 minus 120591120591prime gt 0
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang 120591120591 minus 120591120591prime lt 0
= minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591
primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0) + 119886119886119953119953(0)119886119886119953119953primedagger (0)rang 120591120591 minus 120591120591prime gt 0
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang 120591120591 minus 120591120591prime lt 0
= minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591
primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang + 120575120575119953119953119953119953prime 120591120591 minus 120591120591prime gt 0
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang 120591120591 minus 120591120591prime lt 0
ie
119866119866119953119953119953119953prime0 (120591120591 120591120591prime) =
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang + 120575120575119953119953119953119953prime 120591120591 minus 120591120591prime gt 0
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang 120591120591 minus 120591120591prime lt 0 (413)
where we have used Eq (46) for 119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime) and Commutation relation 119886119886119953119953(0)119886119886119953119953prime
dagger (0) = 120575120575119953119953119953119953prime
To calculate lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang we first prove following equation
119890119890119860119860119861119861119890119890minus119860119860 = sum 1119899119899119860(119899119899)119861119861infin
119899119899=0 (414)
where 119860(119896119896)119861119861 equiv 119860 119860(119896119896minus1)119861119861 To prove this identity we define the operator 119891119891(119905119905) = 119890119890119905119905A119861119861119890119890minus119905119905119860119860
and taylor expand it around 119905119905 = 0
119891119891(119905119905) = 119891119891(0) + 119905119905119891119891prime(0) + 1199051199052
2119891119891primeprime(0) + ⋯ = sum 119905119905119899119899
119899119899119889119889119899119899119891119889119889119905119905119899119899
119905119905=0
infin119899119899=0 (415)
The corresponding derivatives are given by
⎩⎪⎨
⎪⎧ 119891119891prime(119905119905) = 119890119890119905119905119860119860119860119861119861119890119890minus119905119905119860119860 minus 119890119890119905119905119860119860119861119861119860119890119890minus119905119905119860119860 = 119890119890119905119905119860119860119860119861119861119890119890minus119905119905119860119860
119891119891primeprime(119905119905) = 119890119890119905119905119860119860119860119860119861119861 minus 119860119861119861119860119890119890minus119905119905119860119860 = 119890119890119905119905119860119860119860(2)119861119861119890119890minus119905119905119860119860
⋮119891119891(119899119899)(119905119905) = 119890119890119905119905119860119860119860(119899119899)119861119861119890119890minus119905119905119860119860
(416)
14
Substituting Eq (416) into Eq (415) we have
119890119890119905119905A119861119861119890119890minus119905119905119860119860 = sum 119905119905119899119899
119899119899119860(119899119899)119861119861infin
119899119899=0 (417)
Itrsquos clear that Eq (413) is a spectial case of Eq (416) with 119905119905 = 1 With this we can proceed as
follows
lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang =
11198851198851198671198670
Tr119890119890minus1205731205731198671198670119886119886119953119953primedagger (0)119886119886119953119953(0) =
11198851198851198671198670
Tr119890119890minus1205731205731198671198670119886119886119953119953primedagger (0)1198901198901205731205731198671198670119890119890minus1205731205731198671198670119886119886119953119953(0)
=11198851198851198671198670
Tr (minus120573120573)119899119899
1198991198991198671198670
(119899119899)119886119886119953119953primedagger (0)
infin
119899119899=0
119890119890minus1205731205731198671198670119886119886119953119953(0)
Note that 1198671198670119886119886119953119953primedagger (0) = ℏ120596120596119953119953prime119886119886119953119953prime
dagger (0) we have
1198671198670(119899119899)119886119886119953119953prime
dagger (0) = ℏ120596120596119953119953prime119899119899119886119886119953119953primedagger (0) (418)
such that
lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang =
11198851198851198671198670
Tr (minus120573120573)119899119899
1198991198991198671198670
(119899119899)119886119886119953119953primedagger (0)
infin
119899119899=0
119890119890minus1205731205731198671198670119886119886119953119953(0)
=11198851198851198671198670
Tr minus120573120573ℏ120596120596119953119953prime
119899119899
119899119899119886119886119953119953primedagger (0)
infin
119899119899=0
119890119890minus1205731205731198671198670119886119886119953119953(0)
= 119890119890minus120573120573ℏ120596120596119953119953prime11198851198851198671198670
Tr119890119890minus1205731205731198671198670119886119886119953119953(0)119886119886119953119953primedagger (0) = 119890119890minus120573120573ℏ120596120596119953119953primelang119886119886119953119953(0)119886119886119953119953prime
dagger (0)rang
= 119890119890minus120573120573ℏ120596120596119953119953primelang119886119886119953119953primedagger (0)119886119886119953119953(0)rang + 120575120575119953119953119953119953prime
therefore we obtain lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang 119890119890120573120573ℏ120596120596119953119953prime minus 1 = 120575120575119953119953119953119953prime For 119953119953prime ne 119953119953 and general 119890119890120573120573ℏ120596120596119953119953prime we have
lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang = 0 For 119953119953prime = 119953119953 we obtain lang119886119886119953119953
dagger(0)119886119886119953119953(0)rang = 119890119890120573120573ℏ120596120596119953119953 minus 1minus1
Hence we have
lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang = 120575120575119953119953prime119953119953
1119890119890120573120573ℏ120596120596119953119953minus1
equiv 120575120575119953119953prime119953119953119899119899119861119861120596120596119953119953 (419)
where 119899119899119861119861120596120596119953119953 is the Bose-Einstein distribution for phonons Substituting Eq (418) in Eq (413)
we obtain
119866119866119953119953119953119953prime0 (120591120591 120591120591prime) =
minus120575120575119953119953119953119953prime1 + 119899119899119861119861120596120596119953119953119890119890minus120596120596119953119953120591120591minus120591120591prime 120591120591 minus 120591120591prime gt 0
minus120575120575119953119953119953119953prime119899119899119861119861120596120596119953119953119890119890minus120596120596119953119953120591120591minus120591120591prime 120591120591 minus 120591120591prime lt 0 (420)
Note that only those Greenrsquos functions of the form 119866119866119953119953119953119953prime0 (120591120591 120591120591prime) = 1198661198661199531199530(120591120591 minus 120591120591prime)120575120575119953119953119953119953prime are non-zero due
to the orthogonality of normal modes Looking at the non-vanishing terms and setting 120591120591prime = 0
without loss of generality we can calculate the Fourier transform of 1198661198661199531199530(120591120591) from Eq (412)
1198661198661199531199530(119894119894120596120596119899119899) = int d1205911205911198901198901198941198941205961205961198991198991205911205911198661198661199531199530(120591120591)120573120573ℏ0 = minusint d120591120591119890119890119894119894120596120596119899119899120591120591120573120573ℏ
0 1 + 119899119899119861119861120596120596119953119953119890119890minus120596120596119953119953120591120591 = minus1+119899119899119861119861120596120596119953119953119894119894120596120596119899119899minus120596120596119953119953
119890119890119894119894120596120596119899119899minus120596120596119953119953120573120573ℏ minus
15
1 = minus1+ 1
119890119890120573120573ℏ120596120596119953119953minus1
119894119894120596120596119899119899minus120596120596119953119953119890119890119894119894
2120587120587119899119899120573120573ℏ 120573120573ℏ119890119890minus120573120573ℏ120596120596119953119953 minus 1 = minus 119890119890120573120573ℏ120596120596119953119953
119894119894120596120596119899119899minus120596120596119953119953
119890119890minus120573120573ℏ120596120596119953119953minus1
119890119890120573120573ℏ120596120596119953119953minus1= minus 1
119894119894120596120596119899119899minus120596120596119953119953
1minus119890119890120573120573ℏ120596120596119953119953
119890119890120573120573ℏ120596120596119953119953minus1= 1
119894119894120596120596119899119899minus120596120596119953119953
ie
1198661198661199531199530(119894119894120596120596119899119899) == 1119894119894120596120596119899119899minus120596120596119953119953
(421)
5 Fermirsquos golden rule 51 Interaction picture
Next we can proceed with calculating the Greenrsquos function of the coupled system To this end we
define the coupling Hamiltonian operator term in the interaction picture as
119867119867119862119862119868119868 (120591120591) = 1198901198901198671198670120591120591ℏ 119867119867119862119862119890119890
minus1198671198670120591120591ℏ (51)
The time evolution of 119867119867119862119862(120591120591) is then given by
ℏ part119867119867119862119862119868119868 (120591120591)120597120597120591120591
= 1198671198670119867119867119862119862119868119868 (120591120591) (52)
Note that the Greenrsquos function defined above was given in the Heisenberg picture To proceed we
need to transform it to the interaction picture
119866119866119953119953119953119953prime(120591120591 0) = minus lang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953primedagger (0)rang = minusTr 120588120588119867119867119879119879120591120591 119890119890
119867119867120591120591ℏ 119886119886119953119953(0)119890119890minus
119867119867120591120591ℏ 119886119886119953119953prime
dagger (0)
= minusTr 120588120588119867119867119879119879120591120591 119890119890119867119867120591120591ℏ 119890119890minus
1198671198670120591120591ℏ 119890119890
1198671198670120591120591ℏ 119886119886119953119953(0)119890119890minus
1198671198670120591120591ℏ 119890119890
1198671198670120591120591ℏ 119890119890minus
119867119867120591120591ℏ 119886119886119953119953prime
dagger (0)
= minusTr 120588120588119867119867119879119879120591120591 119880119880(0 120591120591)119886119886119953119953119868119868 (120591120591)119880119880(120591120591 0)119886119886119953119953primedagger (0)
where
119886119886119953119953119868119868 (120591120591) equiv 1198901198901198671198670120591120591ℏ 119886119886119953119953(0)119890119890minus
1198671198670120591120591ℏ (53)
is the operator in interaction picture and the operator 119880119880 is defined by
119880119880(1205911205911 1205911205912) equiv 11989011989011986711986701205911205911ℏ 119890119890minus
119867119867 (1205911205911minus1205911205912)ℏ 119890119890minus
11986711986701205911205912ℏ (54)
Note that while 119880119880 is not unitary it satisfies the following group property
119880119880(1205911205911 1205911205912)119880119880(1205911205912 1205911205913) = 119880119880(1205911205911 1205911205913) (55)
and the boundary condition 119880119880(1205911205911 1205911205911) = 1 In addition the 120591120591 derivative of 119880119880 is simply
ℏpart119880119880(120591120591 120591120591prime)
120597120597120591120591= 1198671198670119890119890
1198671198670120591120591ℏ 119890119890minus
119867119867120591120591minus120591120591primeℏ minus 119890119890
1198671198670120591120591ℏ 119867119867119890119890minus
119867119867120591120591minus120591120591primeℏ 119890119890minus
1198671198670120591120591primeℏ
= 1198901198901198671198670120591120591ℏ 1198671198670 minus 119867119867119890119890minus
1198671198670120591120591ℏ 119890119890
1198671198670120591120591ℏ 119890119890minus
119867119867120591120591minus120591120591primeℏ 119890119890minus
1198671198670120591120591primeℏ = 119890119890
1198671198670120591120591ℏ minus119867119867119862119862119890119890
minus1198671198670120591120591ℏ 119880119880(120591120591 120591120591prime)
= minus119867119867119862119862119868119868 (120591120591)119880119880(120591120591 120591120591prime)
16
ie
ℏ part119880119880120591120591120591120591prime120597120597120591120591
= minus119867119867119862119862119868119868 (120591120591)119880119880(120591120591 120591120591prime) (56)
The solution of Eq (56) is (see page 235 of Ref [2])
119880119880(120591120591 120591120591prime) = summinus1ℏ
119899119899
119899119899 int 1198891198891205911205911120591120591120591120591prime ⋯int 119889119889120591120591119899119899
120591120591120591120591prime 119879119879120591120591119867119867119862119862119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119899119899)infin
119899119899=0 (57)
The exact thermal Greens function now may be rewritten in the interaction picture as
119866119866119953119953119953119953prime(120591120591 0) = minus1119885119885119867119867
Tr 119890119890minus120573120573119867119867119879119879120591120591119880119880(0 120591120591)119886119886119953119953119868119868 (120591120591)119880119880(120591120591 0)119886119886119953119953primedagger (0)
= minusTr 119890119890minus120573120573119867119867119879119879120591120591 119880119880(0 120591120591)119886119886119953119953119868119868 (120591120591)119880119880(120591120591 0)119886119886119953119953prime
dagger (0)
Tr119890119890minus120573120573119867119867
= minusTr 119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)119879119879120591120591 119886119886119953119953119868119868 (120591120591)119880119880(0 120591120591)119880119880(120591120591 0)119886119886119953119953prime
dagger (0)
Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)
= minusTr 119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)119879119879120591120591 119886119886119953119953119868119868 (120591120591)119880119880(00)119886119886119953119953prime
dagger (0)
Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)=
= minusTr 119890119890minus1205731205731198671198670119879119879120591120591 119880119880(120573120573ℏ 0)119886119886119953119953119868119868 (120591120591)119886119886119953119953prime
dagger (0)
Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)
Where we used the fact that we are free to change the order of the operators within the time ordering
operation (see pages 241-242 of Ref [2])
52 Wickrsquos theorem
Thus the Greenrsquos function can be expanded as [2]
119866119866119953119953119953119953prime(120591120591 0) = minussum 1
119898119898minus1ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)119886119886119953119953119868119868 (120591120591)119886119886
119953119953primedagger (0)rang0infin
119898119898=0
sum 1119898119898minus
1ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)rang0infin
119898119898=0 (58)
where lang⋯ rang0 represents the ensemble average with respect to the non-interacting basis
Tr119890119890minus1205731205731198671198670(⋯ ) Or explicitly
119866119866119953119953119953119953prime(120591120591 0) = minus119866119866119953119953119953119953prime0 (120591120591)minus1ℏint 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953primedagger (0)rang0+
12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0+⋯
1minus1ℏint 1198891198891205911205911120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)rang0+12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0+⋯
(59)
For consiceness we introduce the following notation
119863119863119898119898 equiv 1119898119898minus 1
ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)rang0 (510)
To simplify the calculation in Eq (58) we adopt Wickrsquos theorem (see pages 237-241 of Ref
[2])which can be expressed as follows
lang119879119879120591120591119860119861119861119862119863119863 ⋯119884119884119885rang0 = lang119879119879120591120591119860119861119861rang0lang119879119879120591120591119862119863119863rang0⋯ lang119879119879120591120591119884119884119885rang0 + lang119879119879120591120591119860119862rang0lang119879119879120591120591119861119861119863119863rang0⋯ lang119879119879120591120591119883119883119885rang0 + ⋯ (511)
17
Here 119860119861119861 hellip 119884119884 119885 represent 119886119886119953119953(120591120591) or 119886119886119953119953dagger(120591120591) In Eq (511) each term corresponds to a particular
pairing of the operators 119860119861119861119862119863119863 ⋯119884119884119885 and all possible pairings are taken into account Here the only
non-vanishing propagators will have the form [see Eq (420)]
lang119879119879120591120591 119886119886119953119953primedagger (120591120591prime)119886119886119953119953(120591120591)rang0 = minuslang119879119879120591120591 119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime)rang0 = 119866119866119953119953119953119953prime0 (120591120591 120591120591prime) = 1198661198661199531199530(120591120591 minus 120591120591prime)120575120575119953119953119953119953prime (512)
Therefore the second term in the numerator of Eq (59) can be calculated as
1198681198682 = minus1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0
= minus1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 lang119879119879120591120591119886119886119953119953prime
dagger (0)119886119886119953119953(120591120591)rang0 minus1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
= 119866119866119953119953119953119953prime0 (120591120591 0) minus
1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 minus
1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
The first term in above equation is called disconnected part since the pairing is performed separately
on 119867119867119862119862(1205911205911) and 119886119886119953119953(120591120591)119886119886119953119953primedagger (0) All other terms have pairs that mix creation and annihilation operators
of the Hamiltonian with 119886119886119953119953(120591120591) or 119886119886119953119953primedagger (0) and are said to have connected party For simplicity we
include all these terms in the notation lang119879119879120591120591119867119867119862119862(1205911205911)119886119886119953119953(120591120591)119886119886119953119953primedagger (0)rang0119888119888 Furthermore we define 119866119866119953119953119953119953prime
(1) (120591120591) equiv
minus1ℏ int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862(1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 Then we have
1198681198683 = minus1ℏint 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 = 119866119866119953119953119953119953prime0 (120591120591 0)1198631198631 + 119866119866119953119953119953119953prime
(1) (120591120591) (513)
Using this method the third term in the numerator of Eq (59) can be calculated as
1198681198683 = 12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 = 12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0 lang119879119879120591120591119886119886119953119953prime
dagger (0)119886119886119953119953(120591120591)rang0 +
12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 + 12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 +
12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 = 119866119866119953119953119953119953prime0 (120591120591 0) 1
2ℏ2 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0 +
12ℏ2 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 + 12ℏ2 int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 +
12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
Here the last term is denoted as 119866119866119953119953119953119953prime(2) (120591120591)
Using Eq (513) the second and third terms on the right hand above equation can be simplified as
follows
18
12ℏ2
1198891198891205911205912120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 +1
2ℏ2 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
=12 minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0
minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
+12 minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0
minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
=12 minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0
minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
+12 minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0
minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
= minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0
minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 = 1198631198631119866119866119953119953119953119953prime(1) (120591120591)
Where we performed the following integration variables interchange 1205911205911 ⟷ 1205911205912 Thus we have
1198681198683 = 119866119866119953119953119953119953prime(2) (120591120591) + 119866119866119953119953119953119953prime
(1) (120591120591)1198631198631 + 119866119866119953119953119953119953prime0 (120591120591 0)1198631198632 (514)
Similarity we can calculate the fourth term of Eq (59) as
1198681198684 = minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 =
minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0 lang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 + 3 times
minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 + 3 times
minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 +
minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 =
119866119866119953119953119953119953prime0 (120591120591 0) minus1
6ℏ3 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0+
12ℏ2 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0
minus1ℏ int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888+
12ℏ2 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 minus1ℏ int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0+ 119866119866119953119953119953119953prime
(3) (120591120591) = 119866119866119953119953119953119953prime(3) (120591120591) +
119866119866119953119953119953119953prime(2) (120591120591)1198631198631 + 119866119866119953119953119953119953prime
(1) (120591120591)1198631198632 + 119866119866119953119953119953119953prime0 (120591120591 0)1198631198633
Higher order terms can be treated in the same manner (see pages 95-96 of Ref [2]) Then when
substituting 1198681198682 1198681198683 1198681198684 and all higher order terms into Eq (59) we can simplify the numerator as
follows
119866119866119953119953119953119953prime0 (120591120591) minus
1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0
+1
2ℏ2 1198891198891205911205911120573120573ℏ
0 1198891198891205911205912120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 + ⋯
= 119866119866119953119953119953119953prime0 (120591120591) + 119866119866119953119953119953119953prime
(1) (120591120591) + 119866119866119953119953119953119953prime(2) (120591120591) + ⋯ (1 + 1198631198631 + 1198631198632 + ⋯ )
Noting that the expression (1 + 1198631198631 + 1198631198632 + ⋯ ) is exactly canceled with the denominator the
Greenrsquos function can be simplified as
19
119866119866119953119953119953119953prime(120591120591) = 119866119866119953119953119953119953prime0 (120591120591) + 119866119866119953119953119953119953prime
(1) (120591120591) + 119866119866119953119953119953119953prime(2) (120591120591) + ⋯ (515)
where
119866119866119953119953119953119953prime
(1) (120591120591) = minus1ℏ int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888
119866119866119953119953119953119953prime(2) (120591120591) = 1
2ℏ2 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888 (516)
53 Calculation of Greenrsquos function
531 First order appriximation In Eq (516) we go back to the full notation 119886119886119954119954120582120582 to distinguish phonons of different branches
then 119866119866119953119953119953119953prime(1) (120591120591) is calculated as
119866119866119953119953119953119953prime(1) (120591120591) = minus
1ℏ
12 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591 119881119881119895119895119895119895prime(119948119948)119876119876119948119948119895119895(1205911205911)119876119876119948119948119895119895prime
dagger (1205911205911)3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ 119881119881119895119895119895119895primelowast (119948119948)119876119876119948119948119895119895prime(1205911205911)119876119876119948119948119895119895
dagger (1205911205911)3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= minus12ℏ
1198891198891205911205911120573120573ℏ
0lang119879119879120591120591
ℏ119881119881119895119895119895119895prime(119948119948)2120596120596119948119948119895119895120596120596119948119948119895119895prime
119886119886119948119948119895119895(1205911205911) + 119886119886minus119948119948119895119895dagger (1205911205911) 119886119886minus119948119948119895119895prime(1205911205911) + 119886119886119948119948119895119895prime
dagger (1205911205911)3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ ℏ119881119881119895119895119895119895prime
lowast (119948119948)
2120596120596119948119948119895119895120596120596119948119948119895119895prime119886119886119948119948119895119895prime(1205911205911) + 119886119886minus119948119948119895119895prime
dagger (1205911205911) 119886119886minus119948119948119895119895(1205911205911) + 119886119886119948119948119895119895dagger (1205911205911)
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
According to Wickrsquos theorem [2] the terms that contain 119886119886119948119948119895119895119886119886minus119948119948119895119895prime and 119886119886minus119948119948119895119895dagger 119886119886119948119948119895119895prime
dagger are equal to zero
thus the first term of above equation are calculated as
11989211989211 = minus14
119881119881119895119895119895119895prime(119948119948)
120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119886119886minus119948119948119895119895
dagger (1205911205911)119886119886minus119948119948119895119895prime(1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ lang119879119879120591120591119886119886119948119948119895119895(1205911205911)119886119886119948119948119895119895primedagger (1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888
= minus14
119881119881119895119895119895119895prime(119948119948)
120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119886119886minus119948119948119895119895
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus119948119948119895119895prime(1205911205911)rang0
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ lang119879119879120591120591119886119886119948119948119895119895primedagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)119886119886119948119948119895119895(1205911205911)rang0
= minus14
119881119881119895119895119895119895prime(119948119948)
120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
01198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954minus119948119948120575120575120582120582119895119895119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119954119954primeminus119948119948120575120575120582120582prime119895119895prime3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ 1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119948119948120575120575120582120582119895119895prime119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119954119954prime119948119948120575120575120582120582prime119895119895
20
Simplification of above equation gives
11989211989211 =
⎩⎪⎨
⎪⎧minus119881119881120582120582120582120582prime(minus119954119954)
4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954120582120582
0 (120591120591 1205911205911)119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903
2lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582(119954119954)
4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954120582120582
0 (120591120591 1205911205911)119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582prime le 3119903119903
2lt 120582120582 le 3119903119903
0 else
(517)
Similarity the second term of 119866119866119953119953119953119953prime(1) (120591120591) is calculated as
11989211989212 = minus14
minus119881119881119895119895119895119895primelowast (119948119948)
4120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119886119886119948119948119895119895prime(1205911205911)119886119886119948119948119895119895
dagger (1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ lang119879119879120591120591119886119886minus119948119948119895119895primedagger (1205911205911)119886119886minus119948119948119895119895(1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888
= minus14
minus119881119881119895119895119895119895primelowast (119948119948)
4120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)119886119886119948119948119895119895prime(1205911205911)rang0 lang119879119879120591120591119886119886119948119948119895119895dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus119948119948119895119895(1205911205911)rang0 lang119879119879120591120591119886119886minus119948119948119895119895prime
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
ie
11989211989212 =
⎩⎪⎨
⎪⎧
minus119881119881120582120582120582120582primelowast (119954119954)
4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954prime120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903
2lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582lowast (minus119954119954)
4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954prime120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119954119954prime 1 le 120582120582prime le 3119903119903
2lt 120582120582 le 3119903119903
0 else
(518)
Note that since 119881119881120582120582120582120582primelowast (119954119954) = 119881119881120582120582120582120582prime(minus119954119954) [Eq (314)] the first order approximation of Greenrsquos function
can be written as
119866119866119954119954120582120582119954119954prime120582120582prime(1) (120591120591) =
⎩⎪⎨
⎪⎧minus119881119881
120582120582120582120582prime(minus119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ
0 1198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582 le3119903119903
2lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582
(119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ
0 1198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582prime le3119903119903
2lt 120582120582 le 3119903119903
0 else
(519)
To derive the Greenrsquos function in frequency domain we use the expression for Fourier series then
we have
1198891198891205911205911120573120573ℏ
01198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0) = 1198891198891205911205911120573120573ℏ
0
1120573120573ℏ
119890119890minus119894119894120596120596119899119899(120591120591minus1205911205911)1198661198661199541199541205821205820 (119894119894120596120596119899119899)
119899119899
1120573120573ℏ
119890119890minus119894119894120596120596119899119899prime1205911205911119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899prime)
119899119899prime
=1
(120573120573ℏ)2119890119890minus119894119894120596120596119899119899120591120591 1198891198891205911205911120573120573ℏ
0119890119890119894119894120596120596119899119899minus120596120596119899119899prime1205911205911 119866119866119954119954120582120582
0 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899prime)
119899119899119899119899prime=
=1
(120573120573ℏ)2119890119890minus119894119894120596120596119899119899120591120591[120573120573ℏ120575120575119899119899119899119899prime]1198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899prime)119899119899119899119899prime
=1120573120573ℏ
119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899)119899119899
According to the definition of Fourier series [see Eq (412)] we get the Fourier coefficient of
21
119866119866119954119954120582120582119954119954prime120582120582prime(1) (120591120591)
119866119866119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) =
⎩⎪⎨
⎪⎧minus119881119881120582120582120582120582prime(minus119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582prime1198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime 1 le 120582120582 le 31199031199032
lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582(119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582prime1198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime 1 le 120582120582prime le 31199031199032
lt 120582120582 le 3119903119903
0 else
(520)
where 119866119866119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) = 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) times Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) times 119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899) and Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) is defined as
Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) =
⎩⎪⎨
⎪⎧minus119881119881120582120582120582120582prime(minus119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582prime120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903
2lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582(119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582prime120575120575119954119954119954119954prime 1 le 120582120582prime le 3119903119903
2lt 120582120582 le 3119903119903
0 else
(521)
532 second order approximation The second order approximation of the Greenrsquos function in Eq (516) 119866119866119954119954120582120582119954119954prime120582120582prime
(2) (120591120591) is calculated as
119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) =
18ℏ2 1198891198891205911205911
120573120573ℏ
0 1198891198891205911205912120573120573ℏ
0lang119879119879120591120591
⎣⎢⎢⎡ 11988111988111989511989511198951198951prime(119948119948120783120783)1198761198761199481199481207831207831198951198951(1205911205911)1198761198761199481199481207831207831198951198951prime
dagger (1205911205911)3119903119903
1198951198951prime=31199031199032 +1
31199031199032
1198951198951=1119948119948120783120783
+ 11988111988111989411989411198941198941primelowast (119948119948120783120783)1198761198761199481199481207831207831198941198941prime(1205911205911)1198761198761199481199481207831207831198941198941
dagger (1205911205911)3119903119903
1198941198941prime=31199031199032 +1
31199031199032
1198941198941=1119948119948120783120783 ⎦⎥⎥⎤
⎣⎢⎢⎡ 11988111988111989511989521198951198952prime(119948119948120784120784)1198761198761199481199481207841207841198951198952(1205911205912)1198761198761199481199481207841207841198951198952prime
dagger (1205911205912)3119903119903
1198951198952prime=31199031199032 +1
31199031199032
1198951198952=1119948119948120784120784
+ 11988111988111989411989421198941198942primelowast (119948119948120784120784)1198761198761199481199481207841207841198941198942prime(1205911205912)1198761198761199481199481207841207841198941198942
dagger (1205911205912)3119903119903
1198941198942prime=31199031199032 +1
31199031199032
1198941198942=1119948119948120784120784 ⎦⎥⎥⎤119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888 = 1198921198921 + 1198921198922 + 1198921198923 + 1198921198924
where
1198921198921 = 18ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0
sum sum 11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989511989521198951198952prime(119948119948120784120784)1le1198951198952le
31199031199032 lt1198951198952
primele311990311990311994811994812078412078411989511989521198951198952prime
1le1198951198951le31199031199032 lt1198951198951
primele311990311990311994811994812078312078311989511989511198951198951prime
times lang1198791198791205911205911198761198761199481199481207831207831198951198951(1205911205911)1198761198761199481199481207831207831198951198951primedagger (1205911205911)1198761198761199481199481207841207841198951198952(1205911205912)1198761198761199481199481207841207841198951198952prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(522)
1198921198922 = 18ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0
sum sum 11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942primelowast (119948119948120784120784)
1le1198941198942le31199031199032 lt1198941198942
primele311990311990311994811994812078412078411989411989421198941198942prime
1le1198951198951le31199031199032 lt1198951198951
primele311990311990311994811994812078312078311989511989511198951198951prime
times lang1198791198791205911205911198761198761199481199481207831207831198951198951(1205911205911)1198761198761199481199481207831207831198951198951primedagger (1205911205911)1198761198761199481199481207841207841198941198942prime (1205911205912)1198761198761199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(523)
1198921198923 = 18ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0
sum sum 11988111988111989411989411198941198941primelowast (119948119948120783120783)11988111988111989511989521198951198952prime(119948119948120784120784)
1le1198951198952le31199031199032 lt1198951198952
primele311990311990311994811994812078412078411989511989521198951198952prime
1le1198941198941le31199031199032 lt1198941198941
primele311990311990311994811994812078312078311989411989411198941198941prime
times lang1198791198791205911205911198761198761199481199481207831207831198941198941prime (1205911205911)1198761198761199481199481207831207831198941198941dagger (1205911205911)1198761198761199481199481207841207841198951198952(1205911205912)1198761198761199481199481207841207841198951198952prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(524)
22
1198921198924 = 18ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0
sum sum 11988111988111989411989411198941198941primelowast (119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)1le1198941198942le
31199031199032 lt1198941198942
primele311990311990311994811994812078412078411989411989421198941198942prime
1le1198941198941le31199031199032 lt1198941198941
primele311990311990311994811994812078312078311989411989411198941198941prime
times lang1198791198791205911205911198761198761199481199481207831207831198941198941prime (1205911205911)1198761198761199481199481207831207831198941198941dagger (1205911205911)1198761198761199481199481207841207841198941198942prime (1205911205912)1198761198761199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(525)
In what follows we will expand the term 1198921198922 The expansion of all other terms follows the same lines
and eventually results in the same expression Expanding the product 11987611987611994811994812078312078311989511989511198761198761199481199481207831207831198951198951primedagger 1198761198761199481199481207841207841198941198942prime1198761198761199481199481207841207841198941198942
dagger we can
rewrite Eq (523) as follows
1198921198922 = 132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)119881119881
11989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032 lt1198941198942
primele311990311990311994811994812078412078411989411989421198941198942prime
1le1198951198951le31199031199032 lt1198951198951
primele311990311990311994811994812078312078311989511989511198951198951prime
times 119868119868 (526)
Where
119868119868 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911) + 119886119886minus1199481199481207831207831198951198951dagger (1205911205911) 119886119886minus1199481199481207831207831198951198951prime(1205911205911) + 1198861198861199481199481207831207831198951198951prime
dagger (1205911205911) 1198861198861199481199481207841207841198941198942prime (1205911205912) + 119886119886minus1199481199481207841207841198941198942primedagger (1205911205912) 119886119886minus1199481199481207841207841198941198942(1205911205912) + 1198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0c
= lang119879119879120591120591 1198861198861199481199481207831207831198951198951119886119886minus1199481199481207831207831198951198951prime + 11988611988611994811994812078312078311989511989511198861198861199481199481207831207831198951198951primedagger + 119886119886minus1199481199481207831207831198951198951
dagger 119886119886minus1199481199481207831207831198951198951prime + 119886119886minus1199481199481207831207831198951198951dagger 1198861198861199481199481207831207831198951198951prime
dagger 12059112059111198861198861199481199481207841207841198941198942prime119886119886minus1199481199481207841207841198941198942 + 1198861198861199481199481207841207841198941198942prime1198861198861199481199481207841207841198941198942
dagger + 119886119886minus1199481199481207841207841198941198942primedagger 119886119886minus1199481199481207841207841198941198942
+ 119886119886minus1199481199481207841207841198941198942primedagger 1198861198861199481199481207841207841198941198942
dagger 1205911205912119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0c
= lang119879119879120591120591 11988611988611994811994812078312078311989511989511198861198861199481199481207831207831198951198951primedagger + 119886119886minus1199481199481207831207831198951198951
dagger 119886119886minus1199481199481207831207831198951198951prime12059112059111198861198861199481199481207841207841198941198942prime1198861198861199481199481207841207841198941198942
dagger + 119886119886minus1199481199481207841207841198941198942primedagger 119886119886minus11994811994812078412078411989411989421205911205912
119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
Where the symbol [hellip ]1205911205911 signifies that the operators in the brackets are given in the interaction
picture The last equality in above equation comes from the fact that the contractions of the product
of the operators are equal to zero when the number of creation and annihilation operators are not the
same (see Wickrsquos theorem in Ref [2]) Using Wickrsquos theorem [2] we are able to calculate the above
equation term by term as follows
⎩⎪⎨
⎪⎧ 1198681198681 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951prime
dagger (1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)1198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888
1198681198682 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
1198681198683 = lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)1198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
1198681198684 = lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(527)
We shall now calculate them term by term
23
1198681198681 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951primedagger (1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)1198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942
dagger (1205911205912)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
+ lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951prime
dagger (1205911205911)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0
+ lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)1198861198861199481199481207841207841198941198942prime(1205911205912)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0119888119888
= 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199481199481207841207841198951198952
0 (1205911205912 1205911205912)12057512057511989411989421198941198942prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582
+ 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 1198661198661199481199481207831207831198951198951
0 (1205911205911 1205911205911)12057512057511989511989511198951198951prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582
+ 1198661198661199481199481207831207831198951198951prime0 (1205911205912 1205911205911)1205751205751199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951prime ∙ 119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582
= 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582
+ 1198661198661199481199481207831207831198951198951prime0 (1205911205912 1205911205911)1205751205751199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951prime ∙ 119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582
The last equality results from the fact that 1198951198951 and 1198951198951prime are indices of different ranges (belonging to
different subsystems) and the same holds for 1198941198942 and 1198941198942prime Similarity
1198681198684 = lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0
= 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 119866119866minus11994811994812078412078411989411989420 (1205911205912 1205911205912)12057512057511989411989421198941198942prime
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 119866119866minus1199481199481207831207831198951198951prime 0 (1205911205911 1205911205912)1205751205751199481199481207831207831199481199481207841207841205751205751198951198951prime1198941198942prime
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205911 1205911205911)12057512057511989511989511198951198951prime
+ 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205912 1205911205911)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942
= 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 119866119866minus1199481199481207831207831198951198951prime 0 (1205911205911 1205911205912)1205751205751199481199481207831207831199481199481207841207841205751205751198951198951prime1198941198942prime
+ 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205912 1205911205911)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942
Where again the last equality results from the fact that 1198951198951 and 1198951198951prime are indices of different ranges
(belonging to different subsystems) and the same holds for 1198941198942 and 1198941198942prime Similarly we obtain 1198681198682 =
1198681198683 = 0 for the same reason as shown below
24
1198681198682 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951prime
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0
+ lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)1198861198861199481199481207831207831198951198951(1205911205911)rang0
+ lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0
= 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582 ∙ 119866119866minus11994811994812078412078411989411989420 (1205911205912 1205911205912)12057512057511989411989421198941198942prime
+ 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)120575120575minus11994811994812078412078411994811994812078312078312057512057511989511989511198941198942prime ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205911)12057512057511989511989511198951198951prime
+ 1198661198661199481199481207831207831198951198951prime0 (1205911205912 1205911205911)120575120575minus11994811994812078412078411994811994812078312078312057512057511989411989421198951198951prime ∙ 119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 = 0
1198681198683 = lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)1198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0
+ lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0
+ lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0
= 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 11986611986611994811994812078412078411989411989420 (1205911205912 1205911205912)12057512057511989411989421198941198942prime
+ 11986611986611994811994812078412078411989411989420 (1205911205911 1205911205912)120575120575minus1199481199481207831207831199481199481207841207841205751205751198951198951prime1198941198942 ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205911 1205911205911)12057512057511989511989511198951198951prime
+ 1198661198661199481199481207841207841198941198942prime0 (1205911205912 1205911205911)120575120575minus1199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951 ∙ 119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 = 0
The two non-vanishing terms (1198681198681 and 1198681198684) produce the following contributions to 1198921198922 of Eq (526)
11989211989221 = 132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime1198681198681 =
132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime1198661198661199481199481207831207831198951198951
0 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙
1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582 + 1
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime1198661198661199481199481207831207831198951198951prime
0 (1205911205912 1205911205911)1205751205751199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951prime ∙
119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 =
⎩⎪⎨
⎪⎧
120575120575119954119954119954119954prime
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast (119954119954prime)
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (1205911205911 1205911205912) ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
120575120575119954119954119954119954prime
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032+1 1198661198661199541199541198951198951prime
0 (1205911205912 1205911205911) ∙ 119866119866119954119954prime120582120582prime0 (1205911205911 0)119866119866119954119954120582120582
0 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032
0 else
and
25
11989211989224 = 132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime1198681198684 =
132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙
119866119866minus1199481199481207831207831198951198951prime 0 (1205911205911 1205911205912)1205751205751199481199481207831207831199481199481207841207841205751205751198951198951prime1198941198942prime + 1
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙
1198661198661199541199541205821205820 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 119866119866minus1199481199481207831207831198951198951
0 (1205911205912 1205911205911)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 =
⎩⎪⎨
⎪⎧120575120575119954119954119954119954prime
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032+1 1198661198661199541199541198951198951prime
0 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime0 (1205911205912 0)119866119866119954119954120582120582
0 (120591120591 1205911205911) 120582120582 120582120582prime isin 1 31199031199032
120575120575119954119954119954119954prime
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast (119954119954prime)
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
0 else
Here we used the following properties 120596120596minus119954119954120582120582 = 120596120596119954119954120582120582 and 1198811198811205821205821198951198951prime(minus119954119954) = 1198811198811205821205821198951198951primelowast (119954119954) [see Eqs (112) and
(314)]
The equivalence of 11989211989221 and 11989211989224 can be seen by changing the integration variables 1205911205912 harr 1205911205911
Substituting 11989211989221 and 11989211989224 into Eq (526) 1198921198922 is given by
1198921198922 =
⎩⎪⎨
⎪⎧
120575120575119954119954119954119954prime
16 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast (119954119954prime)
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime
0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
120575120575119954119954119954119954prime
16 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032+1 1198661198661199541199541198951198951prime
0 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime0 (1205911205911 0)119866119866119954119954120582120582
0 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032
0 else
(528)
By changing the indices and integration variables itrsquos straightforward to show that the other three
terms (119892119892111989211989231198921198924 ) are identical to 1198921198922 thus the final expression of 119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) [Eq (516)) can be
calculated as
119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) = 41198921198922 =
⎩⎪⎨
⎪⎧120575120575119954119954119954119954prime
4 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032
120575120575119954119954119954119954prime
4 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast 119954119954prime
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime
0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 31199031199032
+ 13119903119903
0 else
(529)
To calculate 119866119866119954119954120582120582119954119954prime120582120582prime(2) in frequency space we expand 119866119866119954119954120582120582119954119954prime120582120582prime
(2) (120591120591) in a Fourier series
119866119866119954119954120582120582119954119954prime120582120582prime
(2) (120591120591) = 1120573120573ℏsum 119890119890minus119894119894120596120596119899119899120591120591119866119866119954119954120582120582119954119954prime120582120582prime
(2) (119894119894120596120596119899119899)119899119899
119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = int d120591120591119890119890119894119894120596120596119899119899120591120591119866119866119954119954120582120582119954119954prime120582120582prime
(2) (120591120591)120573120573ℏ0
(530)
Using Eq (421) we get for 120582120582 120582120582prime isin 1 31199031199032
26
119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) =
120575120575119954119954119954119954prime4
1198891198891205911205911120573120573ℏ
0 1198891198891205911205912120573120573ℏ
0
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1198661198661199541199541198951198951prime0 (1205911205912 1205911205911)119866119866119954119954120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912)
=120575120575119954119954119954119954prime
4
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1198891198891205911205911120573120573ℏ
0 1198891198891205911205912120573120573ℏ
0
1120573120573ℏ
119890119890minus1198941198941205961205961198991198992(1205911205912minus1205911205911)1198661198661199541199541198951198951prime0 1198941198941205961205961198991198992
1198991198992
∙1120573120573ℏ
119890119890minus11989411989412059612059611989911989911205911205911119866119866119954119954120582120582prime0 1198941198941205961205961198991198991
1198991198991
∙1120573120573ℏ
119890119890minus119894119894120596120596119899119899(120591120591minus1205911205912)1198661198661199541199541205821205820 (119894119894120596120596119899119899)
119899119899
=120575120575119954119954119954119954prime
4
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1120573120573ℏ
119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954120582120582prime
0 11989411989412059612059611989911989911198661198661199541199541198951198951prime0 1198941198941205961205961198991198992
11989911989911989911989911198991198992⎩⎪⎨
⎪⎧ 1120573120573ℏ
1198891198891205911205911120573120573ℏ
0119890119890minus1198941198941205961205961198991198991minus12059612059611989911989921205911205911
times1120573120573ℏ
1198891198891205911205912120573120573ℏ
0119890119890minus1198941198941205961205961198991198992minus1205961205961198991198991205911205912
⎭⎪⎬
⎪⎫
=120575120575119954119954119954119954prime
4
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1120573120573ℏ
119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)
11989911989911989911989911198991198992
119866119866119954119954120582120582prime0 11989411989412059612059611989911989911198661198661199541199541198951198951prime
0 1198941198941205961205961198991198992120575120575119899119899111989911989921205751205751198991198991198991198992
=120575120575119954119954119954119954prime120573120573ℏ
119890119890minus119894119894120596120596119899119899120591120591
119899119899
1198661198661199541199541205821205820 (119894119894120596120596119899119899)
14
119881119881120582120582prime1198951198951prime(119954119954
prime)1198811198811205821205821198951198951primelowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899)119866119866119954119954120582120582prime
0 (119894119894120596120596119899119899)
Comparing above expression with Eq (530) the Fourier transform of 119866119866119954119954120582120582119954119954120582120582prime(2) (120591120591) reads as
119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) 120575120575119954119954119954119954prime
4sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 119866119866119954119954120582120582prime
0 (119894119894120596120596119899119899) (531)
where we have introduced the notation
Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) equiv
120575120575119954119954119954119954prime
4sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) (532)
and 119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954120582120582prime(2) (119894119894120596120596119899119899) ∙ 119866119866119954119954120582120582prime
0 (119894119894120596120596119899119899) Similarly for 120582120582 120582120582prime isin 31199031199032
+ 13119903119903 we
have
Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) =
120575120575119954119954119954119954prime
4sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast 119954119954prime
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (119894119894120596120596119899119899) (533)
and all together we have for Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899)
Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) =
⎩⎪⎪⎨
⎪⎪⎧120575120575119954119954119954119954prime
4sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 1 3119903119903
2
120575120575119954119954119954119954prime
4sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast 119954119954prime
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
0 else
(534)
27
533 Dysonrsquos equation Substituting Eqs (521) and (534) into Eq (516) and performing a Fourier transform we have
119866119866119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) = 119866119866119954119954120582120582119954119954prime120582120582prime0 (119894119894120596120596119899119899) + 119866119866119954119954120582120582119954119954prime120582120582prime
(1) (119894119894120596120596119899119899) + 119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) + ⋯
= 1198661198661199541199541205821205820 (119894119894120596120596119899119899)120575120575119954119954119954119954prime120575120575120582120582120582120582prime + 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) ∙ 119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899) + 1198661198661199541199541205821205820 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime
(2) (119894119894120596120596119899119899)
∙ 119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899) + ⋯ = 119866119866119954119954120582120582
0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime120575120575120582120582120582120582prime + 1198661198661199541199541205821205820 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) ∙ 119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899)
or
119866119866119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) = 1198661198661199541199541205821205820 (119894119894120596120596119899119899)120575120575119954119954119954119954prime120575120575120582120582120582120582prime + 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) ∙ 119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899) (535)
Eq (535) is the so-called Dysonrsquos equation (see Ref [2]) where
Σ119954119954120582120582119954119954120582120582prime(119894119894120596120596119899119899) = Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) + Σ119954119954120582120582119954119954prime120582120582prime
(2) (119894119894120596120596119899119899) + ⋯
is called self-energy and Σ119954119954120582120582119954119954120582120582prime(119899119899) (119894119894120596120596119899119899)119899119899 = 12⋯ is the self-energy of nth order approximation Up
to the second order approximation the self-energy is written as
Σ119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) =
⎩⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎧ minus
120575120575119954119954119954119954prime
2
119881119881120582120582120582120582primelowast (119954119954)
120596120596119954119954120582120582120596120596119954119954120582120582prime 1 le 120582120582 le 3119903119903
2lt 120582120582prime le 3119903119903
minus120575120575119954119954119954119954prime
2
119881119881120582120582prime120582120582(119954119954)
120596120596119954119954120582120582120596120596119954119954120582120582prime 1 le 120582120582prime le 3119903119903
2lt 120582120582 le 3119903119903
120575120575119954119954119954119954prime
4sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 1 3119903119903
2
120575120575119954119954119954119954prime
4sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast 119954119954prime
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
(536)
54 Fermirsquos golden Rule
To get the Fermirsquos golden Rule we need to calculate the retarded Greenrsquos function which can be
calculated from 119866119866119954119954120582120582119954119954120582120582prime(119894119894120596120596119899119899) by analytic continuation to the real axis via 119894119894120596120596119899119899 rarr 120596120596 + 119894119894119894119894 with an
infinitesimal positive 119894119894 [12]
119866119866119954119954120582120582119954119954120582120582prime119903119903 (120596120596) equiv 119866119866119954119954120582120582119954119954120582120582prime(119894119894120596120596119899119899 rarr 120596120596 + 119894119894119894119894) 119894119894 rarr 0 (537)
Similarity the retarded self-energy is defined as
Σ119954119954120582120582119954119954120582120582prime119903119903 (120596120596) equiv Σ119954119954120582120582119954119954120582120582prime(119894119894120596120596119899119899 rarr 120596120596 + 119894119894119894119894) 119894119894 rarr 0 (538)
The transition rate of phonons of branch 120582120582 at 119954119954 Γ119954119954120582120582(120596120596) is related to the retarded self energy
Σ119954119954120582120582119903119903 (120596120596) via [13]
Γ119954119954120582120582(120596120596) = minus2ImΣ119954119954120582120582119954119954120582120582119903119903 (120596120596) (539)
Using the expression for 1198661198661199541199541205821205820 (119894119894120596120596119899119899) [Eq (421)] and focused on the second order approximation [13]
we have
28
Σ119954119954120582120582119954119954120582120582119903119903 (119894119894120596120596119899119899) =
⎩⎪⎨
⎪⎧14sum
1198811198811205821205821198951198951prime(119954119954)
2
1205961205961199541199541198951198951prime120596120596119954119954120582120582
31199031199031198951198951prime=
31199031199032 +1
1
120596120596minus1205961205961199541199541198951198951prime +119894119894119894119894
120582120582 isin 1 31199031199032
14sum 1198811198811198951198951120582120582(119954119954)2
1205961205961199541199541198951198951120596120596119954119954120582120582
311990311990321198951198951=1
1120596120596minus1205961205961199541199541198951198951+119894119894119894119894
120582120582 isin 31199031199032
+ 13119903119903
(540)
Using the relation
Im 1
120596120596minus1205961205961199541199541198951198951prime +119894119894119894119894
= minus120587120587120575120575120596120596 minus 1205961205961199541199541198951198951prime (541)
The transition rate reads as
Γ119954119954120582120582(120596120596) =
⎩⎪⎨
⎪⎧1205871205872sum
1198811198811205821205821198951198951prime(119954119954)
2
1205961205961199541199541198951198951prime120596120596119954119954120582120582
31199031199031198951198951prime=
31199031199032 +1
120575120575120596120596 minus 1205961205961199541199541198951198951prime 120582120582 isin 1 31199031199032
1205871205872sum 1198811198811198951198951120582120582(119954119954)2
1205961205961199541199541198951198951120596120596119954119954120582120582
311990311990321198951198951=1
120575120575120596120596 minus 1205961205961199541199541198951198951 120582120582 isin 31199031199032
+ 13119903119903
(542)
Or equivalently
Γ119954119954120582120582(119864119864 = ℏ120596120596) =
⎩⎪⎨
⎪⎧120587120587ℏ3
2sum
1198811198811205821205821198951198951prime(119954119954)
2
1198641198641199541199541198951198951prime119864119864119954119954120582120582
31199031199031198951198951prime=
31199031199032 +1
120575120575119864119864 minus 1198641198641199541199541198951198951prime 120582120582 isin 1 31199031199032
120587120587ℏ3
2sum 1198811198811198951198951120582120582(119954119954)2
1198641198641199541199541198951198951119864119864119954119954120582120582
311990311990321198951198951=1
120575120575119864119864 minus 1198641198641199541199541198951198951 120582120582 isin 31199031199032
+ 13119903119903
(543)
Γ119954119954120582120582(119864119864) represents the probability per unit time of a transition with energy 119864119864 from phonon branch 120582120582
at wave number 119954119954 in subsystem I (II) to a set of phonon branches in subsystem II (I) with the same
wave number Here state 119954119954120582120582 in one subsystem is coupled to state 1199541199541198951198951prime in the other subsystem via
1198811198811205821205821198951198951prime(119954119954) Since the two expressions in Eq (543) are completely equivalent just representing
transitions of opposite directions we consider only the first one in what follows Assuming that
subsystem II sufficiently large such that its density of states is nearly continuous the sum over its
states in the above expression can be replaced by an integral over the energy sum (⋯ )119895119895 rarr
int(⋯ )120588120588119864119864119954119954d119864119864119954119954 giving
Γ119954119954120582120582(119864119864) = 120587120587ℏ3
2 intd119864119864119954119954prime120588120588119864119864119954119954prime 119881119881120582120582119895119895prime(119954119954)
2119864119864119954119954119895119895prime=119864119864119954119954prime
119864119864119954119954prime119864119864119954119954120582120582120575120575119864119864 minus 119864119864119954119954prime = 120587120587ℏ3
2120588120588(119864119864)
119881119881120582120582119895119895prime(119954119954)2119864119864119954119954119895119895prime=119864119864
119864119864119864119864119954119954120582120582 (544)
Eq (544) represents the transition rate of the phonons with energy 119864119864 from branch 120582120582 and wave
number 119954119954 in subsystem I to the continuous manifold of states in subsystem II The total transition
rate between the two subsystems is then given by the sum of transition rates from all states in
subsystem I weighted by their phonon populations to the manifold of states in subsystem II
Assuming that the two subsystems are weakly coupled such that the width of state 119954119954120582120582 in subsystem
I is small and the probability to leave it at an energy other than 119864119864119954119954120582120582 is negligible we can now write
29
total transition rate as
Γtot = 1119885119885119890119890minus120573120573119864119864119954119954120582120582Γ119954119954120582120582119864119864119954119954120582120582
119954119954120582120582
=120587120587ℏ3
2
119890119890minus120573120573119864119864119954119954120582120582
119885119885
120588120588119864119864119954119954120582120582 119881119881120582120582119895119895prime(119954119954)2119864119864119954119954119895119895prime=119864119864119954119954120582120582
1198641198641199541199541205821205822119954119954120582120582
=120587120587ℏ3
2
119890119890minus120573120573119864119864119954119954120582120582
119885119885
120588120588119864119864119954119954120582120582 119881119881120582120582120582120582+31199031199032(119954119954)
2
1198641198641199541199541205821205822119954119954120582120582
Finally we get
Γtot = Γ119954119954120582120582(119864119864) =120587120587ℏ3
2sum 119890119890minus120573120573119864119864119954119954120582120582
119885119885
120588120588119864119864119954119954120582120582119881119881120582120582120582120582+31199031199032(119954119954)
2
1198641198641199541199541205821205822119954119954120582120582 (545)
where 119885119885 = sum 119890119890minus120573120573119864119864119954119954120582120582119954119954120582120582 is the partition function Here we choose the index of phonon branches 120582120582
such that the phonon branch with lower energy has a smaller index ie 1198641198641199541199541205821205821 lt 1198641198641199541199541205821205822 for index 1205821205821 lt
1205821205822 Note that phonon branches 120582120582 and 1198951198951prime belong to the subsystem I (with index from 1 to 31199031199032) and
subsystem II (with index from 1 to 1 + 31199031199032 to 3119903119903) respectively we choose 119895119895prime = 120582120582 + 31199031199032
to make
sure that 119864119864119954119954119895119895prime = 119864119864119954119954120582120582 since phonon energy of two uncoupled system is identical Eq (545) is the
Fermirsquos golden rule for inter-phonon coupling
References
[1] Y Xu J-S Wang W Duan B-L Gu and B Li Nonequilibrium Greens function method for phonon-phonon interactions and ballistic-diffusive thermal transport Phys Rev B 78 224303 (2008) [2] A L Fetter and J D Walecka Quantum theory of many-particle systems (Courier Corporation 2012) [3] G E Stedman Fermis Golden RulemdashAn Exercise in Quantum Field Theory Am J Phys 39 205 (1971)
9
derivation of Fermirsquos golden rule and temperature dependence of the cross-plane thermal
conductivity
AUTHOR INFORMATION
Corresponding Author
E-mail urbakhtauextauacil
Notes
The authors declare no competing financial interest
ACKNOWLEDGMENTS
The authors would like to thank Prof Abraham Nitzan Dr Guy Cohen and Dr Yiming Pan for
helpful discussions WO acknowledges the financial support from a fellowship program for
outstanding postdoctoral researchers from China and India in Israeli Universities and the support from
the National Natural Science Foundation of China (Nos 11890673 and 11890674) HQ
acknowledges the financial support from the National Natural Science Foundation of China (No
11890674) MU acknowledges the financial support of the Israel Science Foundation Grant
No 114118 and the ISF-NSFC joint grant 319119 OH is grateful for the generous financial
support of the Israel Science Foundation under grant no 158617 and the Naomi Foundation for
generous financial support via the 2017 Kadar Award This work is supported in part by COST Action
MP1303
10
References
[1] A A Balandin Thermal properties of graphene and nanostructured carbon materials Nat Mater 10 569 (2011) [2] S Ghosh I Calizo D Teweldebrhan E P Pokatilov D L Nika A A Balandin W Bao F Miao and C N Lau Extremely high thermal conductivity of graphene Prospects for thermal management applications in nanoelectronic circuits Appl Phys Lett 92 151911 (2008) [3] A A Balandin S Ghosh W Bao I Calizo D Teweldebrhan F Miao and C N Lau Superior thermal conductivity of single-layer graphene Nano Lett 8 902 (2008) [4] J H Seol et al Two-Dimensional Phonon Transport in Supported Graphene Science 328 213 (2010) [5] R Taylor The thermal conductivity of pyrolytic graphite J Philosophical Magazine 13 157 (1966) [6] S Ghosh W Bao D L Nika S Subrina E P Pokatilov C N Lau and A A Balandin Dimensional crossover of thermal transport in few-layer graphene Nat Mater 9 555 (2010) [7] Z Wang R Xie C T Bui D Liu X Ni B Li and J T L Thong Thermal Transport in Suspended and Supported Few-Layer Graphene Nano Lett 11 113 (2011) [8] Z Wei Z Ni K Bi M Chen and Y Chen Interfacial thermal resistance in multilayer graphene structures Phys Lett A 375 1195 (2011) [9] Y Ni Y Chalopin and S Volz Significant thickness dependence of the thermal resistance between few-layer graphenes Appl Phys Lett 103 061906 (2013) [10] J Chen J H Walther and P Koumoutsakos Strain Engineering of Kapitza Resistance in Few-Layer Graphene Nano Lett 14 819 (2014) [11] Q Fu J Yang Y Chen D Li and D Xu Experimental evidence of very long intrinsic phonon mean free path along the c-axis of graphite Appl Phys Lett 106 031905 (2015) [12] G Fugallo A Cepellotti L Paulatto M Lazzeri N Marzari and F Mauri Thermal Conductivity of Graphene and Graphite Collective Excitations and Mean Free Paths Nano Lett 14 6109 (2014) [13] Z Wei J Yang W Chen K Bi D Li and Y Chen Phonon mean free path of graphite along the c-axis Appl Phys Lett 104 081903 (2014) [14] E K Sichel R E Miller M S Abrahams and C J Buiocchi Heat capacity and thermal conductivity of hexagonal pyrolytic boron nitride Phys Rev B 13 4607 (1976) [15] P Jiang X Qian R Yang and L Lindsay Anisotropic thermal transport in bulk hexagonal boron nitride Phys Rev Mater 2 064005 (2018) [16] M-H Wang Y-E Xie and Y-P Chen Thermal transport in twisted few-layer graphene Chinese Physics B 26 116503 (2017) [17] X Nie L Zhao S Deng Y Zhang and Z Du How interlayer twist angles affect in-plane and cross-plane thermal conduction of multilayer graphene A non-equilibrium molecular dynamics study Int J Heat Mass Tran 137 161 (2019) [18] A N Kolmogorov and V H Crespi Registry-dependent interlayer potential for graphitic systems Phys Rev B 71 235415 (2005) [19] M Reguzzoni A Fasolino E Molinari and M C Righi Potential energy surface for graphene on graphene Ab initio derivation analytical description and microscopic interpretation Phys Rev B 86 (2012) [20] M M van Wijk A Schuring M I Katsnelson and A Fasolino Moire Patterns as a Probe of Interplanar Interactions for Graphene on h-BN Phys Rev Lett 113 135504 (2014) [21] D Mandelli W Ouyang M Urbakh and O Hod The Princess and the Nanoscale Pea Long-Range Penetration of Surface Distortions into Layered Materials Stacks ACS Nano 13 7603 (2019) [22] E Koren E Loumlrtscher C Rawlings A W Knoll and U Duerig Adhesion and friction in mesoscopic graphite contacts Science 348 679 (2015) [23] E Koren I Leven E Loumlrtscher A Knoll O Hod and U Duerig Coherent commensurate electronic states at the interface between misoriented graphene layers Nat Nanotechnol 11 752 (2016)
11
[24] W Ouyang I Azuri D Mandelli A Tkatchenko L Kronik M Urbakh and O Hod Mechanical and Tribological Properties of Layered Materials under High Pressure Assessing the Importance of Many-Body Dispersion Effects J Chem Theory Comput 16 666 (2020) [25] W Ouyang D Mandelli M Urbakh and O Hod Nanoserpents Graphene Nanoribbon Motion on Two-Dimensional Hexagonal Materials Nano Lett 18 6009 (2018) [26] I Leven T Maaravi I Azuri L Kronik and O Hod Interlayer Potential for Grapheneh-BN Heterostructures J Chem Theory Comput 12 2896 (2016) [27] T Maaravi I Leven I Azuri L Kronik and O Hod Interlayer Potential for Homogeneous Graphene and Hexagonal Boron Nitride Systems Reparametrization for Many-Body Dispersion Effects J Phys Chem C 121 22826 (2017) [28] L H Liang and B Li Size-dependent thermal conductivity of nanoscale semiconducting systems Phys Rev B 73 153303 (2006) [29] P K Schelling S R Phillpot and P Keblinski Comparison of atomic-level simulation methods for computing thermal conductivity Phys Rev B 65 144306 (2002) [30] J Shiomi Nonequilirium molecular dynamics methods for lattice heat conduction calculations Annual Review of Heat Transfer 17 (2014) [31] G E Stedman Fermis Golden RulemdashAn Exercise in Quantum Field Theory Am J Phys 39 205 (1971) [32] L T Kong G Bartels C Campantildeaacute C Denniston and M H Muumlser Implementation of Greens function molecular dynamics An extension to LAMMPS Comput Phys Commun 180 1004 (2009) [33] S Plimpton Fast Parallel Algorithms for Short-Range Molecular Dynamics J Comput Phys 117 1 (1995) [34] M Harb et al The c-axis thermal conductivity of graphite film of nanometer thickness measured by time resolved X-ray diffraction Appl Phys Lett 101 233108 (2012) [35] H Zhang X Chen Y-D Jho and A J Minnich Temperature-Dependent Mean Free Path Spectra of Thermal Phonons Along the c-Axis of Graphite Nano Lett 16 1643 (2016) [36] D P Sellan E S Landry J E Turney A J H McGaughey and C H Amon Size effects in molecular dynamics thermal conductivity predictions Phys Rev B 81 214305 (2010) [37] S J Stuart A B Tutein and J A Harrison A reactive potential for hydrocarbons with intermolecular interactions J Chem Phys 112 6472 (2000)
S1
Supporting information for ldquoControllable Thermal Conductivity of
Twisted Graphene and Hexagonal Boron Nitride Interfacesrdquo
Wengen Ouyang1 Huasong Qin2 Michael Urbakh1 and Oded Hod1
1Department of Physical Chemistry School of Chemistry The Raymond and Beverly
Sackler Faculty of Exact Sciences and The Sackler Center for Computational
Molecular and Materials Science Tel Aviv University Tel Aviv 6997801 Israel
2State Key Laboratory for Strength and Vibration of Mechanical Structures School of
Aerospace Xirsquoan Jiaotong University Xirsquoan 710049 China
The Supporting information includes following sections
1 Methodology
2 Thermal conductivity of twisted bilayer graphene
3 Theory for calculating the phonon coupling of twisted bilayer graphene
4 Derivation of Fermirsquos golden rule
5 Temperature dependence of cross-plane thermal conductivity
S2
1 Methodology
11 Model system
The initial interlayer distance across the layered stack was set equal to 34 Aring and 33 Aring for graphite
and bulk h-BN respectively Periodic boundary conditions were applied in all directions It should be
noted that the lattice structure is rigorously periodic only at some specific twist angles the values of
which are listed in Table S1 in section 3 below While the cross-sectional area for each misfit angle
θ is different all systems considered have a contact area exceeding 12 nm2 which was shown to
provide converged results with respect to unit-cell dimensions [1] The intralayer interactions within
each graphene and h-BN layer were modeled via the second generation REBO potential [2] and
Tersoff potential [3] respectively The interlayer interactions between the layers of graphite and bulk
h-BN were described via our dedicated interlayer potential (ILP) [4] which is implemented in the
LAMMPS [5] suite of codes [6]
Fig S1 Schematic representation of the simulation setup (a) and steady-state temperature profile (b) respectively In panel (a) two identical AB-stacked graphite slabs (gray and orange respectively) are twisted with respect to each other to create a stacking fault of misfit angle θ A thermal bias is induced by applying Langevin thermostats to the two layers marked by dashed red (Thot) and green (Tcold) rectangles The arrows indicate the direction of the vertical heat flux Since periodic boundary conditions are applied also in the vertical direction two twisted interfaces are shown across which heat flows in opposite directions The steady-state temperature profiles are illustrated in panel (b) where N is the total number of layers in the model system and RAB dAB and Rθ and dθ mark the interfacial Kapitza resistance [89] and interlayer distance for contacting graphene layers with AB-stacking and misfit angle θ respectively The red lines in panel (b) mark the temperature variation across the twisted interface where the vertical axis corresponds to the position of the various layers along the stack and the horizontal axis marks the temperature of the various layers
S3
12 Simulation Protocol
All MD simulations were performed with the LAMMPS simulation package [5] The velocity-Verlet
algorithm with a time-step of 05 fs was used to propagate the equations of motion A Noseacute-Hoover
thermostat with a time constant of 025 ps was used for constant temperature simulations To maintain
a specified hydrostatic pressure the three translational vectors of the simulation cell were adjusted
independently by a Noseacute-Hoover barostat with a time constant of 10 ps [7] To relax the box we first
equilibrated the systems in the NPT ensemble at a temperature of T = 300 K and zero pressure for
250 ps (see Fig S2) After equilibration Langevin thermostats with damping coefficients 10 ps-1
were applied to the bottom and middle layer of the graphene stack (see Fig S1) with target
temperature Thot = 375 K (hot reservoir) and Tcold = 225 K (cold reservoir) respectively Then the
system was allowed to reach steady-state over a subsequent simulation period of 750 ps (see Fig S2)
during which the dynamics of all non-thermostated layers followed the NVE ensemble For the larger
model systems the length of the NPT and Langevin stages was doubled (for the 32 and 48 layers
systems) or tripled (for the 104 layers graphitic system) to ensure convergence of the obtained steady-
state Once steady-state was obtained the last 500 ps were used to calculate the thermal conductivity
of the twisted graphite and bulk h-BN The statistical errors were estimated using ten different data
sets each calculated over a time interval of 50 ps
Fig S2 Time evolution of the temperature of thermostated layers for 16 layers twisted graphite with misfit angle (a) θ = 0deg (b) θ = 509deg (c) θ = 1518deg and (d) θ = 3016deg Note that the thermal fluctuations increase with increasing the misfit angle due to the growing interfacial thermal resistance that enhances phonon back scattering at the twisted junction
S4
13 Calculation of the interfacial thermal resistance
According to Fourierrsquos law the cross-plane thermal conductivity (120581120581CP) of a twisted graphitic interface
of misfit angle θ can be calculated as
120581120581CP = 119876119860119860Δ119879119879Δ119911119911
(S1)
where 119876 is the heat flux A is the cross-section area and Δ119879119879Δ119911119911 is the temperature gradient along
the direction of heat flux (perpendicular to the basal plane in our case) Fig S1(b) shows a schematic
temperature profile along the z-direction where the vertical axis corresponds to the position of the
various layers along the stack and the horizontal axis marks the temperature of the various layers The
actual temperature profiles extracted from the NEMD simulations for twisted graphite with different
number of layers can be found in Fig S3 For Bernal-stacked graphite (ie θ = 0deg red circles in Fig
S3) only the linear region of the temperature profile was used to calculate 120581120581CP and the points
corresponding to the layers where the thermostats were applied were omitted (marked with green
triangle in Fig S3) The 120581120581CP of the system was calculated using Eq (S1) by averaging over the two
linear regions of the temperature profiles For the twisted case (θ ne 0deg) we found a sudden
temperature decrease Δ119879119879120579120579 at the position of the twisted interface (see black squares in Fig S3) 120581120581CP
in this case was calculated using the temperature gradient calculated for the same layer range as that
for θ = 0deg To characterize the thermal properties of the twisted interface the concept of interfacial
thermal resistance (ITR) ie Kapitza resistance [89] was introduced According the definition of
the Kapitza resistance [8] 119877119877 = 119860119860Δ119879119879119876 and noticing that Δ119879119879tot = (1198731198732 minus 3)Δ119879119879AB + Δ119879119879120579120579 and
Δ119911119911 = (1198731198732 minus 3)119889119889AB + 119889119889120579120579 Eq (1) can be rewritten considering two-resistors in series as [see Fig
S1(b)]
(1198731198732 minus 3)119877119877AB + 119877119877120579120579 = [(1198731198732 minus 3)119889119889AB + 119889119889120579120579] 120581120581CPfrasl (S2)
Here 119877119877AB 119889119889AB Δ119879119879AB and 119877119877120579120579 119889119889120579120579 Δ119879119879120579120579 are the ITR interlayer distance and temperature
difference for adjacent AB-stacked and twisted graphene layers respectively For the aligned contact
(120579120579 = 0deg) the ITR can be simply calculated as 119877119877AB = 119889119889AB120581120581AB Once 119877119877AB is known 119877119877120579120579 can be
calculated from Eq (S2) We note that the sharp temperature drop at the twisted interface indicates
that 119877119877120579120579 should be much larger than 119877119877AB
S5
Fig S3 Temperature profiles for graphitic stacks consisting of (a) 8 and (b) 16 layers The red circles and black squares represent the temperature profiles for the aligned (120579120579 = 0deg) and twisted (120579120579 =3016deg) junctions respectively Green triangles represent data points that were omitted in the 120581120581CP calculation
2 Thermal conductivity of twisted bilayer graphene
As comparison we also calculated the interfacial thermal conductivity (ITC) and ITR of twisted
bilayer graphene (tBLG) with the transient MD simulation approach [15-17] since the NEMD
simulation protocol used in the main text becomes invalid in this case In this protocol the system
was first equilibrated within the NPT ensemble at T = 200 K and zero pressure for 100 ps which was
followed by a 100 ps NVT ensemble equilibration stage and a 100 ps of NVE ensemble equilibration
stage After the system reached steady-state an ultrafast heat impulse was imposed on the top layer
of the t-BLG for 50 fs to increase the temperature of the top layer from 200 K to 400 K while that of
bottom layer of tBLG remained unchanged After the external heat source was removed thermal
energy flowed from the top layer to the bottom layer due to the temperature difference and the
temperature of both layers approached 300 K when quasi-steady-state was reached During the
thermal relaxation time interval (500 ps) the temperature and energy of the system sections were
recorded The ITR could then be extracted using the following equation [15-17]
part119864119864t120597120597120597120597
= 119860119860119877119877119879119879bot(119905119905) minus 119879119879top(119905119905) (S3)
where 119864119864t is the total energy of the top graphene layer R is the ITR of the tBLG A is the interfacial
cross-section area and 119879119879bot and 119879119879top are the instantaneous temperatures measured for the bottom
and top layers of the tBLG respectively Note that in Eq S3 we assume a linear dependence of the
heat flux on the temperature difference between the layers The ITC of the tBLG is simply defined as
ITC equiv 119889119889ITR where d is the average interlayer distance
S6
The ITC and ITR of tBLG as functions of misfit angle calculated with the transient MD simulation
protocol are illustrated in Fig S4 demonstrating similar misfit-angle dependence as that for the
NEMD protocol with Langevin thermostat exercised to obtain the results presented in the main text
This further validates the reliability of the simulation protocol adopted in the main text which is more
suitable to treat thick slabs and allows to obtain a true stead-state
Fig S4 Misfit-angle dependence of (a) ITC and (b) ITR for a twisted bilayer graphene obtained using the transient MD simulation approach
3 Theory for calculating the phonon-phonon coupling of twisted bilayer graphene
31 Brillouin Zone of supercell in tBLG
For tBLG the lattice structure is rigorously periodic only at some specific misfit angles θ where the
lattice vector 1199231199231 = 11989911989911199381199381 + 11989911989921199381199382 in the bottom layer equals the vector 1199231199232 = 11989811989811199381199381 + 11989811989821199381199382 in the
top layer with certain integers m1 m2 and n1 n2 Here 1199381199381 = 119886119886(10) and 1199381199382 = 11988611988612radic32 are
the primitive lattice vectors of the bottom layer and a is the lattice constant of monolayer graphene
Thus the exact superlattice period is then given by [18]
119871119871 = |11989911989911199381199381 + 11989911989921199381199382| = 11988611988611989911989912 + 11989911989922 + 11989911989911198991198992 = |1198991198991minus1198991198992|1198861198862 sin(1205791205792) (S4)
where θ is the angle between two lattice vectors 1199231199231 and 1199231199232 In the simulations below we always
rotated the supercell such that its lattice vector is 1199231199231 = 119871119871(10) and 1199231199232 = 11987111987112radic32 In this case
the corresponding reciprocal lattice vector of the moireacute superlattice satisfies the relation 119918119918119894119894 ∙ 119923119923119895119895 =
2120587120587120575120575119894119894119895119895 such that
1199181199181 = 4120587120587radic3119871119871
radic32
minus12 1199181199182 = 4120587120587
radic3119871119871(01) (S5)
S7
Both the lattice vectors and the corresponding reciprocal lattice vectors of the superlattice of the tBLG
are presented in Fig S5 Table S1 reports the parameters used to construct rhombus periodic
supercells of different misfit angles that can be duplicated to construct a rectangular periodic supercell
Table S1 The parameters used to construct periodic supercells of various misfit angles
120579120579 (deg) 119860119860 (nm2) 1198991198991 1198991198992 1198981198981 1198981198982
0 60383688 24 0 24 0
0696407 709613169 48 47 47 48
1121311 273718420 30 29 29 30
2000628 85648391 17 16 16 17
3006558 152322047 23 21 21 23
4048894 189013524 26 23 23 26
5085849 53255058 14 12 12 14
7926470 49376245 14 11 11 14
9998709 86277388 19 14 14 19
15178179 31554670 15 4 4 15
19932013 42876612 15 8 8 15
25039660 13942761 9 4 4 9
30158276 18974735 11 4 4 11
32204228 21805221 12 4 4 12
32 Special points for Brillouin Zone integration
The calculation of the sum over wave vector q in Eq (4) in the main text can be transformed to an
integral using the relation sum (⋯ )119954119954 = 1119881119881119887119887int (⋯ )BZ d119954119954 where 119881119881119887119887 = (2120587120587)3119881119881 is the volume of the
Brillouin Zone (BZ) and V is volume of the real-space unit-cell The calculation of integral is usually
inefficient since it requires calculating the value of the function over a large set of k points in the first
BZ To calculate such integrations more efficiently simple k-point meshes can be replaced by a
carefully selected set of special points in the BZ 119954119954119894119894 [19-22] over which the function is evaluated
The integral can then be estimated via
119868119868 = 1119881119881119887119887int 119891119891(119954119954)BZ d119954119954 asymp 1
119873119873sum 119908119908119894119894119891119891(119954119954119894119894)119894119894 (S6)
S8
where 119881119881119887119887 is the volume of the BZ 119908119908119894119894 is the weight of the ith data point and N normalizes the
weighting factors to unity 119873119873 = sum 119908119908119894119894119894119894 The set of selected 119954119954119894119894 forms a grid in the irreducible
Brillouin zone (IBZ) as is illustrated by the red points in Fig S5(b) The coordinates of these points
for a hexagonal lattice are presented in Eq (S7)
Fig S5 (a) Twisted bilayer graphene of misfit angle θ = 509deg L1 and L2 are the superlattice vectors (b) The corresponding first Brillouin Zone of (a) G1 and G2 are the reciprocal lattice vectors of the superlattice The triangle ΔΓMK represents the irreducible Brillouin zone Red circles mark the position of the special points used to evaluate the integral over the first Brillouin Zone
⎩⎪⎨
⎪⎧ 1199541199541 = 1
9 1205971205973 1199541199542 = 2
9 1205971205973 1199541199543 = 1
18 512059712059718
1199541199544 = 518
512059712059718 1199541199545 = 1
9 21205971205979 1199541199546 = 2
9 21205971205979
1199541199547 = 118
312059712059718 1199541199548 = 1
9 1205971205979 1199541199549 = 1
18 12059712059718
(S7)
Here 119905119905 = radic3 and the unit of the coordinates is 2120587120587119871119871 The weighting factors 119908119908119894119894 are [19]
119908119908124689 = 112
119908119908357 = 16 (S8)
Using Eqs (S6-S8) Eq (4) in the main text can be evaluated as follows
Γtot = 120587120587ℏ3
2sum 119908119908119896119896 sum
119890119890minus120573120573119864119864119954119954119948119948120582120582
119885119885
120588120588119864119864119954119954119948119948120582120582119881119881120582120582120582120582+31199031199032(119954119954119948119948)
2
1198641198641199541199541199481199481205821205822120582120582
9119896119896=1 (S9)
This equation was used to calculate the transition rate presented in Fig 3 in the main text
4 Derivation of Fermirsquos golden rule
The derivation of Fermirsquos golden rule is provided in a separate file (Fermirsquos Golden Rule for
phononspdf) due to its extent
S9
5 Temperature dependence of interfacial thermal conductivity
In the main text the target temperatures of the Langevin thermostats for the bottom and middle layers
of graphene and h-BN were set to 225 K and 375 K respectively After reaching the steady-state the
average temperature of the system was found to be ~300 K To check the effect of average temperature
on our results we calculated the cross-plane thermal conductivity (120581120581CP ) and the corresponding
interfacial thermal resistance (ITR) at a different temperature gradient (325 K ndash 475 K) resulting the
average steady-state temperature of ~400K The protocol described in Section 1 above was used to
perform these calculations as well Both ILP and Lennard-Jones (LJ) potential were tested for
graphite whereas for the bulk h-BN simulations only the ILP was used The results for graphite and
bulk h-BN are illustrated in Fig S6 and Fig S7 respectively For the ILP we find that the overall
values of 120581120581CP (ITR) decrease (increase) slightly with increasing average temperature which is
consistent with a recent experiment [10] The LJ potential calculations as well exhibit very week
dependence on average temperature within the range studied Altogether the layer dependences of
both quantities remain mostly insensitive to the average temperature The reason is that the thickness
of the graphite and h-BN model systems was chosen to be much smaller than their phonon mean free
path (~200 nm for graphite [110-13] and ~100 nm for bulk h-BN [14]) such that phonon transport is
dominated by phonon-boundary scattering and the Umklapp process only makes marginal
contribution [10]
S10
Fig S6 Layer dependence of 120581120581CP (a c) and ITR (b d) for Bernal-stacked graphite at average steady-state temperatures of 300 K (red circles) and 400 K (black squares) The left and right columns correspond to the 120581120581CP and ITR calculated with ILP and LJ potential respectively
Fig S7 Layer dependence of 120581120581CP (a) and ITR (b) for AArsquo-stacked h-BN at average temperatures of 300 K (red circles) and 400 K (blue squares)
S11
References [1] Z Wei J Yang W Chen K Bi D Li and Y Chen Phonon mean free path of graphite along the c-axis Appl Phys Lett 104 081903 (2014) [2] D W Brenner O A Shenderova J A Harrison S J Stuart B Ni and S B Sinnott A second-generation reactive empirical bond order (REBO) potential energy expression for hydrocarbons J Phys Condens Matter 14 783 (2002) [3] A Kınacı J B Haskins C Sevik and T Ccedilağın Thermal conductivity of BN-C nanostructures Phys Rev B 86 (2012) [4] W Ouyang I Azuri D Mandelli A Tkatchenko L Kronik M Urbakh and O Hod Mechanical and Tribological Properties of Layered Materials under High Pressure Assessing the Importance of Many-Body Dispersion Effects J Chem Theory Comput 16 666 (2020) [5] S Plimpton Fast Parallel Algorithms for Short-Range Molecular Dynamics J Comput Phys 117 1 (1995) [6] W Ouyang D Mandelli M Urbakh and O Hod Nanoserpents Graphene Nanoribbon Motion on Two-Dimensional Hexagonal Materials Nano Lett 18 6009 (2018) [7] W Shinoda M Shiga and M Mikami Rapid estimation of elastic constants by molecular dynamics simulation under constant stress Phys Rev B 69 134103 (2004) [8] G L Pollack Kapitza Resistance Rev Mod Phys 41 48 (1969) [9] H-S Yang G R Bai L J Thompson and J A Eastman Interfacial thermal resistance in nanocrystalline yttria-stabilized zirconia Acta Mater 50 2309 (2002) [10] Q Fu J Yang Y Chen D Li and D Xu Experimental evidence of very long intrinsic phonon mean free path along the c-axis of graphite Appl Phys Lett 106 031905 (2015) [11] J Chen J H Walther and P Koumoutsakos Strain Engineering of Kapitza Resistance in Few-Layer Graphene Nano Lett 14 819 (2014) [12] H Zhang X Chen Y-D Jho and A J Minnich Temperature-Dependent Mean Free Path Spectra of Thermal Phonons Along the c-Axis of Graphite Nano Lett 16 1643 (2016) [13] D P Sellan E S Landry J E Turney A J H McGaughey and C H Amon Size effects in molecular dynamics thermal conductivity predictions Phys Rev B 81 214305 (2010) [14] P Jiang X Qian R Yang and L Lindsay Anisotropic thermal transport in bulk hexagonal boron nitride Phys Rev Mater 2 064005 (2018) [15] J Zhang Y Hong and Y Yue Thermal transport across graphene and single layer hexagonal boron nitride J Appl Phys 117 134307 (2015) [16] B Liu F Meng C D Reddy J A Baimova N Srikanth S V Dmitriev and K Zhou Thermal transport in a graphenendashMoS2 bilayer heterostructure a molecular dynamics study RSC Advances 5 29193 (2015) [17] Z Ding Q-X Pei J-W Jiang W Huang and Y-W Zhang Interfacial thermal conductance in grapheneMoS2 heterostructures Carbon 96 888 (2016) [18] N N T Nam and M Koshino Lattice relaxation and energy band modulation in twisted bilayer graphene Phys Rev B 96 075311 (2017)
S12
[19] R Ramiacutearez and M C Boumlhm Simple geometric generation of special points in brillouin-zone integrations Two-dimensional bravais lattices Int J Quantum Chem 30 391 (1986) [20]S L Cunningham Special points in the two-dimensional Brillouin zone Phys Rev B 10 4988 (1974) [21] H J Monkhorst and J D Pack Special points for Brillouin-zone integrations Phys Rev B 13 5188 (1976) [22] D J Chadi Special points for Brillouin-zone integrations Phys Rev B 16 1746 (1977)
1
Fermirsquos golden rule for phonons
1 Basic theory for phonons 11 Basic notations
Let us consider a 3D crystal with a total of 119873119873 = 119873119873111987311987321198731198733 unit cells and periodic boundary conditions
To be specific let 119938119938119894119894 119894119894 = 123 be the lattice vectors that define the unit cell We index unit cells
with n = (n1n2n3) where each ni = 12 ∙∙∙ Ni and their locations are 119929119929119899119899 = sum 1198991198991198941198941199381199381198941198943119894119894=1 Assume that
there are r atoms in each unit cell which are indexed with 119904119904 = 1⋯ 119903119903 The mass and the equilibrium
distance of the sth atom are notated as 119872119872s and 119929119929s0 respectively Then the location of the sth atom in
the nth unit cell at time t can be expressed as
119955119955119899119899119899119899(119905119905) = 119929119929119899119899 + 119929119929s0 + 119958119958119899119899119899119899(119905119905) (11)
where 119958119958119899119899119899119899(119905119905) is its displacement from its equilibrium position
The Lagrangian for this classical problem can be written as
ℒ = sum sum 119872119872119904119904|119955119899119899119904119904|2(119905119905)2
119903119903119899119899=1
119873119873119899119899=1 minus 119881119881 (12)
where the second term is the sum of interactions between all pairs of atoms
Under the harmonic approximation ie expanding the total potential energy 119881119881 around the
equilibrium positions The Lagrangian can be simplified as
ℒ = sum sum 119872119872119904119904|119958119899119899119904119904|2(119905119905)2
119903119903119899119899=1
119873119873119899119899=1 minus 1
2sum sum sum 120601120601120572120572120572120572prime
119899119899119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime120572120572120572120572prime119899119899119899119899prime119899119899119899119899prime (13)
where 119906119906119899119899119899119899120572120572 120572120572 = 123 are the Cartesian coordinates of the displacement 119958119958119899119899119899119899(119905119905) and
120601120601120572120572120572120572prime 119899119899119899119899prime119904119904 119904119904prime = 1205971205972119881119881
120597120597119903119903119899119899119904119904119899119899119903119903119899119899prime119904119904prime119899119899primeeq
= 120601120601120572120572prime120572120572 119899119899prime119899119899119904119904prime 119904119904 = 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime (14)
Note that the first order term vanishes because we are expanding around the equilibrium positions
120601120601120572120572120572120572prime 119899119899119899119899prime119904119904 119904119904prime represents the component of the force acted on the sth atom in the nth unit cell along 120572120572
direction when the atom 119904119904prime in the unit cell 119899119899prime moves a unit displacement along 120572120572prime direction The
symmetries of 120601120601120572120572120572120572prime 119899119899 119899119899prime119904119904 119904119904prime appearing in Eq (14) arise from the intechangability of the second
derivative and the translational invariance of the interactions
12 Dynamical matrix
The equation of motion of the sth atom in the nth unit cell can be derived using the EulerndashLagrange
equation as follows
2
119872119872119899119899119906119899119899119899119899120572120572 = minussum 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119899119899prime119899119899prime120572120572prime 119906119906119899119899prime119899119899prime120572120572prime (15)
If we displace all atoms equally ie shifting 119906119906119899119899prime119899119899prime120572120572prime to 119906119906119899119899prime119899119899prime120572120572prime + 120575120575 the total force on the sth atom in
the nth unit cell does not change From the above equation we have
sum 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119899119899prime120572120572prime = 0 (16)
We are looking for normal modes (because any general solution can be written as a linear combination
of them) these are solutions where all atoms oscillate with the same frequency Moreover because
of the lattice structure we expect solutions to reflect this periodicity So we guess solutions of the
form
119906119906119899119899119899119899120572120572(119905119905) = 1119872119872119904119904
119890119890119899119899120572120572119890119890119894119894(119954119954∙119929119929119899119899minus120596120596119905119905) (17)
where 119890119890119899119899120572120572 are real-space solutions that will be determined later and 119954119954 is the wave-vector in
reciprocal space Substituting Eq (17) into Eq (15) we can get
1205961205962119890119890119899119899120572120572(119954119954) = sum 119863119863119899119899120572120572119899119899prime120572120572prime(119954119954)119890119890119899119899prime120572120572prime(119954119954)119899119899prime120572120572prime (18)
where
119863119863119899119899120572120572119899119899prime120572120572prime(119954119954) = sum 1
119872119872119904119904119872119872119904119904prime120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119890119890
minus119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime119899119899prime = sum 1
119872119872119904119904119872119872119904119904prime120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime 119890119890
minus119894119894119954119954∙119929119929119897119897119897119897 (19)
is called dynamical matrix (dimension 3119903119903 times 3119903119903) Note that we have defined the relative cell distance
vector 119929119929119897119897 equiv 119929119929119899119899 minus 119929119929119899119899prime and number index 119897119897 equiv 119899119899 minus 119899119899prime where the infinite sum over 119899119899prime can be
replaced by the sum over 119897119897 for any value of the index 119899119899 Note that dynamical matrix is Hermitian
symmetric (ie 119863119863119899119899120572120572119899119899prime120572120572prime(119954119954)
lowast= 119863119863119899119899prime120572120572prime
119899119899120572120572 (minus119954119954)) because ϕ is symmetric so all the eigenvalues of Eq (18)
(120596120596120582120582(119954119954) 120582120582 = 12⋯ 3119903119903) are real for each 119954119954 in the Brillouin zone (BZ) which is determined by
det1205961205961205821205822(119954119954)120575120575119899119899120572120572120575120575119899119899prime120572120572prime minus 119863119863119899119899120572120572119899119899prime120572120572prime(119954119954) = 0 (110)
Taking the conjugate of this equation we have
0 = det 1205961205961205821205822(119954119954)lowast120575120575119899119899120572120572120575120575119899119899prime120572120572prime minus 119863119863119899119899120572120572119899119899
prime120572120572prime(119954119954)lowast = det1205961205961205821205822(119954119954)120575120575119899119899120572120572120575120575119899119899prime120572120572prime minus 119863119863119899119899120572120572119899119899
prime120572120572prime(minus119954119954) (111)
while replacing 119954119954 by minus119954119954 in Eq (110) we have det1205961205961205821205822(minus119954119954)120575120575119899119899120572120572120575120575119899119899prime120572120572prime minus 119863119863119899119899120572120572119899119899prime120572120572prime(minus119954119954) = 0
Itrsquos clear that 1205961205961205821205822(119954119954) and 1205961205961205821205822(minus119954119954) obey the same equation thus we have
120596120596120582120582(minus119954119954) = 120596120596120582120582(119954119954) (112)
The corresponding eigenvectors are orthonormal
sum 119890119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582
prime119899119899120572120572 = 120575120575120582120582120582120582prime (113)
The complex conjugate of Eq (18) gives
1205961205962119890119890119899119899120572120572lowast (119954119954) = sum 119863119863119899119899120572120572119899119899prime120572120572prime(119954119954)
lowast119890119890119899119899prime120572120572primelowast (119954119954)119899119899prime120572120572prime = sum 119863119863119899119899prime120572120572prime
119899119899120572120572 (minus119954119954)119890119890119899119899prime120572120572primelowast (119954119954)119899119899prime120572120572prime (114)
3
While replacing 119954119954 by minus119954119954 in Eq (18) we get the following equation
1205961205962119890119890119899119899120572120572(minus119954119954) = sum 119863119863119899119899120572120572119899119899prime120572120572prime(minus119954119954)119890119890119899119899prime120572120572prime(minus119954119954)119899119899prime120572120572prime (115)
From Eq (114) and Eq (115) we see that eigenvectors 119890119890119899119899120572120572120582120582 (minus119954119954)lowast and 119890119890119899119899120572120572120582120582 (119954119954) obey the same
eigenvalue equation Since the eigenvectors are normalized we get the following property
119890119890119899119899120572120572120582120582 (minus119954119954)lowast
= 119890119890119899119899120572120572120582120582 (119954119954) (116)
2 Second quantization The general solution is a linear combination of all these normal modes thus we have
119906119906119899119899119899119899120572120572(119905119905) = sum 119862119862120582120582(119954119954)119873119873119872119872119904119904
119890119890119899119899120572120572120582120582 (119954119954)119890119890minus119894119894120596120596119954119954120582120582119905119905119890119890119894119894119954119954∙119929119929119899119899119954119954120582120582 = sum 119876119876120582120582(119954119954119905119905)119873119873119872119872119904119904
119890119890119899119899120572120572120582120582 (119954119954)119890119890119894119894119954119954∙119929119929119899119899119954119954120582120582 (21)
where the we define the normal coordinates as 119876119876120582120582(119954119954 119905119905) equiv 119862119862120582120582(119954119954)119890119890minus119894119894120596120596120582120582(119954119954)119905119905 in the eigenvectors
representation To ensure that the displacements 119906119906119899119899119899119899120572120572(119905119905) are real (namely 119906119906119899119899119899119899120572120572lowast (119905119905) = 119906119906119899119899119899119899120572120572(119905119905)) the
following relation on 119876119876120582120582(119954119954 119905119905) and 119890119890119899119899120572120572120582120582 is enforced
119876119876120582120582(119954119954 119905119905)119890119890119899119899120572120572120582120582 (119954119954)lowast
= 119876119876120582120582(minus119954119954 119905119905)119890119890119899119899120572120572120582120582 (minus119954119954) (22)
where we used the fact that the sum over 119954119954 runs symmetrically over both negative and positive
values Using Eq (116) we have
[119876119876120582120582(119954119954 119905119905)]lowast = 119876119876120582120582(minus119954119954 119905119905) (23)
21 Kinetic energy term
Using Eq (21) the kinetic energy of the system can be expressed as
119879119879 = 12sum 1198721198721198991198991199061198991198991198991198991205721205722 (119905119905)119899119899119899119899120572120572 = 1
2sum 119872119872119904119904
119873119873119872119872119904119904sum sum 119876120582120582(119954119954 119905119905)119876120582120582prime(119954119954prime 119905119905)119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582
prime(119954119954prime) sum 119890119890119894119894119954119954+119954119954prime∙119929119929119899119899119899119899119954119954prime120582120582prime119954119954120582120582119899119899120572120572 =
12sum sum 119876120582120582(119954119954 119905119905)119876120582120582prime(minus119954119954 119905119905)119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582
prime(minus119954119954)119954119954119954119954prime120582120582prime =119899119899120572120572
12sum 119876120582120582(119954119954 119905119905)119876120582120582prime
lowast (119954119954 119905119905) sum 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582prime(119954119954)
lowast119899119899120572120572 =119954119954120582120582120582120582prime
12sum 119876120582120582(119954119954 119905119905)119876120582120582lowast(119954119954 119905119905)119954119954120582120582 =
12sum 119876120582120582(119954119954 119905119905)119876120582120582(minus119954119954 119905119905)119954119954120582120582
ie
119879119879 = 12sum 119876120582120582(119954119954 119905119905)119876120582120582(minus119954119954 119905119905)119954119954120582120582 (24)
To derive Eq (24) we used Eqs (113) and (116) as well as the following equations
1119873119873sum 119890119890119894119894119954119954+119954119954prime∙119929119929119899119899119899119899 = 120575120575119954119954+119954119954prime120782120782 (25)
4
22 Potential energy term
Similarity the potential energy can be rewritten in terms of the normal coordinates as follows
119880119880 =12120601120601119899119899119899119899120572120572119899119899prime119899119899prime120572120572prime119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
120572120572120572120572prime119899119899119899119899prime119899119899119899119899prime=
12120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119899119899+119954119954prime∙119929119929119899119899prime
119873119873119872119872119899119899119872119872119899119899prime
[119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(119954119954prime 119905119905)] 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime120582120582prime (119954119954prime)
119954119954prime120582120582prime119954119954120582120582119899119899119899119899prime120572120572120572120572prime119899119899119899119899prime
=12120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894(119954119954+119954119954prime)∙119929119929119899119899119890119890minus119894119894119954119954prime∙119929119929119897119897
119873119873119872119872119899119899119872119872119899119899prime
[119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(119954119954prime 119905119905)] 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime120582120582prime (119954119954prime)
119954119954prime120582120582prime119954119954120582120582119899119899119897119897120572120572120572120572prime119899119899119899119899prime
=12120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890minus119894119894119954119954prime∙119929119929119897119897
119872119872119899119899119872119872119899119899prime[119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(119954119954prime 119905119905)] 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (119954119954prime)1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899
119899119899=120575120575119902119902+119902119902prime=0
119954119954prime120582120582prime119954119954120582120582119897119897120572120572120572120572prime119899119899119899119899prime
=12120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894119954119954119929119929119897119897
119872119872119899119899119872119872119899119899prime [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)] 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (minus119954119954)119954119954120582120582120582120582prime119897119897120572120572120572120572prime119899119899119899119899prime
=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]119890119890119899119899120572120572120582120582 (119954119954)
119899119899120572120572
119863119863119899119899120572120572119899119899prime120572120572prime(minus119954119954)119890119890119899119899prime120572120572prime
120582120582prime (minus119954119954)119899119899prime120572120572prime119954119954120582120582120582120582prime
=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]119890119890119899119899120572120572120582120582 (119954119954)
119899119899120572120572
1205961205961205821205822(minus119954119954)119890119890119899119899120572120572120582120582prime (minus119954119954)
119954119954120582120582120582120582prime
=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]1205961205961205821205822(119954119954)120575120575120582120582120582120582prime =119954119954120582120582120582120582prime
121205961205961205821205822(119954119954)119876119876120582120582(119954119954 119905119905)119876119876120582120582(minus119954119954 119905119905)
119954119954120582120582
ie
119880119880 = 12sum 1205961205961205821205822(119954119954)119876119876120582120582(119954119954 119905119905)119876119876120582120582(minus119954119954 119905119905)119954119954120582120582 (26)
where in above deriviation the orthogonality of its eigenvectores and the symmetry of its eigenvalues
are used Thus the Lagrangian reads as
ℒ = 119879119879 minus 119881119881 = 12sum 119876120582120582(minus119954119954 119905119905)119876120582120582(119954119954 119905119905) minus 120596120596120582120582
2(119954119954)119876119876120582120582(minus119954119954 119905119905)119876119876120582120582(119954119954 119905119905)119954119954120582120582 (27)
Using the relation 119875119875120582120582(119954119954 119905119905) = 120597120597ℒ part119876120582120582(119954119954 119905119905) the Hamiltonian of the system then can be written as
119867119867 = 119879119879 + 119881119881 = 12sum [119875119875120582120582(minus119954119954 119905119905)119875119875120582120582(119954119954 119905119905) + 120596120596120582120582
2(119954119954)119876119876120582120582(minus119954119954 119905119905)119876119876120582120582(119954119954 119905119905)]119954119954120582120582 (28)
Now we quantize H by asking the momenta and coordinates to be operators
119867119867 = 12sum 119875119875minus119954119954120582120582119875119875119954119954120582120582 + 120596120596119954119954120582120582
2 119876119876minus119954119954120582120582119876119876119954119954120582120582119954119954120582120582 (29)
here 119875119875119954119954120582120582 and 119876119876119954119954120582120582 obey the following commutation relations
119876119876119954119954120582120582119875119875119954119954prime120582120582prime = 119894119894ℏ120575120575119954119954119954119954prime120575120575120582120582120582120582prime
119876119876119954119954120582120582119876119876119954119954prime120582120582prime = 0 119875119875119954119954120582120582119875119875119954119954prime120582120582prime = 0 (210)
Similar to the case of ordinary quantum harmonic oscillators it is convenient to define ladder
operators for each mode as follows
119876119876119954119954120582120582 = ℏ
2120596120596119954119954120582120582119887119887119954119954120582120582 + 119887119887minus119954119954120582120582
dagger 119875119875119954119954120582120582 = 119894119894ℏ120596120596119954119954120582120582
2119887119887119954119954120582120582
dagger minus 119887119887minus119954119954120582120582 (211)
where 119887119887119954119954120582120582dagger and 119887119887119954119954120582120582 are Bosonic creation and annihilation operators for phonons with momentum
119954119954 branch index 120582120582 and frequency 120596120596119954119954120582120582 which obeys the Bosonic commutation relation
5
119887119887119954119954120582120582 119887119887119954119954prime120582120582prime
dagger = 120575120575119954119954119954119954prime120575120575120582120582120582120582prime
119887119887119954119954120582120582 119887119887119954119954prime120582120582prime = 0 119887119887119954119954120582120582dagger 119887119887119954119954prime120582120582prime
dagger = 0 (212)
Substituting Eqs (211) into (29) and using the properties Eq (210) and (212) we have
119867119867 =12119875119875minus119954119954120582120582119875119875119954119954120582120582 +120596120596119954119954120582120582
2 119876119876minus119954119954120582120582119876119876119954119954120582120582119954119954120582120582
=12minus
ℏ120596120596119954119954120582120582
2119887119887minus119954119954120582120582
dagger minus 119887119887119954119954120582120582119887119887119954119954120582120582dagger minus 119887119887minus119954119954120582120582+ 120596120596119954119954120582120582
2 ℏ2120596120596119954119954120582120582
119887119887minus119954119954120582120582 + 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 + 119887119887minus119954119954120582120582
dagger 119954119954120582120582
=ℏ4120596120596119954119954120582120582minus119887119887minus119954119954120582120582
dagger 119887119887119954119954120582120582dagger minus 119887119887minus119954119954120582120582
dagger 119887119887minus119954119954120582120582 minus 119887119887119954119954120582120582119887119887119954119954120582120582dagger + 119887119887119954119954120582120582119887119887minus119954119954120582120582
119954119954120582120582
+ 119887119887minus119954119954120582120582119887119887119954119954120582120582 + 119887119887minus119954119954120582120582119887119887minus119954119954120582120582dagger + 119887119887119954119954120582120582
dagger 119887119887119954119954120582120582 + 119887119887119954119954120582120582dagger 119887119887minus119954119954120582120582
dagger
=ℏ4120596120596119954119954120582120582119887119887minus119954119954120582120582
dagger 119887119887minus119954119954120582120582 + 119887119887minus119954119954120582120582119887119887minus119954119954120582120582dagger + 119887119887119954119954120582120582119887119887119954119954120582120582
dagger + 119887119887119954119954120582120582dagger 119887119887119954119954120582120582
119954119954120582120582
=ℏ41205961205961199541199541205821205822119887119887minus119954119954120582120582
dagger 119887119887minus119954119954120582120582 + 1+ 2119887119887119954119954120582120582dagger 119887119887119954119954120582120582 + 1
119954119954120582120582
= ℏ120596120596119954119954120582120582 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 +
12
119954119954120582120582
Therefore we finally get the quantized representation of non-interacting phonons
119867119867 = sum ℏ120596120596119954119954120582120582 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 + 1
2119954119954120582120582 (213)
The operator of atom displacements (Eq 114) is expressed in terms of the phonon operators by
119906119906119899119899119899119899120572120572 = sum ℏ
2119873119873119872119872119904119904120596120596119954119954120582120582119890119890119899119899120572120572120582120582 119890119890119894119894119954119954∙119929119929119899119899119887119887119954119954120582120582 + 119887119887minus119954119954120582120582
dagger 119954119954120582120582 (214)
These equations will be used below
3 Inter-phonon coupling within harmonic approximation 31 Hamiltonian with inter-phonon coupling
In this section we consider systems that consist of two (or more) covalently bonded units that are
weakly coupled between them Unlike previous studies that considered phonon-phonon couplings
resulting from anharmonicity effects [1] here all phonon mode considered are Harmonic and the
couplings arise from the division of the entire system into subunits The Hamiltonian of the whole
system can be written as
119867119867 = 1198671198671 + 1198671198672 + 11986711986712 (31)
To derive the expressions of 1198671198671 1198671198672 and 11986711986712 in Eq (31) we consider the Hamiltonian of the
whole system written as the function of the atomic displacements
119867119867 = 119879119879 + 119881119881 = sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)2119899119899119899119899120572120572 + 1
2sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime120572120572120572120572prime119899119899119899119899prime119899119899119899119899prime (32)
6
Letrsquos assume that subsystem I and II contain atoms with indeices ranging from 119904119904 = 12⋯ 1199031199032 and
119904119904 = 1199031199032
+ 1 1199031199032
+ 2⋯ 119903119903 respectively Then the kinetic energy term in Eq (32) can be rewritten as
119879119879 = sum sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)2
1199031199032119899119899=1 + sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)
2119903119903119899119899=1199031199032+1
119899119899120572120572 (33)
While the potential energy term is
119881119881 =12 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
1199031199032
119899119899119899119899prime=1
+ 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
119903119903
119899119899119899119899prime=1199031199032+1
+ 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime119899119899119899119899prime
+ 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
1199031199032
119899119899prime=1
119903119903
119899119899=1199031199032+1
= 11988111988111 + 11988111988122 + 11988111988112 + 11988111988121
Or equivalently
⎩⎪⎪⎨
⎪⎪⎧ 11988111988111 = 1
2sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
1199031199032119899119899119899119899prime=1120572120572120572120572prime119899119899119899119899prime
11988111988122 = 12sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
119903119903119899119899119899119899prime=1199031199032+1120572120572120572120572prime119899119899119899119899prime
11988111988112 = 12sum sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
119903119903119899119899prime=1199031199032+1
1199031199032119899119899=1120572120572120572120572prime119899119899119899119899prime
11988111988121 = 12sum sum sum sum 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime
1199031199032119899119899prime=1
119903119903119899119899=1199031199032+1120572120572120572120572prime119899119899119899119899prime
(34)
32 Second quantization
We may now quantize this Hamiltonian and write it in the basis of the eigenstates of the coupled
subunits In second quantization the atomic displacement operators of the two subunits are given in
the following form
119906119906119899119899119899119899120572120572 =
⎩⎨
⎧ sum ℏ
2119873119873119872119872119904119904ω119954119954120582120582119890119890119899119899120572120572120582120582 119890119890119894119894119954119954∙119929119929119899119899119886119886119954119954120582120582 + 119886119886minus119954119954120582120582
dagger 119954119954120582120582 119904119904 isin 1 1199031199032
sum ℏ
2119873119873119872119872119904119904120596120596119954119954prime120582120582prime119890119899119899120572120572120582120582
prime119890119890119894119894119954119954prime∙119929119929119899119899 119886119886 119954119954prime120582120582prime + 119886119886minus119954119954prime120582120582primedagger 119954119954prime120582120582prime 119904119904 isin 119903119903
2+ 1 119903119903
(35)
Here we use different notations for the creation and annihilation operators (119886119886119954119954120582120582119886119886minus119954119954120582120582dagger and 119886119886119954119954120582120582119886119886minus119954119954120582120582
dagger )
eigenvalues (ω119954119954120582120582120596120596119954119954120582120582) and eigenvectors (119890119890119899119899120572120572120582120582 119890119899119899120572120572120582120582 ) for the two subunits Note that 119890119890119899119899120572120572120582120582 and 119890119899119899120572120572120582120582 are
of the dimensions of the whole system however their non-zero elements appear only on the relevant
subunits such that they obey the following relations sum 119890119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582
prime119899119899120572120572 = 120575120575120582120582120582120582prime sum 119890119899119899120572120572120582120582
lowast119890119899119899120572120572120582120582
prime119899119899120572120572 =
120575120575120582120582120582120582prime sum 119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582
prime119899119899120572120572 = sum 119890119890119899119899120572120572120582120582
lowast119890119899119899120572120572120582120582
prime119899119899120572120572 = 0 Defining the normal coordinates of the two subunits as
119876119876119954119954120582120582 equiv ℏ
2ω119954119954120582120582119886119886119954119954120582120582 + 119886119886minus119954119954120582120582
dagger 119876119876119954119954120582120582 equiv ℏ
2ω119954119954120582120582119886119886119954119954120582120582 + 119886119886minus119954119954120582120582
dagger (36)
and the corresponding momenta operators as
7
119875119875119954119954120582120582 = 119894119894ℏω119954119954120582120582
2119886119886119954119954120582120582
dagger minus 119886119886minus119954119954120582120582 119875119875119954119954120582120582 = 119894119894ℏω1199541199541205821205822
119886119886119954119954120582120582dagger minus 119886119886minus119954119954120582120582 (37)
Eq (35) reads
119906119906119899119899119899119899120572120572 = sum 119890119890119904119904119899119899120582120582 (119954119954)
119873119873119872119872119904119904119890119890119894119894119954119954∙119929119929119899119899119876119876119954119954120582120582119954119954120582120582 119904119904 isin 1 119903119903
2
sum 119890119904119904119899119899120582120582 (119954119954)119873119873119872119872119904119904
119890119890119894119894119954119954∙119929119929119899119899119876119876119954119954120582120582119954119954120582120582 119904119904 isin 1199031199032
+ 1 119903119903 (38)
The second quantized kinetic energy operator is then written as
119879119879 = 1198791198791 + 1198791198792 = 12sum 119875119875119954119954120582120582119875119875minus119954119954120582120582119954119954120582120582 + 1
2sum 119875119875119954119954prime120582120582prime119875119875minus119954119954prime120582120582prime119954119954prime120582120582prime (39)
Correspondingly the various potential energy terms are obtained by substituting Eq (38) in Eq (34)
11988111988111 =12 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119899119899119899119899prime
119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime120582120582prime (119954119954prime)
119872119872119899119899119872119872119899119899prime
1119873119873119890119890119894119894119954119954∙119929119929119899119899+119894119894119954119954prime∙119929119929119899119899prime119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954120582120582119954119954prime120582120582prime
1199031199032
119899119899119899119899prime=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954119954119954prime120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (119954119954prime)1119873119873 119890119890
119894119894(119954119954+119954119954prime)119929119929119899119899prime
119899119899prime120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime
119872119872119899119899119872119872119899119899prime119899119899
1199031199032
119899119899119899119899prime=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954119954119954prime120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (119954119954prime) 120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime
1199031199032
119899119899119899119899prime=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime
119954119954120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (minus119954119954)120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
1199031199032
119899119899119899119899prime=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime
119954119954120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954) 119863119863119899119899120572120572119899119899
prime120572120572prime(minus119954119954)119890119890119899119899prime120572120572prime120582120582prime (minus119954119954)
120572120572prime
1199031199032
119899119899prime=1
1199031199032
119899119899=1120572120572
=12 ω119954119954120582120582prime
2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582120582120582prime
119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582prime (119954119954)
lowast
1199031199032
119899119899=1
=120572120572
12ω119954119954120582120582
2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582
ie
11988111988111 = 12sum ω119954119954120582120582
2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582 (310)
Going from the second line to the third line we used the fact that the sum over 119899119899prime runs between plusmninfin
and the summand depends only on the difference between 119899119899 and 119899119899prime hence the sum is independent
of the value of the index 119899119899 Therefore we can replace the sum over 119899119899prime by a sum over 119897119897 equiv 119899119899 minus 119899119899prime
amd define 119929119929119897119897 equiv 119929119929119899119899 minus 119929119929119899119899prime Following the same procedure we can get the corresponding
expressions for the second diagonal term
11988111988122 = 12sum 1205961205961199541199541205821205822 119876119876119954119954120582120582119876119876minus119954119954120582120582119954119954120582120582 (311)
Similarly for the off-diagonal terms we get
8
11988111988112 =12 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119899119899119899119899prime
119903119903
119899119899prime=1199031199032+1
119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime
120582120582prime (119954119954prime)
119872119872119899119899119872119872119899119899prime
1119873119873119890119890119894119894119954119954∙119929119929119899119899+119894119894119954119954prime∙119929119929119899119899prime119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954120582120582119954119954prime120582120582prime
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954119954119954prime120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime
120582120582prime (119954119954prime)1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime
119872119872119899119899119872119872119899119899prime119899119899
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954119954119954prime120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime
120582120582prime (119954119954prime) 120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime
119954119954120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime
120582120582prime (minus119954119954)120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954
119890119899119899prime120572120572prime120582120582prime (119954119954)
lowast120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119890119890119899119899120572120572120582120582 (119954119954)119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954
Where we have defined
119881119881120582120582120582120582prime(119954119954) equiv sum sum sum 119890119899119899prime120572120572prime120582120582prime (119954119954)
lowast119881119881119899119899120572120572119899119899
prime120572120572prime(119954119954)119890119890119899119899120572120572120582120582 (119954119954)119903119903119899119899prime=1199031199032+1
1199031199032119899119899=1120572120572120572120572prime (312)
where
119881119881119899119899120572120572119899119899prime120572120572prime(119954119954) equiv sum 119890119890119894119894119954119954∙119929119929119897119897
119872119872119904119904119872119872119904119904prime119897119897 120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime (313)
Itrsquos easy to show that 119881119881120582120582120582120582prime(119954119954) has the following property
119881119881120582120582120582120582primelowast (119954119954) = 119881119881120582120582120582120582prime(minus119954119954) (314)
Following the same procedure we have
9
11988111988121 =12 120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119899119899119899119899prime
119903119903
119899119899=1199031199032+1
119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime
120582120582prime (119954119954prime)
119872119872119899119899119872119872119899119899prime
1119873119873119890119890119894119894119954119954∙119929119929119899119899+119894119894119954119954prime∙119929119929119899119899prime119876119876119954119954120582120582119876119876119954119954prime120582120582prime
119954119954120582120582119954119954prime120582120582prime
1199031199032
119899119899prime=1120572120572120572120572prime
=12 120601120601120572120572prime120572120572
119899119899prime minus 119899119899119904119904prime 119904119904
119899119899prime119899119899
119903119903
119899119899prime=1199031199032+1
119890119899119899prime120572120572prime120582120582prime (119954119954prime)119890119890119899119899120572120572120582120582 (119954119954)
119872119872119899119899prime119872119872119899119899
1119873119873 119890119890
119894119894119954119954prime∙119929119929119899119899prime+119894119894119954119954∙119929119929119899119899119876119876119954119954prime120582120582prime119876119876119954119954120582120582119954119954prime120582120582prime119954119954120582120582
1199031199032
119899119899=1120572120572prime120572120572
=12 119876119876119954119954prime120582120582prime119876119876119954119954120582120582
119954119954119954119954prime120582120582120582120582prime119890119899119899prime120572120572prime120582120582prime (119954119954prime)119890119890119899119899120572120572120582120582 (119954119954)
1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime120601120601120572120572120572120572prime
119899119899 minus 119899119899prime119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime
119872119872119899119899119872119872119899119899prime119899119899
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954prime120582120582prime119876119876119954119954120582120582
119954119954119954119954prime120582120582120582120582prime119890119899119899prime120572120572prime120582120582prime (119954119954prime)119890119890119899119899120572120572120582120582 (119954119954) 120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
1119873119873 119890119890
119894119894(119954119954+119954119954prime)∙119929119929119899119899prime
119899119899prime
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954prime120582120582prime119876119876minus119954119954prime120582120582119890119899119899prime120572120572prime
120582120582prime (119954119954prime)119890119890119899119899120572120572120582120582 (minus119954119954prime)120601120601120572120572120572120572prime 119897119897
119904119904 119904119904prime119890119890minus119894119894119954119954prime∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897119954119954prime120582120582120582120582prime
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954prime120582120582prime119876119876119954119954prime120582120582
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954prime 119890119890119899119899120572120572120582120582 (119954119954prime)
lowast120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890minus119894119894119954119954prime∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119890119899119899prime120572120572prime120582120582prime (119954119954prime)
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582prime119876119876119954119954120582120582
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954
119890119890119899119899120572120572120582120582 (119954119954)lowast120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890minus119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119890119899119899prime120572120572prime120582120582prime (119954119954)
119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
=12 119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger dagger
3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954
119890119899119899prime120572120572prime120582120582prime (119954119954)
lowast120601120601120572120572120572120572prime
119897119897119904119904 119904119904prime
119890119890119894119894119954119954∙119929119929119897119897
119872119872119899119899119872119872119899119899prime119897119897
119890119890119899119899120572120572120582120582 (119954119954)119903119903
119899119899prime=1199031199032+1
1199031199032
119899119899=1120572120572120572120572prime
lowast
=
⎩⎨
⎧12 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903
120582120582prime=31199031199032 +1
31199031199032
120582120582=1119954119954⎭⎬
⎫dagger
= 11988111988112dagger
Define
1198671198671 = 11987911987911 + 119881119881111198671198672 = 11987911987922 + 1198811198812211986711986712 = 11988111988112 + 11988111988121
(315)
we finally get the expressions of 1198671198671 1198671198672 and 11986711986712 in Eq (31) as follows
⎩⎪⎨
⎪⎧ 1198671198671 = sum sum ℏ120596120596119954119954120582120582 119886119886119954119954120582120582
dagger 119886119886119954119954120582120582 + 1231199031199032
120582120582=1119954119954
1198671198672 = sum sum ℏ120596120596119954119954120582120582prime 119886119886119954119954120582120582primedagger 119886119886119954119954120582120582prime + 1
23119903119903
120582120582prime=1+31199031199032119954119954
11986711986712 = 12sum sum sum 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903120582120582prime=31199031199032 +1
31199031199032120582120582=1 + h c 119954119954
(316)
Here hc means the Hermitian conjugate Since the indices of 119886119886119954119954120582120582 119886119886119954119954120582120582prime and 119876119876119954119954120582120582 119876119876119954119954120582120582prime belong to
the two system sections we can define an abbreviated notation 119886119886119954119954120582120582 and 119876119876119954119954120582120582 using the index 120582120582 to
identify which subsystem they belong to In this case the Hamiltonian of the coupled system can be
10
simplified as follows
119867119867 = 1198671198670 + 119867119867119862119862 (317)
where
1198671198670 = sum sum ℏ120596120596119954119954120582120582 119886119886119954119954120582120582
dagger 119886119886119954119954120582120582 + 123119903119903
120582120582=1119954119954
119867119867119862119862 = 12sum sum sum 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime
dagger3119903119903120582120582prime=31199031199032 +1
31199031199032120582120582=1 + ℎ 119888119888 119954119954
(318)
4 Greenrsquos function 41 Dynamics of the ladder operators
In order to describe the dynamics of the ladder operators appearing in the Hamiltonian of Eq (317)
we express them in the Heisenberg picture as follows
119886119886119901119901(120591120591) = 119890119890119894119894ℏ119867119867119905119905119886119886119901119901119890119890
minus119894119894ℏ119867119867119905119905 = 119890119890
119867119867120591120591ℏ 119886119886119901119901119890119890
minus119867119867120591120591ℏ (41)
where we define the imaginary time 120591120591 equiv 119894119894119905119905 and introduce the notation 119953119953 equiv (119954119954 120582120582) and 119953119953 equiv (minus119954119954 120582120582)
The corresponding equation of motion for the ladder operators is given by
ℏ part119886119886119953119953(120591120591)
120597120597120591120591= 119867119867119886119886119953119953(120591120591) (42)
For the uncoupled system (119867119867119862119862 = 0) this gives
ℏpart119886119886119953119953(120591120591)120597120597120591120591
= 1198671198670119886119886119953119953(120591120591) = 1198671198670 1198901198901198671198670120591120591ℏ 119886119886119953119953119890119890
minus1198671198670120591120591ℏ = 1198671198670119890119890
1198671198670120591120591ℏ 119886119886119953119953119890119890
minus1198671198670120591120591ℏ minus 119890119890
1198671198670120591120591ℏ 119886119886119953119953119890119890
minus1198671198670120591120591ℏ 1198671198670
= 1198901198901198671198670120591120591ℏ 1198671198670119886119886119953119953 minus 1198861198861199531199531198671198670119890119890
minus1198671198670120591120591ℏ = 119890119890
1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890
minus1198671198670120591120591ℏ
ie
ℏ part119886119886119953119953(120591120591)
120597120597120591120591= 119890119890
1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890
minus1198671198670120591120591ℏ (43)
The commutator on the right-hand-side of Eq (43) can be evaluated as follows
1198671198670119886119886119953119953prime = sum ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953 + 1
2119953119953 119886119886119953119953prime = sum ℏ120596120596119953119953119886119886119953119953
dagger119886119886119953119953119886119886119953119953prime119953119953 = sum ℏ120596120596119953119953119886119886119953119953dagger119886119886119953119953119886119886119953119953prime minus 119886119886119953119953prime119886119886119953119953
dagger119886119886119953119953119953119953 =
sum ℏ120596120596119953119953119886119886119953119953dagger119886119886119953119953119886119886119953119953prime minus 120575120575119953119953prime119953119953 + 119886119886119953119953
dagger119886119886119953119953prime119886119886119953119953119953119953 = minusℏ120596120596119953119953prime119886119886119953119953prime + sum ℏ120596120596119953119953119886119886119953119953dagger119886119886119953119953119886119886119953119953prime minus 119886119886119953119953
dagger119886119886119953119953119886119886119953119953prime119953119953 = minusℏ120596120596119953119953prime119886119886119953119953prime
ie
1198671198670119886119886119953119953prime = minusℏ120596120596119953119953prime119886119886119953119953prime (44)
Therefore we have
ℏ part119886119886119953119953(120591120591)
120597120597120591120591= 119890119890
1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890
minus1198671198670120591120591ℏ = minusℏ120596120596119953119953119890119890
1198671198670120591120591ℏ 119886119886119953119953119890119890
minus1198671198670120591120591ℏ = minusℏ120596120596119953119953119886119886119953119953(120591120591) (45)
11
The solution of Eq (45) is 119886119886119953119953(120591120591) = 119886119886119953119953(0)119890119890minus120596120596119953119953120591120591 = 119886119886119953119953119890119890minus120596120596119953119953120591120591 For 119886119886119953119953dagger(120591120591) we have
1198671198670119886119886119953119953primedagger (0) = 1198671198670119886119886119953119953prime
dagger = ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953 +
12
119953119953
119886119886119953119953primedagger = ℏ120596120596119953119953119886119886119953119953
dagger119886119886119953119953119886119886119953119953primedagger
119953119953
= ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953119886119886119953119953prime
dagger minus 119886119886119953119953primedagger 119886119886119953119953
dagger119886119886119953119953119953119953
= ℏ120596120596119953119953 119886119886119953119953dagger 120575120575119953119953119953119953prime + 119886119886119953119953prime
dagger 119886119886119953119953 minus 119886119886119953119953primedagger 119886119886119953119953
dagger119886119886119953119953119953119953
= ℏ120596120596119953119953prime119886119886119953119953primedagger + ℏ120596120596119953119953 119886119886119953119953
dagger119886119886119953119953primedagger 119886119886119953119953 minus 119886119886119953119953prime
dagger 119886119886119953119953dagger119886119886119953119953
119953119953
= ℏ120596120596119953119953prime119886119886119953119953primedagger
In summary we have
119886119886119953119953(120591120591) = 119886119886119953119953(0)119890119890minus120596120596119953119953120591120591 = 119886119886119953119953119890119890minus120596120596119953119953120591120591
119886119886119953119953dagger(120591120591) = 119886119886119953119953
dagger(0)119890119890120596120596119953119953120591120591 = 119886119886119953119953dagger119890119890120596120596119953119953120591120591 (46)
42 Greenrsquos function
To describe thermal transport properties between different system sections we use the formalism of
thermal (or imaginary time) phonon Greenrsquos function [2] To this end we define the thermal Greenrsquos
function for phonons as
119866119866119953119953119953119953prime(120591120591 120591120591prime) = minuslang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime)rang = minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime) (47)
where 119879119879120591120591 is the time ordering operator and 120588120588119867119867 is the statistical operator for the grand canonical
ensemble (note that the chemical potential for phonons is zero)
120588120588119867119867 = 119890119890minus120573120573119867119867119885119885119867119867 (48)
where the partition function is given by 119885119885119867119867 equiv Tr119890119890minus120573120573119867119867 and 120573120573 = 1119896119896119861119861119879119879 with 119896119896119861119861 being
Boltzmannrsquos constant and 119879119879 the temperature For time independent Hamiltonians 119866119866119953119953119953119953prime(120591120591 120591120591prime)
depends only on 120591120591 minus 120591120591prime ie
119866119866119953119953119953119953prime(120591120591 120591120591prime) = 119866119866119953119953119953119953prime(120591120591 minus 120591120591prime 0) (49)
To show this we shall first assume that 120591120591 gt 120591120591prime such that
119866119866119953119953119953119953prime(120591120591 120591120591prime) = minuslang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime)rang = minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime) = minusTr 120588120588119867119867119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime)
= minusTr 120588120588119867119867119890119890119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591ℏ 119890119890
119867119867120591120591primeℏ 119886119886119953119953prime
dagger 119890119890minus119867119867120591120591primeℏ = minusTr 120588120588119867119867119890119890
minus119867119867120591120591primeℏ 119890119890
119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591minus120591120591prime
ℏ 119886119886119953119953primedagger
= minusTr 120588120588119867119867119890119890119867119867120591120591minus120591120591prime
ℏ 119886119886119953119953119890119890minus119867119867120591120591minus120591120591prime
ℏ 119886119886119953119953primedagger = minusTr 120588120588119867119867119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953prime
dagger (0)
= minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953primedagger (0) = minus lang119879119879120591120591119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953prime
dagger (0)rang = 119866119866119953119953119953119953prime(120591120591 minus 120591120591prime 0)
Where we have used the commutativity of 120588120588119867119867 and 119890119890plusmn119867119867120591120591prime
ℏ and the invariance of the trace operation
12
towards cyclic permutations Similarly for 120591120591 lt 120591120591prime we have
119866119866119953119953119953119953prime(120591120591 120591120591prime) = minuslang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime)rang = minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime) = minusTr 120588120588119867119867119886119886119953119953primedagger (120591120591prime)119886119886119953119953(120591120591)
= minusTr 120588120588119867119867119890119890119867119867120591120591primeℏ 119886119886119953119953prime
dagger 119890119890minus119867119867120591120591primeℏ 119890119890
119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591ℏ = minusTr 120588120588119867119867119886119886119953119953prime
dagger 119890119890119867119867120591120591minus120591120591prime
ℏ 119886119886119953119953119890119890minus119867119867120591120591ℏ 119890119890
119867119867120591120591primeℏ
= minusTr 120588120588119867119867119886119886119953119953primedagger 119890119890
119867119867120591120591minus120591120591primeℏ 119886119886119953119953119890119890
minus119867119867120591120591minus120591120591prime
ℏ = minusTr 120588120588119867119867119886119886119953119953primedagger (0)119886119886119953119953(120591120591 minus 120591120591prime)
= minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953primedagger (0)119886119886119953119953(120591120591 minus 120591120591prime) = minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953prime
dagger (0)
= minuslang119879119879120591120591119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953primedagger (0)rang = 119866119866119953119953119953119953prime(120591120591 minus 120591120591prime 0)
For simplicity we will introduce the notation 119866119866119953119953119953119953prime(120591120591 0) equiv 119866119866119953119953119953119953prime(120591120591) We further note that when using
imaginary time 119866119866119953119953119953119953prime(120591120591) is a periodic functions in the domain [minus120573120573ℏ120573120573ℏ] with a period of 120573120573ℏ (see
Page 236 of Ref [2])
119866119866119953119953119953119953prime(120591120591 + 120573120573ℏ 0) = 119866119866119953119953119953119953prime(120591120591 0) 120591120591 lt 0119866119866119953119953119953119953prime(120591120591 minus 120573120573ℏ 0) = 119866119866119953119953119953119953prime(120591120591 0) 120591120591 gt 0 (410)
To show this we shall again assume first that minus120573120573ℏ lt 120591120591 lt 0 to write that is
119866119866119953119953119953119953prime(120591120591 + 120573120573ℏ 0) = minus lang119879119879120591120591119886119886119953119953(120591120591 + 120573120573ℏ)119886119886119953119953primedagger (0)rang = minuslang119886119886119953119953(120591120591 + 120573120573ℏ)119886119886119953119953prime
dagger (0)rang
= minusTr 120588120588119867119867119890119890119867119867(120591120591+120573120573ℏ)
ℏ 119886119886119953119953119890119890minus119867119867(120591120591+120573120573ℏ)
ℏ 119886119886119953119953prime = minusTr 120588120588119867119867119890119890120573120573119867119867119890119890
119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591ℏ 119890119890minus120573120573119867119867119886119886119953119953prime
= minusTr120588120588119867119867119890119890120573120573119867119867119886119886119953119953(120591120591)119890119890minus120573120573119867119867119886119886119953119953prime(0) = minus
1119885119885119867119867
Tr119890119890minus120573120573119867119867119890119890120573120573119867119867119886119886119953119953(120591120591)119890119890minus120573120573119867119867119886119886119953119953prime(0)
= minus1119885119885119867119867
Tr119886119886119953119953(120591120591)119890119890minus120573120573119867119867119886119886119953119953prime(0) = minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119886119886119953119953prime(0)119886119886119953119953(120591120591)
= minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime(0) = minusTr120588120588119867119867119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime = 119866119866119953119953119953119953prime(120591120591 0)
Similarly for 120573120573ℏ gt 120591120591 gt 0 we have
119866119866119953119953119953119953prime(120591120591 minus 120573120573ℏ 0) = minus lang119879119879120591120591119886119886119953119953(120591120591 minus 120573120573ℏ)119886119886119953119953primedagger (0)rang = minuslang119886119886119953119953prime
dagger (0)119886119886119953119953(120591120591 minus 120573120573ℏ)rang
= minusTr 120588120588119867119867119886119886119953119953prime119890119890119867119867(120591120591minus120573120573ℏ)
ℏ 119886119886119953119953119890119890minus119867119867(120591120591minus120573120573ℏ)
ℏ = minusTr 120588120588119867119867119886119886119953119953prime119890119890minus120573120573119867119867119890119890
119867119867120591120591ℏ 119886119886119953119953119890119890
minus119867119867120591120591ℏ 119890119890120573120573119867119867
= minusTr120588120588119867119867119886119886119953119953prime(0)119890119890minus120573120573119867119867119886119886119953119953(120591120591)119890119890120573120573119867119867 = minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119886119886119953119953prime(0)119890119890minus120573120573119867119867119886119886119953119953(120591120591)119890119890120573120573119867119867
= minus1119885119885119867119867
Tr119886119886119953119953prime(0)119890119890minus120573120573119867119867119886119886119953119953(120591120591) = minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119886119886119953119953(120591120591)119886119886119953119953prime(0)
= minus1119885119885119867119867
Tr119890119890minus120573120573119867119867119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime(0) = minusTr120588120588119867119867119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime = 119866119866119953119953119953119953prime(120591120591 0)
13
Therefore 119866119866119953119953119953119953prime(120591120591) can be expanded as a Fourier series in the domain [0120573120573ℏ] as follows
119866119866119953119953119953119953prime(120591120591) = 1120573120573ℏsum 119890119890minus119894119894120596120596119899119899120591120591119866119866119953119953119953119953prime(119894119894120596120596119899119899)infinminusinfin (411)
where 120596120596119899119899 = 2120587120587119899119899120573120573ℏ
and the associated Fourier coefficient is given by
119866119866119953119953119953119953prime(119894119894120596120596119899119899) = int d120591120591119890119890119894119894120596120596119899119899120591120591119866119866119953119953119953119953prime(120591120591)120573120573ℏ0 120596120596119899119899 = 2119899119899120587120587
120573120573ℏ (412)
Having proven the translational time invariance and the periodicity of the Greenrsquos functios we can
now calculate it for the uncoupled system
119866119866119953119953119953119953prime0 (120591120591 120591120591prime) = minuslang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime)rang0 == minus lang119886119886119953119953(0)119890119890minus120596120596119953119953120591120591119886119886119953119953prime
dagger (0)119890119890120596120596119953119953prime120591120591primerang 120591120591 minus 120591120591prime gt 0
minus lang119886119886119953119953primedagger (0)119890119890120596120596119953119953prime120591120591
prime119886119886119953119953(0)119890119890minus120596120596119953119953120591120591rang 120591120591 minus 120591120591prime lt 0
= minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591
primelang119886119886119953119953(0)119886119886119953119953prime
dagger (0)rang 120591120591 minus 120591120591prime gt 0
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang 120591120591 minus 120591120591prime lt 0
= minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591
primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0) + 119886119886119953119953(0)119886119886119953119953primedagger (0)rang 120591120591 minus 120591120591prime gt 0
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang 120591120591 minus 120591120591prime lt 0
= minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591
primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang + 120575120575119953119953119953119953prime 120591120591 minus 120591120591prime gt 0
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang 120591120591 minus 120591120591prime lt 0
ie
119866119866119953119953119953119953prime0 (120591120591 120591120591prime) =
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang + 120575120575119953119953119953119953prime 120591120591 minus 120591120591prime gt 0
minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime
dagger (0)119886119886119953119953(0)rang 120591120591 minus 120591120591prime lt 0 (413)
where we have used Eq (46) for 119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime) and Commutation relation 119886119886119953119953(0)119886119886119953119953prime
dagger (0) = 120575120575119953119953119953119953prime
To calculate lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang we first prove following equation
119890119890119860119860119861119861119890119890minus119860119860 = sum 1119899119899119860(119899119899)119861119861infin
119899119899=0 (414)
where 119860(119896119896)119861119861 equiv 119860 119860(119896119896minus1)119861119861 To prove this identity we define the operator 119891119891(119905119905) = 119890119890119905119905A119861119861119890119890minus119905119905119860119860
and taylor expand it around 119905119905 = 0
119891119891(119905119905) = 119891119891(0) + 119905119905119891119891prime(0) + 1199051199052
2119891119891primeprime(0) + ⋯ = sum 119905119905119899119899
119899119899119889119889119899119899119891119889119889119905119905119899119899
119905119905=0
infin119899119899=0 (415)
The corresponding derivatives are given by
⎩⎪⎨
⎪⎧ 119891119891prime(119905119905) = 119890119890119905119905119860119860119860119861119861119890119890minus119905119905119860119860 minus 119890119890119905119905119860119860119861119861119860119890119890minus119905119905119860119860 = 119890119890119905119905119860119860119860119861119861119890119890minus119905119905119860119860
119891119891primeprime(119905119905) = 119890119890119905119905119860119860119860119860119861119861 minus 119860119861119861119860119890119890minus119905119905119860119860 = 119890119890119905119905119860119860119860(2)119861119861119890119890minus119905119905119860119860
⋮119891119891(119899119899)(119905119905) = 119890119890119905119905119860119860119860(119899119899)119861119861119890119890minus119905119905119860119860
(416)
14
Substituting Eq (416) into Eq (415) we have
119890119890119905119905A119861119861119890119890minus119905119905119860119860 = sum 119905119905119899119899
119899119899119860(119899119899)119861119861infin
119899119899=0 (417)
Itrsquos clear that Eq (413) is a spectial case of Eq (416) with 119905119905 = 1 With this we can proceed as
follows
lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang =
11198851198851198671198670
Tr119890119890minus1205731205731198671198670119886119886119953119953primedagger (0)119886119886119953119953(0) =
11198851198851198671198670
Tr119890119890minus1205731205731198671198670119886119886119953119953primedagger (0)1198901198901205731205731198671198670119890119890minus1205731205731198671198670119886119886119953119953(0)
=11198851198851198671198670
Tr (minus120573120573)119899119899
1198991198991198671198670
(119899119899)119886119886119953119953primedagger (0)
infin
119899119899=0
119890119890minus1205731205731198671198670119886119886119953119953(0)
Note that 1198671198670119886119886119953119953primedagger (0) = ℏ120596120596119953119953prime119886119886119953119953prime
dagger (0) we have
1198671198670(119899119899)119886119886119953119953prime
dagger (0) = ℏ120596120596119953119953prime119899119899119886119886119953119953primedagger (0) (418)
such that
lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang =
11198851198851198671198670
Tr (minus120573120573)119899119899
1198991198991198671198670
(119899119899)119886119886119953119953primedagger (0)
infin
119899119899=0
119890119890minus1205731205731198671198670119886119886119953119953(0)
=11198851198851198671198670
Tr minus120573120573ℏ120596120596119953119953prime
119899119899
119899119899119886119886119953119953primedagger (0)
infin
119899119899=0
119890119890minus1205731205731198671198670119886119886119953119953(0)
= 119890119890minus120573120573ℏ120596120596119953119953prime11198851198851198671198670
Tr119890119890minus1205731205731198671198670119886119886119953119953(0)119886119886119953119953primedagger (0) = 119890119890minus120573120573ℏ120596120596119953119953primelang119886119886119953119953(0)119886119886119953119953prime
dagger (0)rang
= 119890119890minus120573120573ℏ120596120596119953119953primelang119886119886119953119953primedagger (0)119886119886119953119953(0)rang + 120575120575119953119953119953119953prime
therefore we obtain lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang 119890119890120573120573ℏ120596120596119953119953prime minus 1 = 120575120575119953119953119953119953prime For 119953119953prime ne 119953119953 and general 119890119890120573120573ℏ120596120596119953119953prime we have
lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang = 0 For 119953119953prime = 119953119953 we obtain lang119886119886119953119953
dagger(0)119886119886119953119953(0)rang = 119890119890120573120573ℏ120596120596119953119953 minus 1minus1
Hence we have
lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang = 120575120575119953119953prime119953119953
1119890119890120573120573ℏ120596120596119953119953minus1
equiv 120575120575119953119953prime119953119953119899119899119861119861120596120596119953119953 (419)
where 119899119899119861119861120596120596119953119953 is the Bose-Einstein distribution for phonons Substituting Eq (418) in Eq (413)
we obtain
119866119866119953119953119953119953prime0 (120591120591 120591120591prime) =
minus120575120575119953119953119953119953prime1 + 119899119899119861119861120596120596119953119953119890119890minus120596120596119953119953120591120591minus120591120591prime 120591120591 minus 120591120591prime gt 0
minus120575120575119953119953119953119953prime119899119899119861119861120596120596119953119953119890119890minus120596120596119953119953120591120591minus120591120591prime 120591120591 minus 120591120591prime lt 0 (420)
Note that only those Greenrsquos functions of the form 119866119866119953119953119953119953prime0 (120591120591 120591120591prime) = 1198661198661199531199530(120591120591 minus 120591120591prime)120575120575119953119953119953119953prime are non-zero due
to the orthogonality of normal modes Looking at the non-vanishing terms and setting 120591120591prime = 0
without loss of generality we can calculate the Fourier transform of 1198661198661199531199530(120591120591) from Eq (412)
1198661198661199531199530(119894119894120596120596119899119899) = int d1205911205911198901198901198941198941205961205961198991198991205911205911198661198661199531199530(120591120591)120573120573ℏ0 = minusint d120591120591119890119890119894119894120596120596119899119899120591120591120573120573ℏ
0 1 + 119899119899119861119861120596120596119953119953119890119890minus120596120596119953119953120591120591 = minus1+119899119899119861119861120596120596119953119953119894119894120596120596119899119899minus120596120596119953119953
119890119890119894119894120596120596119899119899minus120596120596119953119953120573120573ℏ minus
15
1 = minus1+ 1
119890119890120573120573ℏ120596120596119953119953minus1
119894119894120596120596119899119899minus120596120596119953119953119890119890119894119894
2120587120587119899119899120573120573ℏ 120573120573ℏ119890119890minus120573120573ℏ120596120596119953119953 minus 1 = minus 119890119890120573120573ℏ120596120596119953119953
119894119894120596120596119899119899minus120596120596119953119953
119890119890minus120573120573ℏ120596120596119953119953minus1
119890119890120573120573ℏ120596120596119953119953minus1= minus 1
119894119894120596120596119899119899minus120596120596119953119953
1minus119890119890120573120573ℏ120596120596119953119953
119890119890120573120573ℏ120596120596119953119953minus1= 1
119894119894120596120596119899119899minus120596120596119953119953
ie
1198661198661199531199530(119894119894120596120596119899119899) == 1119894119894120596120596119899119899minus120596120596119953119953
(421)
5 Fermirsquos golden rule 51 Interaction picture
Next we can proceed with calculating the Greenrsquos function of the coupled system To this end we
define the coupling Hamiltonian operator term in the interaction picture as
119867119867119862119862119868119868 (120591120591) = 1198901198901198671198670120591120591ℏ 119867119867119862119862119890119890
minus1198671198670120591120591ℏ (51)
The time evolution of 119867119867119862119862(120591120591) is then given by
ℏ part119867119867119862119862119868119868 (120591120591)120597120597120591120591
= 1198671198670119867119867119862119862119868119868 (120591120591) (52)
Note that the Greenrsquos function defined above was given in the Heisenberg picture To proceed we
need to transform it to the interaction picture
119866119866119953119953119953119953prime(120591120591 0) = minus lang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953primedagger (0)rang = minusTr 120588120588119867119867119879119879120591120591 119890119890
119867119867120591120591ℏ 119886119886119953119953(0)119890119890minus
119867119867120591120591ℏ 119886119886119953119953prime
dagger (0)
= minusTr 120588120588119867119867119879119879120591120591 119890119890119867119867120591120591ℏ 119890119890minus
1198671198670120591120591ℏ 119890119890
1198671198670120591120591ℏ 119886119886119953119953(0)119890119890minus
1198671198670120591120591ℏ 119890119890
1198671198670120591120591ℏ 119890119890minus
119867119867120591120591ℏ 119886119886119953119953prime
dagger (0)
= minusTr 120588120588119867119867119879119879120591120591 119880119880(0 120591120591)119886119886119953119953119868119868 (120591120591)119880119880(120591120591 0)119886119886119953119953primedagger (0)
where
119886119886119953119953119868119868 (120591120591) equiv 1198901198901198671198670120591120591ℏ 119886119886119953119953(0)119890119890minus
1198671198670120591120591ℏ (53)
is the operator in interaction picture and the operator 119880119880 is defined by
119880119880(1205911205911 1205911205912) equiv 11989011989011986711986701205911205911ℏ 119890119890minus
119867119867 (1205911205911minus1205911205912)ℏ 119890119890minus
11986711986701205911205912ℏ (54)
Note that while 119880119880 is not unitary it satisfies the following group property
119880119880(1205911205911 1205911205912)119880119880(1205911205912 1205911205913) = 119880119880(1205911205911 1205911205913) (55)
and the boundary condition 119880119880(1205911205911 1205911205911) = 1 In addition the 120591120591 derivative of 119880119880 is simply
ℏpart119880119880(120591120591 120591120591prime)
120597120597120591120591= 1198671198670119890119890
1198671198670120591120591ℏ 119890119890minus
119867119867120591120591minus120591120591primeℏ minus 119890119890
1198671198670120591120591ℏ 119867119867119890119890minus
119867119867120591120591minus120591120591primeℏ 119890119890minus
1198671198670120591120591primeℏ
= 1198901198901198671198670120591120591ℏ 1198671198670 minus 119867119867119890119890minus
1198671198670120591120591ℏ 119890119890
1198671198670120591120591ℏ 119890119890minus
119867119867120591120591minus120591120591primeℏ 119890119890minus
1198671198670120591120591primeℏ = 119890119890
1198671198670120591120591ℏ minus119867119867119862119862119890119890
minus1198671198670120591120591ℏ 119880119880(120591120591 120591120591prime)
= minus119867119867119862119862119868119868 (120591120591)119880119880(120591120591 120591120591prime)
16
ie
ℏ part119880119880120591120591120591120591prime120597120597120591120591
= minus119867119867119862119862119868119868 (120591120591)119880119880(120591120591 120591120591prime) (56)
The solution of Eq (56) is (see page 235 of Ref [2])
119880119880(120591120591 120591120591prime) = summinus1ℏ
119899119899
119899119899 int 1198891198891205911205911120591120591120591120591prime ⋯int 119889119889120591120591119899119899
120591120591120591120591prime 119879119879120591120591119867119867119862119862119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119899119899)infin
119899119899=0 (57)
The exact thermal Greens function now may be rewritten in the interaction picture as
119866119866119953119953119953119953prime(120591120591 0) = minus1119885119885119867119867
Tr 119890119890minus120573120573119867119867119879119879120591120591119880119880(0 120591120591)119886119886119953119953119868119868 (120591120591)119880119880(120591120591 0)119886119886119953119953primedagger (0)
= minusTr 119890119890minus120573120573119867119867119879119879120591120591 119880119880(0 120591120591)119886119886119953119953119868119868 (120591120591)119880119880(120591120591 0)119886119886119953119953prime
dagger (0)
Tr119890119890minus120573120573119867119867
= minusTr 119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)119879119879120591120591 119886119886119953119953119868119868 (120591120591)119880119880(0 120591120591)119880119880(120591120591 0)119886119886119953119953prime
dagger (0)
Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)
= minusTr 119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)119879119879120591120591 119886119886119953119953119868119868 (120591120591)119880119880(00)119886119886119953119953prime
dagger (0)
Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)=
= minusTr 119890119890minus1205731205731198671198670119879119879120591120591 119880119880(120573120573ℏ 0)119886119886119953119953119868119868 (120591120591)119886119886119953119953prime
dagger (0)
Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)
Where we used the fact that we are free to change the order of the operators within the time ordering
operation (see pages 241-242 of Ref [2])
52 Wickrsquos theorem
Thus the Greenrsquos function can be expanded as [2]
119866119866119953119953119953119953prime(120591120591 0) = minussum 1
119898119898minus1ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)119886119886119953119953119868119868 (120591120591)119886119886
119953119953primedagger (0)rang0infin
119898119898=0
sum 1119898119898minus
1ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)rang0infin
119898119898=0 (58)
where lang⋯ rang0 represents the ensemble average with respect to the non-interacting basis
Tr119890119890minus1205731205731198671198670(⋯ ) Or explicitly
119866119866119953119953119953119953prime(120591120591 0) = minus119866119866119953119953119953119953prime0 (120591120591)minus1ℏint 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953primedagger (0)rang0+
12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0+⋯
1minus1ℏint 1198891198891205911205911120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)rang0+12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862
119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0+⋯
(59)
For consiceness we introduce the following notation
119863119863119898119898 equiv 1119898119898minus 1
ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)rang0 (510)
To simplify the calculation in Eq (58) we adopt Wickrsquos theorem (see pages 237-241 of Ref
[2])which can be expressed as follows
lang119879119879120591120591119860119861119861119862119863119863 ⋯119884119884119885rang0 = lang119879119879120591120591119860119861119861rang0lang119879119879120591120591119862119863119863rang0⋯ lang119879119879120591120591119884119884119885rang0 + lang119879119879120591120591119860119862rang0lang119879119879120591120591119861119861119863119863rang0⋯ lang119879119879120591120591119883119883119885rang0 + ⋯ (511)
17
Here 119860119861119861 hellip 119884119884 119885 represent 119886119886119953119953(120591120591) or 119886119886119953119953dagger(120591120591) In Eq (511) each term corresponds to a particular
pairing of the operators 119860119861119861119862119863119863 ⋯119884119884119885 and all possible pairings are taken into account Here the only
non-vanishing propagators will have the form [see Eq (420)]
lang119879119879120591120591 119886119886119953119953primedagger (120591120591prime)119886119886119953119953(120591120591)rang0 = minuslang119879119879120591120591 119886119886119953119953(120591120591)119886119886119953119953prime
dagger (120591120591prime)rang0 = 119866119866119953119953119953119953prime0 (120591120591 120591120591prime) = 1198661198661199531199530(120591120591 minus 120591120591prime)120575120575119953119953119953119953prime (512)
Therefore the second term in the numerator of Eq (59) can be calculated as
1198681198682 = minus1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0
= minus1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 lang119879119879120591120591119886119886119953119953prime
dagger (0)119886119886119953119953(120591120591)rang0 minus1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
= 119866119866119953119953119953119953prime0 (120591120591 0) minus
1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 minus
1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
The first term in above equation is called disconnected part since the pairing is performed separately
on 119867119867119862119862(1205911205911) and 119886119886119953119953(120591120591)119886119886119953119953primedagger (0) All other terms have pairs that mix creation and annihilation operators
of the Hamiltonian with 119886119886119953119953(120591120591) or 119886119886119953119953primedagger (0) and are said to have connected party For simplicity we
include all these terms in the notation lang119879119879120591120591119867119867119862119862(1205911205911)119886119886119953119953(120591120591)119886119886119953119953primedagger (0)rang0119888119888 Furthermore we define 119866119866119953119953119953119953prime
(1) (120591120591) equiv
minus1ℏ int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862(1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 Then we have
1198681198683 = minus1ℏint 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 = 119866119866119953119953119953119953prime0 (120591120591 0)1198631198631 + 119866119866119953119953119953119953prime
(1) (120591120591) (513)
Using this method the third term in the numerator of Eq (59) can be calculated as
1198681198683 = 12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 = 12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0 lang119879119879120591120591119886119886119953119953prime
dagger (0)119886119886119953119953(120591120591)rang0 +
12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 + 12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 +
12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 = 119866119866119953119953119953119953prime0 (120591120591 0) 1
2ℏ2 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0 +
12ℏ2 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 + 12ℏ2 int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 +
12ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
Here the last term is denoted as 119866119866119953119953119953119953prime(2) (120591120591)
Using Eq (513) the second and third terms on the right hand above equation can be simplified as
follows
18
12ℏ2
1198891198891205911205912120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 +1
2ℏ2 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
=12 minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0
minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
+12 minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0
minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
=12 minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0
minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
+12 minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0
minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888
= minus1ℏ 1198891198891205911205912
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0
minus1ℏ 1198891198891205911205911
120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 = 1198631198631119866119866119953119953119953119953prime(1) (120591120591)
Where we performed the following integration variables interchange 1205911205911 ⟷ 1205911205912 Thus we have
1198681198683 = 119866119866119953119953119953119953prime(2) (120591120591) + 119866119866119953119953119953119953prime
(1) (120591120591)1198631198631 + 119866119866119953119953119953119953prime0 (120591120591 0)1198631198632 (514)
Similarity we can calculate the fourth term of Eq (59) as
1198681198684 = minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 =
minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0 lang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 + 3 times
minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 + 3 times
minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 +
minus16ℏ3 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 =
119866119866119953119953119953119953prime0 (120591120591 0) minus1
6ℏ3 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0+
12ℏ2 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0
minus1ℏ int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888+
12ℏ2 int 1198891198891205911205912
120573120573ℏ0 int 1198891198891205911205913
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0119888119888 minus1ℏ int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0+ 119866119866119953119953119953119953prime
(3) (120591120591) = 119866119866119953119953119953119953prime(3) (120591120591) +
119866119866119953119953119953119953prime(2) (120591120591)1198631198631 + 119866119866119953119953119953119953prime
(1) (120591120591)1198631198632 + 119866119866119953119953119953119953prime0 (120591120591 0)1198631198633
Higher order terms can be treated in the same manner (see pages 95-96 of Ref [2]) Then when
substituting 1198681198682 1198681198683 1198681198684 and all higher order terms into Eq (59) we can simplify the numerator as
follows
119866119866119953119953119953119953prime0 (120591120591) minus
1ℏ 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0
+1
2ℏ2 1198891198891205911205911120573120573ℏ
0 1198891198891205911205912120573120573ℏ
0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime
dagger (0)rang0 + ⋯
= 119866119866119953119953119953119953prime0 (120591120591) + 119866119866119953119953119953119953prime
(1) (120591120591) + 119866119866119953119953119953119953prime(2) (120591120591) + ⋯ (1 + 1198631198631 + 1198631198632 + ⋯ )
Noting that the expression (1 + 1198631198631 + 1198631198632 + ⋯ ) is exactly canceled with the denominator the
Greenrsquos function can be simplified as
19
119866119866119953119953119953119953prime(120591120591) = 119866119866119953119953119953119953prime0 (120591120591) + 119866119866119953119953119953119953prime
(1) (120591120591) + 119866119866119953119953119953119953prime(2) (120591120591) + ⋯ (515)
where
119866119866119953119953119953119953prime
(1) (120591120591) = minus1ℏ int 1198891198891205911205911
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888
119866119866119953119953119953119953prime(2) (120591120591) = 1
2ℏ2 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888 (516)
53 Calculation of Greenrsquos function
531 First order appriximation In Eq (516) we go back to the full notation 119886119886119954119954120582120582 to distinguish phonons of different branches
then 119866119866119953119953119953119953prime(1) (120591120591) is calculated as
119866119866119953119953119953119953prime(1) (120591120591) = minus
1ℏ
12 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591 119881119881119895119895119895119895prime(119948119948)119876119876119948119948119895119895(1205911205911)119876119876119948119948119895119895prime
dagger (1205911205911)3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ 119881119881119895119895119895119895primelowast (119948119948)119876119876119948119948119895119895prime(1205911205911)119876119876119948119948119895119895
dagger (1205911205911)3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= minus12ℏ
1198891198891205911205911120573120573ℏ
0lang119879119879120591120591
ℏ119881119881119895119895119895119895prime(119948119948)2120596120596119948119948119895119895120596120596119948119948119895119895prime
119886119886119948119948119895119895(1205911205911) + 119886119886minus119948119948119895119895dagger (1205911205911) 119886119886minus119948119948119895119895prime(1205911205911) + 119886119886119948119948119895119895prime
dagger (1205911205911)3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ ℏ119881119881119895119895119895119895prime
lowast (119948119948)
2120596120596119948119948119895119895120596120596119948119948119895119895prime119886119886119948119948119895119895prime(1205911205911) + 119886119886minus119948119948119895119895prime
dagger (1205911205911) 119886119886minus119948119948119895119895(1205911205911) + 119886119886119948119948119895119895dagger (1205911205911)
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
According to Wickrsquos theorem [2] the terms that contain 119886119886119948119948119895119895119886119886minus119948119948119895119895prime and 119886119886minus119948119948119895119895dagger 119886119886119948119948119895119895prime
dagger are equal to zero
thus the first term of above equation are calculated as
11989211989211 = minus14
119881119881119895119895119895119895prime(119948119948)
120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119886119886minus119948119948119895119895
dagger (1205911205911)119886119886minus119948119948119895119895prime(1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ lang119879119879120591120591119886119886119948119948119895119895(1205911205911)119886119886119948119948119895119895primedagger (1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888
= minus14
119881119881119895119895119895119895prime(119948119948)
120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119886119886minus119948119948119895119895
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus119948119948119895119895prime(1205911205911)rang0
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ lang119879119879120591120591119886119886119948119948119895119895primedagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)119886119886119948119948119895119895(1205911205911)rang0
= minus14
119881119881119895119895119895119895prime(119948119948)
120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
01198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954minus119948119948120575120575120582120582119895119895119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119954119954primeminus119948119948120575120575120582120582prime119895119895prime3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ 1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119948119948120575120575120582120582119895119895prime119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119954119954prime119948119948120575120575120582120582prime119895119895
20
Simplification of above equation gives
11989211989211 =
⎩⎪⎨
⎪⎧minus119881119881120582120582120582120582prime(minus119954119954)
4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954120582120582
0 (120591120591 1205911205911)119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903
2lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582(119954119954)
4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954120582120582
0 (120591120591 1205911205911)119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582prime le 3119903119903
2lt 120582120582 le 3119903119903
0 else
(517)
Similarity the second term of 119866119866119953119953119953119953prime(1) (120591120591) is calculated as
11989211989212 = minus14
minus119881119881119895119895119895119895primelowast (119948119948)
4120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119886119886119948119948119895119895prime(1205911205911)119886119886119948119948119895119895
dagger (1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ lang119879119879120591120591119886119886minus119948119948119895119895primedagger (1205911205911)119886119886minus119948119948119895119895(1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888
= minus14
minus119881119881119895119895119895119895primelowast (119948119948)
4120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ
0lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)119886119886119948119948119895119895prime(1205911205911)rang0 lang119879119879120591120591119886119886119948119948119895119895dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
3119903119903
119895119895prime=31199031199032 +1
31199031199032
119895119895=1119948119948
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus119948119948119895119895(1205911205911)rang0 lang119879119879120591120591119886119886minus119948119948119895119895prime
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
ie
11989211989212 =
⎩⎪⎨
⎪⎧
minus119881119881120582120582120582120582primelowast (119954119954)
4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954prime120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903
2lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582lowast (minus119954119954)
4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954prime120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119954119954prime 1 le 120582120582prime le 3119903119903
2lt 120582120582 le 3119903119903
0 else
(518)
Note that since 119881119881120582120582120582120582primelowast (119954119954) = 119881119881120582120582120582120582prime(minus119954119954) [Eq (314)] the first order approximation of Greenrsquos function
can be written as
119866119866119954119954120582120582119954119954prime120582120582prime(1) (120591120591) =
⎩⎪⎨
⎪⎧minus119881119881
120582120582120582120582prime(minus119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ
0 1198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582 le3119903119903
2lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582
(119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ
0 1198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582prime le3119903119903
2lt 120582120582 le 3119903119903
0 else
(519)
To derive the Greenrsquos function in frequency domain we use the expression for Fourier series then
we have
1198891198891205911205911120573120573ℏ
01198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0) = 1198891198891205911205911120573120573ℏ
0
1120573120573ℏ
119890119890minus119894119894120596120596119899119899(120591120591minus1205911205911)1198661198661199541199541205821205820 (119894119894120596120596119899119899)
119899119899
1120573120573ℏ
119890119890minus119894119894120596120596119899119899prime1205911205911119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899prime)
119899119899prime
=1
(120573120573ℏ)2119890119890minus119894119894120596120596119899119899120591120591 1198891198891205911205911120573120573ℏ
0119890119890119894119894120596120596119899119899minus120596120596119899119899prime1205911205911 119866119866119954119954120582120582
0 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899prime)
119899119899119899119899prime=
=1
(120573120573ℏ)2119890119890minus119894119894120596120596119899119899120591120591[120573120573ℏ120575120575119899119899119899119899prime]1198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899prime)119899119899119899119899prime
=1120573120573ℏ
119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899)119899119899
According to the definition of Fourier series [see Eq (412)] we get the Fourier coefficient of
21
119866119866119954119954120582120582119954119954prime120582120582prime(1) (120591120591)
119866119866119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) =
⎩⎪⎨
⎪⎧minus119881119881120582120582120582120582prime(minus119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582prime1198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime 1 le 120582120582 le 31199031199032
lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582(119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582prime1198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime 1 le 120582120582prime le 31199031199032
lt 120582120582 le 3119903119903
0 else
(520)
where 119866119866119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) = 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) times Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) times 119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899) and Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) is defined as
Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) =
⎩⎪⎨
⎪⎧minus119881119881120582120582120582120582prime(minus119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582prime120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903
2lt 120582120582prime le 3119903119903
minus119881119881120582120582prime120582120582(119954119954)
2120596120596119954119954120582120582120596120596119954119954120582120582prime120575120575119954119954119954119954prime 1 le 120582120582prime le 3119903119903
2lt 120582120582 le 3119903119903
0 else
(521)
532 second order approximation The second order approximation of the Greenrsquos function in Eq (516) 119866119866119954119954120582120582119954119954prime120582120582prime
(2) (120591120591) is calculated as
119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) =
18ℏ2 1198891198891205911205911
120573120573ℏ
0 1198891198891205911205912120573120573ℏ
0lang119879119879120591120591
⎣⎢⎢⎡ 11988111988111989511989511198951198951prime(119948119948120783120783)1198761198761199481199481207831207831198951198951(1205911205911)1198761198761199481199481207831207831198951198951prime
dagger (1205911205911)3119903119903
1198951198951prime=31199031199032 +1
31199031199032
1198951198951=1119948119948120783120783
+ 11988111988111989411989411198941198941primelowast (119948119948120783120783)1198761198761199481199481207831207831198941198941prime(1205911205911)1198761198761199481199481207831207831198941198941
dagger (1205911205911)3119903119903
1198941198941prime=31199031199032 +1
31199031199032
1198941198941=1119948119948120783120783 ⎦⎥⎥⎤
⎣⎢⎢⎡ 11988111988111989511989521198951198952prime(119948119948120784120784)1198761198761199481199481207841207841198951198952(1205911205912)1198761198761199481199481207841207841198951198952prime
dagger (1205911205912)3119903119903
1198951198952prime=31199031199032 +1
31199031199032
1198951198952=1119948119948120784120784
+ 11988111988111989411989421198941198942primelowast (119948119948120784120784)1198761198761199481199481207841207841198941198942prime(1205911205912)1198761198761199481199481207841207841198941198942
dagger (1205911205912)3119903119903
1198941198942prime=31199031199032 +1
31199031199032
1198941198942=1119948119948120784120784 ⎦⎥⎥⎤119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888 = 1198921198921 + 1198921198922 + 1198921198923 + 1198921198924
where
1198921198921 = 18ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0
sum sum 11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989511989521198951198952prime(119948119948120784120784)1le1198951198952le
31199031199032 lt1198951198952
primele311990311990311994811994812078412078411989511989521198951198952prime
1le1198951198951le31199031199032 lt1198951198951
primele311990311990311994811994812078312078311989511989511198951198951prime
times lang1198791198791205911205911198761198761199481199481207831207831198951198951(1205911205911)1198761198761199481199481207831207831198951198951primedagger (1205911205911)1198761198761199481199481207841207841198951198952(1205911205912)1198761198761199481199481207841207841198951198952prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(522)
1198921198922 = 18ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0
sum sum 11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942primelowast (119948119948120784120784)
1le1198941198942le31199031199032 lt1198941198942
primele311990311990311994811994812078412078411989411989421198941198942prime
1le1198951198951le31199031199032 lt1198951198951
primele311990311990311994811994812078312078311989511989511198951198951prime
times lang1198791198791205911205911198761198761199481199481207831207831198951198951(1205911205911)1198761198761199481199481207831207831198951198951primedagger (1205911205911)1198761198761199481199481207841207841198941198942prime (1205911205912)1198761198761199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(523)
1198921198923 = 18ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0
sum sum 11988111988111989411989411198941198941primelowast (119948119948120783120783)11988111988111989511989521198951198952prime(119948119948120784120784)
1le1198951198952le31199031199032 lt1198951198952
primele311990311990311994811994812078412078411989511989521198951198952prime
1le1198941198941le31199031199032 lt1198941198941
primele311990311990311994811994812078312078311989411989411198941198941prime
times lang1198791198791205911205911198761198761199481199481207831207831198941198941prime (1205911205911)1198761198761199481199481207831207831198941198941dagger (1205911205911)1198761198761199481199481207841207841198951198952(1205911205912)1198761198761199481199481207841207841198951198952prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(524)
22
1198921198924 = 18ℏ2 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0
sum sum 11988111988111989411989411198941198941primelowast (119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)1le1198941198942le
31199031199032 lt1198941198942
primele311990311990311994811994812078412078411989411989421198941198942prime
1le1198941198941le31199031199032 lt1198941198941
primele311990311990311994811994812078312078311989411989411198941198941prime
times lang1198791198791205911205911198761198761199481199481207831207831198941198941prime (1205911205911)1198761198761199481199481207831207831198941198941dagger (1205911205911)1198761198761199481199481207841207841198941198942prime (1205911205912)1198761198761199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(525)
In what follows we will expand the term 1198921198922 The expansion of all other terms follows the same lines
and eventually results in the same expression Expanding the product 11987611987611994811994812078312078311989511989511198761198761199481199481207831207831198951198951primedagger 1198761198761199481199481207841207841198941198942prime1198761198761199481199481207841207841198941198942
dagger we can
rewrite Eq (523) as follows
1198921198922 = 132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)119881119881
11989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032 lt1198941198942
primele311990311990311994811994812078412078411989411989421198941198942prime
1le1198951198951le31199031199032 lt1198951198951
primele311990311990311994811994812078312078311989511989511198951198951prime
times 119868119868 (526)
Where
119868119868 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911) + 119886119886minus1199481199481207831207831198951198951dagger (1205911205911) 119886119886minus1199481199481207831207831198951198951prime(1205911205911) + 1198861198861199481199481207831207831198951198951prime
dagger (1205911205911) 1198861198861199481199481207841207841198941198942prime (1205911205912) + 119886119886minus1199481199481207841207841198941198942primedagger (1205911205912) 119886119886minus1199481199481207841207841198941198942(1205911205912) + 1198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0c
= lang119879119879120591120591 1198861198861199481199481207831207831198951198951119886119886minus1199481199481207831207831198951198951prime + 11988611988611994811994812078312078311989511989511198861198861199481199481207831207831198951198951primedagger + 119886119886minus1199481199481207831207831198951198951
dagger 119886119886minus1199481199481207831207831198951198951prime + 119886119886minus1199481199481207831207831198951198951dagger 1198861198861199481199481207831207831198951198951prime
dagger 12059112059111198861198861199481199481207841207841198941198942prime119886119886minus1199481199481207841207841198941198942 + 1198861198861199481199481207841207841198941198942prime1198861198861199481199481207841207841198941198942
dagger + 119886119886minus1199481199481207841207841198941198942primedagger 119886119886minus1199481199481207841207841198941198942
+ 119886119886minus1199481199481207841207841198941198942primedagger 1198861198861199481199481207841207841198941198942
dagger 1205911205912119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0c
= lang119879119879120591120591 11988611988611994811994812078312078311989511989511198861198861199481199481207831207831198951198951primedagger + 119886119886minus1199481199481207831207831198951198951
dagger 119886119886minus1199481199481207831207831198951198951prime12059112059111198861198861199481199481207841207841198941198942prime1198861198861199481199481207841207841198941198942
dagger + 119886119886minus1199481199481207841207841198941198942primedagger 119886119886minus11994811994812078412078411989411989421205911205912
119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
Where the symbol [hellip ]1205911205911 signifies that the operators in the brackets are given in the interaction
picture The last equality in above equation comes from the fact that the contractions of the product
of the operators are equal to zero when the number of creation and annihilation operators are not the
same (see Wickrsquos theorem in Ref [2]) Using Wickrsquos theorem [2] we are able to calculate the above
equation term by term as follows
⎩⎪⎨
⎪⎧ 1198681198681 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951prime
dagger (1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)1198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime
dagger (0)rang0119888119888
1198681198682 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
1198681198683 = lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)1198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
1198681198684 = lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
(527)
We shall now calculate them term by term
23
1198681198681 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951primedagger (1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)1198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942
dagger (1205911205912)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
+ lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951prime
dagger (1205911205911)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0
+ lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)1198861198861199481199481207841207841198941198942prime(1205911205912)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0119888119888
= 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199481199481207841207841198951198952
0 (1205911205912 1205911205912)12057512057511989411989421198941198942prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582
+ 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 1198661198661199481199481207831207831198951198951
0 (1205911205911 1205911205911)12057512057511989511989511198951198951prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582
+ 1198661198661199481199481207831207831198951198951prime0 (1205911205912 1205911205911)1205751205751199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951prime ∙ 119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582
= 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582
+ 1198661198661199481199481207831207831198951198951prime0 (1205911205912 1205911205911)1205751205751199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951prime ∙ 119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582
The last equality results from the fact that 1198951198951 and 1198951198951prime are indices of different ranges (belonging to
different subsystems) and the same holds for 1198941198942 and 1198941198942prime Similarity
1198681198684 = lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0
= 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 119866119866minus11994811994812078412078411989411989420 (1205911205912 1205911205912)12057512057511989411989421198941198942prime
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 119866119866minus1199481199481207831207831198951198951prime 0 (1205911205911 1205911205912)1205751205751199481199481207831207831199481199481207841207841205751205751198951198951prime1198941198942prime
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205911 1205911205911)12057512057511989511989511198951198951prime
+ 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205912 1205911205911)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942
= 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 119866119866minus1199481199481207831207831198951198951prime 0 (1205911205911 1205911205912)1205751205751199481199481207831207831199481199481207841207841205751205751198951198951prime1198941198942prime
+ 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205912 1205911205911)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942
Where again the last equality results from the fact that 1198951198951 and 1198951198951prime are indices of different ranges
(belonging to different subsystems) and the same holds for 1198941198942 and 1198941198942prime Similarly we obtain 1198681198682 =
1198681198683 = 0 for the same reason as shown below
24
1198681198682 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951prime
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0
+ lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942prime
dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)1198861198861199481199481207831207831198951198951(1205911205911)rang0
+ lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0
= 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582 ∙ 119866119866minus11994811994812078412078411989411989420 (1205911205912 1205911205912)12057512057511989411989421198941198942prime
+ 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)120575120575minus11994811994812078412078411994811994812078312078312057512057511989511989511198941198942prime ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205911)12057512057511989511989511198951198951prime
+ 1198661198661199481199481207831207831198951198951prime0 (1205911205912 1205911205911)120575120575minus11994811994812078412078411994811994812078312078312057512057511989411989421198951198951prime ∙ 119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 = 0
1198681198683 = lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)1198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888
= lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951
dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0
+ lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0
+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942
dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0
+ lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime
dagger (0)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0
= 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 11986611986611994811994812078412078411989411989420 (1205911205912 1205911205912)12057512057511989411989421198941198942prime
+ 11986611986611994811994812078412078411989411989420 (1205911205911 1205911205912)120575120575minus1199481199481207831207831199481199481207841207841205751205751198951198951prime1198941198942 ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582
+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205911 1205911205911)12057512057511989511989511198951198951prime
+ 1198661198661199481199481207841207841198941198942prime0 (1205911205912 1205911205911)120575120575minus1199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951 ∙ 119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 = 0
The two non-vanishing terms (1198681198681 and 1198681198684) produce the following contributions to 1198921198922 of Eq (526)
11989211989221 = 132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime1198681198681 =
132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime1198661198661199481199481207831207831198951198951
0 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙
1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582 + 1
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime1198661198661199481199481207831207831198951198951prime
0 (1205911205912 1205911205911)1205751205751199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951prime ∙
119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 119866119866119954119954120582120582
0 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 =
⎩⎪⎨
⎪⎧
120575120575119954119954119954119954prime
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast (119954119954prime)
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (1205911205911 1205911205912) ∙ 119866119866119954119954prime120582120582prime
0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
120575120575119954119954119954119954prime
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032+1 1198661198661199541199541198951198951prime
0 (1205911205912 1205911205911) ∙ 119866119866119954119954prime120582120582prime0 (1205911205911 0)119866119866119954119954120582120582
0 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032
0 else
and
25
11989211989224 = 132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime1198681198684 =
132 int 1198891198891205911205911
120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime119866119866119954119954prime120582120582prime
0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙
119866119866minus1199481199481207831207831198951198951prime 0 (1205911205911 1205911205912)1205751205751199481199481207831207831199481199481207841207841205751205751198951198951prime1198941198942prime + 1
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum sum
11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime
lowast (119948119948120784120784)
12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime
1le1198941198942le31199031199032lt1198941198942primele3119903119903
11994811994812078412078411989411989421198941198942prime1le1198951198951le
31199031199032lt1198951198951primele3119903119903
11994811994812078312078311989511989511198951198951prime119866119866119954119954prime120582120582prime
0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙
1198661198661199541199541205821205820 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 119866119866minus1199481199481207831207831198951198951
0 (1205911205912 1205911205911)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 =
⎩⎪⎨
⎪⎧120575120575119954119954119954119954prime
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032+1 1198661198661199541199541198951198951prime
0 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime0 (1205911205912 0)119866119866119954119954120582120582
0 (120591120591 1205911205911) 120582120582 120582120582prime isin 1 31199031199032
120575120575119954119954119954119954prime
32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast (119954119954prime)
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
0 else
Here we used the following properties 120596120596minus119954119954120582120582 = 120596120596119954119954120582120582 and 1198811198811205821205821198951198951prime(minus119954119954) = 1198811198811205821205821198951198951primelowast (119954119954) [see Eqs (112) and
(314)]
The equivalence of 11989211989221 and 11989211989224 can be seen by changing the integration variables 1205911205912 harr 1205911205911
Substituting 11989211989221 and 11989211989224 into Eq (526) 1198921198922 is given by
1198921198922 =
⎩⎪⎨
⎪⎧
120575120575119954119954119954119954prime
16 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast (119954119954prime)
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime
0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
120575120575119954119954119954119954prime
16 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032+1 1198661198661199541199541198951198951prime
0 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime0 (1205911205911 0)119866119866119954119954120582120582
0 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032
0 else
(528)
By changing the indices and integration variables itrsquos straightforward to show that the other three
terms (119892119892111989211989231198921198924 ) are identical to 1198921198922 thus the final expression of 119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) [Eq (516)) can be
calculated as
119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) = 41198921198922 =
⎩⎪⎨
⎪⎧120575120575119954119954119954119954prime
4 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032
120575120575119954119954119954119954prime
4 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912
120573120573ℏ0 sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast 119954119954prime
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime
0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 31199031199032
+ 13119903119903
0 else
(529)
To calculate 119866119866119954119954120582120582119954119954prime120582120582prime(2) in frequency space we expand 119866119866119954119954120582120582119954119954prime120582120582prime
(2) (120591120591) in a Fourier series
119866119866119954119954120582120582119954119954prime120582120582prime
(2) (120591120591) = 1120573120573ℏsum 119890119890minus119894119894120596120596119899119899120591120591119866119866119954119954120582120582119954119954prime120582120582prime
(2) (119894119894120596120596119899119899)119899119899
119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = int d120591120591119890119890119894119894120596120596119899119899120591120591119866119866119954119954120582120582119954119954prime120582120582prime
(2) (120591120591)120573120573ℏ0
(530)
Using Eq (421) we get for 120582120582 120582120582prime isin 1 31199031199032
26
119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) =
120575120575119954119954119954119954prime4
1198891198891205911205911120573120573ℏ
0 1198891198891205911205912120573120573ℏ
0
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1198661198661199541199541198951198951prime0 (1205911205912 1205911205911)119866119866119954119954120582120582prime
0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912)
=120575120575119954119954119954119954prime
4
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1198891198891205911205911120573120573ℏ
0 1198891198891205911205912120573120573ℏ
0
1120573120573ℏ
119890119890minus1198941198941205961205961198991198992(1205911205912minus1205911205911)1198661198661199541199541198951198951prime0 1198941198941205961205961198991198992
1198991198992
∙1120573120573ℏ
119890119890minus11989411989412059612059611989911989911205911205911119866119866119954119954120582120582prime0 1198941198941205961205961198991198991
1198991198991
∙1120573120573ℏ
119890119890minus119894119894120596120596119899119899(120591120591minus1205911205912)1198661198661199541199541205821205820 (119894119894120596120596119899119899)
119899119899
=120575120575119954119954119954119954prime
4
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1120573120573ℏ
119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954120582120582prime
0 11989411989412059612059611989911989911198661198661199541199541198951198951prime0 1198941198941205961205961198991198992
11989911989911989911989911198991198992⎩⎪⎨
⎪⎧ 1120573120573ℏ
1198891198891205911205911120573120573ℏ
0119890119890minus1198941198941205961205961198991198991minus12059612059611989911989921205911205911
times1120573120573ℏ
1198891198891205911205912120573120573ℏ
0119890119890minus1198941198941205961205961198991198992minus1205961205961198991198991205911205912
⎭⎪⎬
⎪⎫
=120575120575119954119954119954119954prime
4
119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1120573120573ℏ
119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)
11989911989911989911989911198991198992
119866119866119954119954120582120582prime0 11989411989412059612059611989911989911198661198661199541199541198951198951prime
0 1198941198941205961205961198991198992120575120575119899119899111989911989921205751205751198991198991198991198992
=120575120575119954119954119954119954prime120573120573ℏ
119890119890minus119894119894120596120596119899119899120591120591
119899119899
1198661198661199541199541205821205820 (119894119894120596120596119899119899)
14
119881119881120582120582prime1198951198951prime(119954119954
prime)1198811198811205821205821198951198951primelowast (119954119954)
1205961205961199541199541198951198951prime120596120596119954119954120582120582120596120596119954119954prime120582120582prime
3119903119903
1198951198951prime=31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899)119866119866119954119954120582120582prime
0 (119894119894120596120596119899119899)
Comparing above expression with Eq (530) the Fourier transform of 119866119866119954119954120582120582119954119954120582120582prime(2) (120591120591) reads as
119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) 120575120575119954119954119954119954prime
4sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 119866119866119954119954120582120582prime
0 (119894119894120596120596119899119899) (531)
where we have introduced the notation
Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) equiv
120575120575119954119954119954119954prime
4sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) (532)
and 119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954120582120582prime(2) (119894119894120596120596119899119899) ∙ 119866119866119954119954120582120582prime
0 (119894119894120596120596119899119899) Similarly for 120582120582 120582120582prime isin 31199031199032
+ 13119903119903 we
have
Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) =
120575120575119954119954119954119954prime
4sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast 119954119954prime
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (119894119894120596120596119899119899) (533)
and all together we have for Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899)
Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) =
⎩⎪⎪⎨
⎪⎪⎧120575120575119954119954119954119954prime
4sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 1 3119903119903
2
120575120575119954119954119954119954prime
4sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast 119954119954prime
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
0 else
(534)
27
533 Dysonrsquos equation Substituting Eqs (521) and (534) into Eq (516) and performing a Fourier transform we have
119866119866119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) = 119866119866119954119954120582120582119954119954prime120582120582prime0 (119894119894120596120596119899119899) + 119866119866119954119954120582120582119954119954prime120582120582prime
(1) (119894119894120596120596119899119899) + 119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) + ⋯
= 1198661198661199541199541205821205820 (119894119894120596120596119899119899)120575120575119954119954119954119954prime120575120575120582120582120582120582prime + 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) ∙ 119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899) + 1198661198661199541199541205821205820 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime
(2) (119894119894120596120596119899119899)
∙ 119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899) + ⋯ = 119866119866119954119954120582120582
0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime120575120575120582120582120582120582prime + 1198661198661199541199541205821205820 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) ∙ 119866119866119954119954prime120582120582prime
0 (119894119894120596120596119899119899)
or
119866119866119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) = 1198661198661199541199541205821205820 (119894119894120596120596119899119899)120575120575119954119954119954119954prime120575120575120582120582120582120582prime + 119866119866119954119954120582120582
0 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) ∙ 119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899) (535)
Eq (535) is the so-called Dysonrsquos equation (see Ref [2]) where
Σ119954119954120582120582119954119954120582120582prime(119894119894120596120596119899119899) = Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) + Σ119954119954120582120582119954119954prime120582120582prime
(2) (119894119894120596120596119899119899) + ⋯
is called self-energy and Σ119954119954120582120582119954119954120582120582prime(119899119899) (119894119894120596120596119899119899)119899119899 = 12⋯ is the self-energy of nth order approximation Up
to the second order approximation the self-energy is written as
Σ119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) =
⎩⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎧ minus
120575120575119954119954119954119954prime
2
119881119881120582120582120582120582primelowast (119954119954)
120596120596119954119954120582120582120596120596119954119954120582120582prime 1 le 120582120582 le 3119903119903
2lt 120582120582prime le 3119903119903
minus120575120575119954119954119954119954prime
2
119881119881120582120582prime120582120582(119954119954)
120596120596119954119954120582120582120596120596119954119954120582120582prime 1 le 120582120582prime le 3119903119903
2lt 120582120582 le 3119903119903
120575120575119954119954119954119954prime
4sum
119881119881120582120582prime1198951198951prime119954119954prime119881119881
1205821205821198951198951prime
lowast (119954119954)
1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime
31199031199031198951198951prime=
31199031199032 +1
1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 1 3119903119903
2
120575120575119954119954119954119954prime
4sum
1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast 119954119954prime
1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime
311990311990321198951198951=1
11986611986611995411995411989511989510 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 3119903119903
2+ 13119903119903
(536)
54 Fermirsquos golden Rule
To get the Fermirsquos golden Rule we need to calculate the retarded Greenrsquos function which can be
calculated from 119866119866119954119954120582120582119954119954120582120582prime(119894119894120596120596119899119899) by analytic continuation to the real axis via 119894119894120596120596119899119899 rarr 120596120596 + 119894119894119894119894 with an
infinitesimal positive 119894119894 [12]
119866119866119954119954120582120582119954119954120582120582prime119903119903 (120596120596) equiv 119866119866119954119954120582120582119954119954120582120582prime(119894119894120596120596119899119899 rarr 120596120596 + 119894119894119894119894) 119894119894 rarr 0 (537)
Similarity the retarded self-energy is defined as
Σ119954119954120582120582119954119954120582120582prime119903119903 (120596120596) equiv Σ119954119954120582120582119954119954120582120582prime(119894119894120596120596119899119899 rarr 120596120596 + 119894119894119894119894) 119894119894 rarr 0 (538)
The transition rate of phonons of branch 120582120582 at 119954119954 Γ119954119954120582120582(120596120596) is related to the retarded self energy
Σ119954119954120582120582119903119903 (120596120596) via [13]
Γ119954119954120582120582(120596120596) = minus2ImΣ119954119954120582120582119954119954120582120582119903119903 (120596120596) (539)
Using the expression for 1198661198661199541199541205821205820 (119894119894120596120596119899119899) [Eq (421)] and focused on the second order approximation [13]
we have
28
Σ119954119954120582120582119954119954120582120582119903119903 (119894119894120596120596119899119899) =
⎩⎪⎨
⎪⎧14sum
1198811198811205821205821198951198951prime(119954119954)
2
1205961205961199541199541198951198951prime120596120596119954119954120582120582
31199031199031198951198951prime=
31199031199032 +1
1
120596120596minus1205961205961199541199541198951198951prime +119894119894119894119894
120582120582 isin 1 31199031199032
14sum 1198811198811198951198951120582120582(119954119954)2
1205961205961199541199541198951198951120596120596119954119954120582120582
311990311990321198951198951=1
1120596120596minus1205961205961199541199541198951198951+119894119894119894119894
120582120582 isin 31199031199032
+ 13119903119903
(540)
Using the relation
Im 1
120596120596minus1205961205961199541199541198951198951prime +119894119894119894119894
= minus120587120587120575120575120596120596 minus 1205961205961199541199541198951198951prime (541)
The transition rate reads as
Γ119954119954120582120582(120596120596) =
⎩⎪⎨
⎪⎧1205871205872sum
1198811198811205821205821198951198951prime(119954119954)
2
1205961205961199541199541198951198951prime120596120596119954119954120582120582
31199031199031198951198951prime=
31199031199032 +1
120575120575120596120596 minus 1205961205961199541199541198951198951prime 120582120582 isin 1 31199031199032
1205871205872sum 1198811198811198951198951120582120582(119954119954)2
1205961205961199541199541198951198951120596120596119954119954120582120582
311990311990321198951198951=1
120575120575120596120596 minus 1205961205961199541199541198951198951 120582120582 isin 31199031199032
+ 13119903119903
(542)
Or equivalently
Γ119954119954120582120582(119864119864 = ℏ120596120596) =
⎩⎪⎨
⎪⎧120587120587ℏ3
2sum
1198811198811205821205821198951198951prime(119954119954)
2
1198641198641199541199541198951198951prime119864119864119954119954120582120582
31199031199031198951198951prime=
31199031199032 +1
120575120575119864119864 minus 1198641198641199541199541198951198951prime 120582120582 isin 1 31199031199032
120587120587ℏ3
2sum 1198811198811198951198951120582120582(119954119954)2
1198641198641199541199541198951198951119864119864119954119954120582120582
311990311990321198951198951=1
120575120575119864119864 minus 1198641198641199541199541198951198951 120582120582 isin 31199031199032
+ 13119903119903
(543)
Γ119954119954120582120582(119864119864) represents the probability per unit time of a transition with energy 119864119864 from phonon branch 120582120582
at wave number 119954119954 in subsystem I (II) to a set of phonon branches in subsystem II (I) with the same
wave number Here state 119954119954120582120582 in one subsystem is coupled to state 1199541199541198951198951prime in the other subsystem via
1198811198811205821205821198951198951prime(119954119954) Since the two expressions in Eq (543) are completely equivalent just representing
transitions of opposite directions we consider only the first one in what follows Assuming that
subsystem II sufficiently large such that its density of states is nearly continuous the sum over its
states in the above expression can be replaced by an integral over the energy sum (⋯ )119895119895 rarr
int(⋯ )120588120588119864119864119954119954d119864119864119954119954 giving
Γ119954119954120582120582(119864119864) = 120587120587ℏ3
2 intd119864119864119954119954prime120588120588119864119864119954119954prime 119881119881120582120582119895119895prime(119954119954)
2119864119864119954119954119895119895prime=119864119864119954119954prime
119864119864119954119954prime119864119864119954119954120582120582120575120575119864119864 minus 119864119864119954119954prime = 120587120587ℏ3
2120588120588(119864119864)
119881119881120582120582119895119895prime(119954119954)2119864119864119954119954119895119895prime=119864119864
119864119864119864119864119954119954120582120582 (544)
Eq (544) represents the transition rate of the phonons with energy 119864119864 from branch 120582120582 and wave
number 119954119954 in subsystem I to the continuous manifold of states in subsystem II The total transition
rate between the two subsystems is then given by the sum of transition rates from all states in
subsystem I weighted by their phonon populations to the manifold of states in subsystem II
Assuming that the two subsystems are weakly coupled such that the width of state 119954119954120582120582 in subsystem
I is small and the probability to leave it at an energy other than 119864119864119954119954120582120582 is negligible we can now write
29
total transition rate as
Γtot = 1119885119885119890119890minus120573120573119864119864119954119954120582120582Γ119954119954120582120582119864119864119954119954120582120582
119954119954120582120582
=120587120587ℏ3
2
119890119890minus120573120573119864119864119954119954120582120582
119885119885
120588120588119864119864119954119954120582120582 119881119881120582120582119895119895prime(119954119954)2119864119864119954119954119895119895prime=119864119864119954119954120582120582
1198641198641199541199541205821205822119954119954120582120582
=120587120587ℏ3
2
119890119890minus120573120573119864119864119954119954120582120582
119885119885
120588120588119864119864119954119954120582120582 119881119881120582120582120582120582+31199031199032(119954119954)
2
1198641198641199541199541205821205822119954119954120582120582
Finally we get
Γtot = Γ119954119954120582120582(119864119864) =120587120587ℏ3
2sum 119890119890minus120573120573119864119864119954119954120582120582
119885119885
120588120588119864119864119954119954120582120582119881119881120582120582120582120582+31199031199032(119954119954)
2
1198641198641199541199541205821205822119954119954120582120582 (545)
where 119885119885 = sum 119890119890minus120573120573119864119864119954119954120582120582119954119954120582120582 is the partition function Here we choose the index of phonon branches 120582120582
such that the phonon branch with lower energy has a smaller index ie 1198641198641199541199541205821205821 lt 1198641198641199541199541205821205822 for index 1205821205821 lt
1205821205822 Note that phonon branches 120582120582 and 1198951198951prime belong to the subsystem I (with index from 1 to 31199031199032) and
subsystem II (with index from 1 to 1 + 31199031199032 to 3119903119903) respectively we choose 119895119895prime = 120582120582 + 31199031199032
to make
sure that 119864119864119954119954119895119895prime = 119864119864119954119954120582120582 since phonon energy of two uncoupled system is identical Eq (545) is the
Fermirsquos golden rule for inter-phonon coupling
References
[1] Y Xu J-S Wang W Duan B-L Gu and B Li Nonequilibrium Greens function method for phonon-phonon interactions and ballistic-diffusive thermal transport Phys Rev B 78 224303 (2008) [2] A L Fetter and J D Walecka Quantum theory of many-particle systems (Courier Corporation 2012) [3] G E Stedman Fermis Golden RulemdashAn Exercise in Quantum Field Theory Am J Phys 39 205 (1971)