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Lecture notes on many-body theory
Michele Fabrizio
updated to 2013, but still work in progress!!All comments are most welcome!
Contents
1 Landau-Fermi-liquid theory: Phenomenology 51.1 The Landau energy functional and the concept of quasiparticles . . . . . . . . . . 51.2 Quasiparticle thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.1 Specific heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.2.2 Compressibility and magnetic susceptibility . . . . . . . . . . . . . . . . . 13
1.3 Quasiparticle transport equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.3.1 Quasiparticle current density . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4 Collective excitations in the collisionless regime . . . . . . . . . . . . . . . . . . . 201.4.1 Zero sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.5 Collective excitations in the collision regime: first sound . . . . . . . . . . . . . . 251.6 Linear response functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.6.1 Formal solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.6.2 Low frequency limit of the response functions . . . . . . . . . . . . . . . . 311.6.3 High frequency limit of the response functions . . . . . . . . . . . . . . . . 32
1.7 Charged Fermi-liquids: the Landau-Silin theory . . . . . . . . . . . . . . . . . . . 341.7.1 Formal solution of the transport equation . . . . . . . . . . . . . . . . . . 37
1.8 Dirty quasiparticle gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391.8.1 Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411.8.2 Diffusive behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2 Second Quantization 482.1 Fock states and space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.2 Fermionic operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.2.1 Second quantization of multifermion-operators . . . . . . . . . . . . . . . 522.2.2 Fermi fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.3 Bosonic operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572.3.1 Bose fields and multiparticle operators . . . . . . . . . . . . . . . . . . . . 57
2.4 Canonical transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592.4.1 More general canonical transformations . . . . . . . . . . . . . . . . . . . 60
1
2.5 Examples and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622.6 Application: fermionic lattice models and the emergence of magnetism . . . . . . 65
2.6.1 Hubbard models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692.6.2 The Mott insulator within the Hubbard model . . . . . . . . . . . . . . . 71
2.7 Spin wave theory in the Heisenberg model . . . . . . . . . . . . . . . . . . . . . . 732.7.1 More rigorous derivation: the Holstein-Primakoff transformation . . . . . 81
3 Linear Response Theory 833.1 Linear Response Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.2 Kramers-Kronig relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.2.1 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873.3 Fluctuation-Dissipation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 883.4 Spectral Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 903.5 Power dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.5.1 Absorption/Emission Processes . . . . . . . . . . . . . . . . . . . . . . . . 933.5.2 Thermodynamic Susceptibilities . . . . . . . . . . . . . . . . . . . . . . . 94
4 Hartree-Fock Approximation 984.1 Hartree-Fock Approximation for Fermions at Zero Temperature . . . . . . . . . . 98
4.1.1 Alternative approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024.2 Hartree-Fock approximation for fermions at finite temperature . . . . . . . . . . 106
4.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1064.2.2 Variational approach at T 6= 0 . . . . . . . . . . . . . . . . . . . . . . . . 110
4.3 Mean-Field approximation for bosons and superfluidity . . . . . . . . . . . . . . . 1144.3.1 Superfluid properties of the gauge symmetry breaking wavefunction . . . 119
4.4 Time-dependent Hartree-Fock approximation for fermions . . . . . . . . . . . . . 1244.4.1 Bosonic representation of the low-energy excitations . . . . . . . . . . . . 126
4.5 Application: antiferromagnetism in the half-filled Hubbard model . . . . . . . . . 1304.5.1 Spin-wave spectrum by time dependent Hartree-Fock . . . . . . . . . . . . 134
4.6 Linear response of an electron gas . . . . . . . . . . . . . . . . . . . . . . . . . . 1394.6.1 Response to an external charge . . . . . . . . . . . . . . . . . . . . . . . . 1424.6.2 Response to a transverse field . . . . . . . . . . . . . . . . . . . . . . . . . 1474.6.3 Limiting values of the response functions . . . . . . . . . . . . . . . . . . 1484.6.4 Power dissipated by the electromagnetic field . . . . . . . . . . . . . . . . 1564.6.5 Reflectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
4.7 Random Phase Approximation for the electron gas . . . . . . . . . . . . . . . . . 160
5 Feynman diagram technique 1635.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
5.1.1 Imaginary-time ordered products . . . . . . . . . . . . . . . . . . . . . . . 163
2
5.1.2 Matsubara frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1645.1.3 Single-particle Green’s functions . . . . . . . . . . . . . . . . . . . . . . . 168
5.2 Perturbation expansion in imaginary time . . . . . . . . . . . . . . . . . . . . . . 1725.2.1 Wick’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
5.3 Perturbation theory for the single-particle Green’s function and Feynman diagrams1745.3.1 Diagram technique in momentum and frequency space . . . . . . . . . . . 1775.3.2 The Dyson equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
5.4 Other kinds of perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1855.4.1 Scalar potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1855.4.2 Coupling to bosonic modes . . . . . . . . . . . . . . . . . . . . . . . . . . 186
5.5 Two-particle Green’s functions and correlation functions . . . . . . . . . . . . . . 1895.5.1 Diagrammatic representation of the two-particle Green’s function . . . . . 1905.5.2 Correlation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
5.6 Coulomb interaction and proper and improper response functions . . . . . . . . . 1945.6.1 Screened interaction and corresponding Dyson equation . . . . . . . . . . 195
5.7 Irreducible vertices and the Bethe-Salpeter equations . . . . . . . . . . . . . . . . 1975.7.1 Bethe-Salpeter equation for the vertex functions . . . . . . . . . . . . . . 198
5.8 The Ward identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2005.9 Consistent approximation schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 204
5.9.1 Example: the Hartree-Fock approximation . . . . . . . . . . . . . . . . . . 2065.10 Some additional properties and useful results . . . . . . . . . . . . . . . . . . . . 208
5.10.1 The occupation number and the Luttinger-Ward functional . . . . . . . . 2085.10.2 The thermodynamic potential . . . . . . . . . . . . . . . . . . . . . . . . . 2115.10.3 The Luttinger theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
6 Landau-Fermi liquid theory: a microscopic justification 2176.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
6.1.1 Vertex and Ward identities . . . . . . . . . . . . . . . . . . . . . . . . . . 2206.2 Correlation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
6.2.1 Conserved quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2256.3 Coulomb interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
7 Kondo effect and the physics of the Anderson impurity model 2287.1 Brief introduction to scattering theory . . . . . . . . . . . . . . . . . . . . . . . . 229
7.1.1 General analysis of the phase-shifts . . . . . . . . . . . . . . . . . . . . . . 2347.2 The Anderson Impurity Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
7.2.1 Variation of the electron number . . . . . . . . . . . . . . . . . . . . . . . 2397.2.2 Energy variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2407.2.3 Mean-field analysis of the interaction . . . . . . . . . . . . . . . . . . . . . 242
7.3 From the Anderson to the Kondo model . . . . . . . . . . . . . . . . . . . . . . . 244
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7.3.1 The emergence of logarithmic singularities and the Kondo temperature . 2457.3.2 Anderson’s scaling theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
8 Introduction to abelian Bosonization 2528.1 Interacting spinless fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
8.1.1 Spin wave theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2558.1.2 Construction of the effective Hamiltonian . . . . . . . . . . . . . . . . . . 256
8.2 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2638.2.1 Umklapp processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
8.3 Bosonization of the Heisenberg model . . . . . . . . . . . . . . . . . . . . . . . . 2688.4 The Hubbard model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
4
Chapter 1
Landau-Fermi-liquid theory:Phenomenology
One of the first thing that one learns in a Solid State Physics course is that the thermodynamicand transport properties of metals are well described in terms of non-interacting electrons; theDrude-Sommerfeld-Boltzmann theory of metals. Yet, this evidence is somehow surprising inview of the fact that actual electrons interact mutually via Coulomb repulsion, which is notweak at all.
This puzzle was solved brilliantly in the end of the 50’s by Landau, as we are going to discussin what follows. We will start by analyzing the case of a neutral Fermi system, like 3He. Laterwe shall discuss charged Fermi systems, relevant to metals.
1.1 The Landau energy functional and the concept of quasipar-ticles
Let us start from a non-interacting Fermi gas at very low temperature T . Right at T = 0 weknow that the ground state is the Fermi sea that is obtained by filling with two opposite-spinelectrons all momentum states with energy smaller than the chemical potential µ (the Fermienergy). In other words, if nkσ is the occupation number of each momentum state – nkσ beingzero or one because of Pauli principle – then the Fermi-sea occupation numbers are
n(0)kσ =
1 if ε
(0)k ≤ µ
0 if ε(0)k > µ
.
At finite temperature, the equilibrium values of the occupation numbers are given by the Fermi-Dirac distribution
n(0)kσ = f
(ε(0)k − µ
)=(
1 + eβ (ε(0)k −µ)
)−1
,
5
with β = 1/KBT the inverse temperature. Any excited state can be uniquely identified by thevariation of the occupation numbers with respect to equilibrium, namely through
δnkσ = nkσ − n(0)kσ , (1.1)
and costs an energy
δE(0) [δnpα] =∑kσ
(ε(0)k − µ
)δnkσ, (1.2)
that is a simple functional of the δnkσ’s.Now suppose that we switch on smoothly the interaction. Each non-interacting excited
state will evolve smoothly into a fully-interacting one. The Landau hypothesis was that thenon-interacting and the fully interacting states are adiabatically connected; in other words theyare in one-to-one correspondence. This implies in particular that each fully-interacting excitedstate can be uniquely identified, like its non-interacting partner, by the deviation with respectto equilibrium of the occupation numbers δnkσ. Consequently, its excitation energy δE must bea functional of the δnkσ’s, i.e.
δE [δnpα] . (1.3)
———Remark
This hypothesis may look simple but it is actually deeply counter-intuitive. Suppose wehave a “non-interacting” Hamiltonian H0 and a fully-interacting one H = H0 +H1, and supposeto follow the spectrum, that we assume discrete, within a specific symmetry-subspace of theHamiltonian
H(λ) = H0 + λH1,
with λ that increases smoothly from 0 to 1. The evolution of the spectrum must resemblethat one drawn schematically in Fig. 1.1 with a series of avoided crossings that allow to followadiabatically the n-th excited level at λ = 0 into the n-th excited level at λ = 1. Note thatthe crossing are avoided because the states have the same symmetry. If we were to considerlevels of different symmetries nothing would prevent crossings. However, while the adiabaticevolution does occur and it is a well accepted phenomenon in models with discrete spectra, itis equally common wisdom that it can not happen when the spectrum is continuous, as in abulk metal. Landau’s revolutionary hypothesis that the adiabatic evolution also occurs in a bulksystem, at least for the low energy excited states, and the fact that, by this simple assumption,one can justify and predict a lot of physical properties is one of the major achievements of thecontemporary Condensed Matter Theory.
———
Let us therefore assume that the excitation energy of the fully-interacting Fermi systemis indeed a functional of the same δnkσ’s as its non-interacting partner, and further suppose
6
E
0 1λ
Figure 1.1: Adiabatic evolution of a discrete spectrum within a specific symmetry-subspace.
that the temperature is very small. In this case, any weak deviation from equilibrium mustcorrespond to δnkσ 1, that justifies a Taylor expansion of (1.3), namely
δE [δnpα] =∑kσ
(εkσ − µ) δnkσ +1
2
∑kk′
∑σσ′
fkσ k′σ′ δnkσ δnk′σ′ +O(δn3). (1.4)
This is the famous Landau’s energy functional. Note that
• even though the two terms are apparently of different orders, in reality they are of thesame order since the excitation energies εkσ − µ are of the same order as the deviations ofthe occupation numbers;
• εkσ should not be confused with the non-interacting energies ε(0)kσ .
Rigorously speaking, the δnkσ’s that appear in (1.4) only serve as labels to identify excitedstates and they do not correspond to deviations of the occupation numbers of the real particles,as in the non-interacting case (1.2). Just for this reason, Landau coined for δnkσ the definition“quasiparticle” occupation number, as if the real-particle excitations were substituted in thepresence of interaction by “quasiparticle” excitations. The Fermi gas of these quasiparticles isthe so-called “Landau-Fermi liquid”.
———Remark
This idea is actually ubiquitous in all Solid State Physics. Let us for instance recall brieflyhow phonons arise. One starts from a model of interacting ions and electrons. Because of theirlarger mass with respect to the electrons and of the Coulomb interaction among them and withthe electrons, the ions localize, thus forming a lattice, and the low energy excitations become thesmall fluctuations around the equilibrium positions, whose quantization gives rise to phonons,which are bosons. In other words, one begins with real particles, the ions, yet their low-energy
7
excitations in the presence of interaction are new bosonic “quasiparticles”, the phonons, thatturns out to be generally weakly interacting among each other and with the electrons. Thisallows to treat such an interaction within perturbation theory, unlike the original Coulombinteraction. In a more general context, since the only many-body systems that can be solvedexactly in any dimensions are free bosons or fermions, it is a very natural, and fortunatelysuccessful, approach to attempt a description of the low-energy excitations of, even strongly,interacting models in terms of effective “quasiparticles”, bosons or fermions, whose interactionis weak enough to be treated perturbatively.
———
Going back to the Landau energy functional (1.4), in the case in which spin isotropy ispreserved the following equivalences hold:
fk↑k′↑ = fk↓k′↓, fk↑k′↓ = fk↓k′↑.
In this case, the choice of a spin quantization axis must not be influential, hence it is convenientto rewrite (1.4) in a form that is explicitly spin isotropic. For that purpose, let us introducecharge deviations from equilibrium
δρk = δnk↑ + δnk↓, (1.5)
and spin onesδσk =
(δσxk, δσ
yk, δσ
zk
), (1.6)
where the z-component is simplyδσzk = δnk↑ − δnk↓.
With these definitions the second term of the energy functional (1.4) can be written as
1
2
∑kk′
fSkk′ δρk δρk′ + fAkk′ δσk · δσk′ , (1.7)
where
fS =1
2(f↑↑ + f↑↓) , fA =
1
2(f↑↑ − f↑↓) , (1.8)
as can be readily demonstrated by equating (1.4) and (1.7) assuming a deviation from equilibriumof the form
δσk = (0, 0, δσzk) .
8
1.2 Quasiparticle thermodynamics
Let us derive, starting from the Landau’s energy functional, the thermodynamical properties ofthe quasiparticles. We note that, if we consider a deviation from equilibrium that consists, inthe absence of interaction, to add δN particles in particular momentum-states, i.e. an excitedstate identified by δnkσ such that ∑
kσ
δnkσ = δ N, (1.9)
such a state must evolve in the Landau’s hypothesis into a state with the same “quasiparticle”excitations δnkσ and furthermore, since the total number of particles is conserved along theadiabatic evolution, with the same deviation δN from equilibrium. In other words, althoughquasiparticles have not to be confused with real particles, still the sum of all quasiparticledeviations of the occupation numbers gives the variation of the total number of real particles,i.e. Eq. (1.9).
———Remark
It is important to note that the adiabatic evolution hypothesis guarantees that only trueconserved quantities, like the total number of particles or the total spin, keep the same expres-sions in terms of quasiparticle occupation numbers as for the real particles. Conserved quantitiesrefer, rigorously speaking, only to the fully interacting Hamiltonian, whereas the non-interactingone has generally much larger symmetry. In other words, the Landau’s theory of Fermi liquidhas only to do with conserved quantities but its use is not at all justified for non-conserved ones,even if they are conserved in the non-interacting limit. This fact is often underestimated andmay lead easily to wrong conclusions.
———
Since the labeling of the states is identical to non-interacting electrons, the phase-spacevolume counting is the same, hence also the formal expression of the entropy-change δS:
δS = δ
[−KB
∑kσ
nkσ lnnkσ + (1− nkσ) ln (1− nkσ)
]. (1.10)
Thermodynamic equilibrium implies an extremum of the free-energy F , namely that
δF = δE − T δS − µ δN = 0. (1.11)
Let us solve this equation by assuming that the deviation from equilibrium is induced by adeviation of the quasiparticle occupation numbers δnkσ, that amounts to impose(
δF
δnkσ
)δn=0
= 0.
9
By Eq. (1.4) we find that
δE
δnkσ= εk +
∑k′σ
fkσ k′σ′ δnk′σ′ ≡ εkσ, (1.12)
is the quasiparticle energy in the presence of excited quasiparticles, including itself, while whatit is needed is (
δE
δnkσ
)δn=0
= εk,
that is obviously independent of δn. Since all terms in (1.11) have exactly the same expression asfor non-interacting particles, one readily realizes that the quasiparticle equilibrium distributionis the Fermi-Dirac distribution
n0kσ = n0
k = f (εk − µ) =(
1 + eβ (εk−µ))−1
, (1.13)
the only difference with respect to non-interacting real particles being the renormalized banddispersion εk.
We must point out that f (εk − µ) corresponds to global equilibrium – all quasiparticlesare in equilibrium among themselves. Analogously, we could define a distribution function forthe “local” equilibrium that minimizes the free-energy of a single excited quasiparticle in thepresence of other excited quasiparticles. It is clear from Eq. (1.12) that the local-equilibriumdistribution function is the Fermi-Dirac distribution with argument the quasiparticle energy εkσ,i.e.
n0kσ = f (εkσ − µ) =
(1 + eβ (εkσ−µ)
)−1,
which is a functional of the occupation number deviations.In accordance, we can define a deviation with respect to local equilibrium as
δnkσ = nkσ − f (εkσ − µ) , (1.14)
as opposed to that one with respect to global equilibrium
δnkσ = nkσ − n0k
= δnkσ + n0kσ − n0
k
= δnkσ +∂n0
k
∂εk(εkσ − εkσ)
= δnkσ +∂f (εk − µ)
∂εk(εkσ − εkσ)
' δnkσ − δ (εk − µ)∑k′σ
fkσ k′σ′ δnk′σ′ , (1.15)
10
where we have assumed to be close to equilibrium and at very low temperature.
Now let us consider, as conventionally done, a space-isotropic system, where εk = ε (|k|) anduse the parametrization
δnkσ = −∂n0k
∂εk
∑lm
Ylm (Ωk) δnlmσ ' δ (εk − µ)∑lm
Ylm (Ωk) δnlmσ, (1.16)
where Ylm (Ωk) are spherical harmonics identified by the Euler angles of the unit vector in thedirection of k, being its modulus fixed by the δ-function to be right on the Fermi sphere. (Theδ-function in the right hand side of (1.16) derives from the fact that both sides of that equationmust consistently be of first order in the deviation from equilibrium.)
In addition, we introduce the Legendre decomposition of fkk′ assuming that it depends onlyon the relative angle θkk′ between the two momenta, their modula being on the Fermi sphere.Therefore
fS(A)kk′ =
∑l
fS(A)l Pl (θkk′) , (1.17)
where Pl (θkk′) are Legendre polynomials. 1 In the non-abelian representation in which the en-ergy functional is explicitly spin-isotropic, one readily finds through (1.15) the following relationsbetween the charge/spin deviations at global and local equilibrium:
δρk = δρk + 2∂n0
k
∂εk
∑k′
fSkk′ δρk′ , (1.18)
δσk = δσk + 2∂n0
k
∂εk
∑k′
fAkk′ δσk′ . (1.19)
One realizes that, because of εk = ε (|k|) and of the identity∫dΩk′
4πPl (θkk′) Yl′m′ (Ωk′) = δll′
1
2l′ + 1Yl′m′ (Ωk) ,
it follows that
−∑k′
∑l
fSl Pl (θkk′)∂n0
k
∂εkYl′m′ (Ωk′) δρl′m′
'∑k′
∑l
fSl Pl (θkk′) δ (εk′ − µ)∑l′m′
Yl′m′ (Ωk′) δρl′m′
1Conventionally, the argument of the Legendre polynomials is indicated as cos θ ∈ [−1 : 1]. Here, in order tosimplify notations, we have decided to use as argument θ
11
=V
2N∑ll′m′
fSl δρl′m′
∫dΩk′
4πPl (θkk′) Yl′m′ (Ωk′)
=V
2N∑l′m′
fSl′ δρl′m′1
2l′ + 1Yl′m′ (Ωk) ,
and analogously for δσk with fS → fA, where N = N (µ) is the quasiparticle density of states
N (ε) =1
V
∑kσ
δ (εk − ε) ,
at the chemical potential. Therefore Eqs. (1.18) and (1.19) have the simple solution
δρlm =δρlm
1 +FSl
2l + 1
, (1.20)
δσlm =δσlm
1 +FAl
2l + 1
, , (1.21)
where the Landau F -parameters are defined through
FS(A)l = V N f
S(A)l . (1.22)
It also common to define A-parameters through
AS(A)l =
FS(A)l
1 +FAl
2l + 1
, (1.23)
so that
FSl δρlm = ASl δρlm, (1.24)
FAl δσlm = AAl δσlm., (1.25)
We note that the above formulas are valid only for an isotropic system, like 3He, or for a metalwith an approximately spherical Fermi surface. In general, one must use, instead of sphericalharmonics and Legendre polynomials, other basis functions appropriate to the symmetry of thelattice. However, in what follows we just take into account the isotropic case, unless otherwisestated.
12
1.2.1 Specific heat
Let us start the calculation of the quasiparticle thermodynamic properties at low temperaturefrom the simplest one: the specific heat. It is easy to show through (1.10) (we set KB = 1 aswell as ~ = 1) that
cV =T
V
(∂S
∂T
)N,V
' T
V
(∂S
∂T
)µ,V
+O(T 2)
= −TV
∑kσ
∂n0k
∂Tln
n0kσ
1− n0kσ
= − 1
TV
∑kσ
(εk − µ)2 ∂n0k
∂εk
= − 1
T
∫dεN (ε+ µ) ε2
∂f(ε)
∂ε=π2
3N T. (1.26)
This is the same expression as in the absence of interaction with the quasiparticle density ofstates N instead of the free-particle one N (0)
N (0) = N (0)(µ) =∑kσ
δ(ε(0)k − µ
).
Therefore, if c(0)V is the specific heat of the free particles, then
cV
c(0)V
=NN (0)
. (1.27)
1.2.2 Compressibility and magnetic susceptibility
Suppose that we perturb the system with a static homogeneous field that may modify thechemical potentials, µσ = µ→ µ+δµσ, for spin up and spin down real particles. At equilibrium,the quasiparticle occupation numbers will change accordingly as
δnkσ =∂n0
k
∂εk(δεkσ − δµσ) , (1.28)
whereδεkσ =
∑k′σ′
fkσ k′σ′ δnk′σ′ ,
is the change in the quasiparticle energy due to the fact that the external field changes all quasi-particle occupation numbers. It is important to note that δµσ that appears in (1.28) coincideswith the field that acts on the real particles because the latter couples to conserved quantities,otherwise nothing could guarantee that the two are equal. Once again this demonstrates thatthis theory only addresses the response of conserved quantities.
13
By means of Eq. (1.15) we find that
δnkσ = δnkσ −∂n0
k
∂εk
∑k′σ′
fkσ k′σ′ δnk′σ′ = −∂n0
k
∂εkδµσ, (1.29)
showing that the deviation from local equilibrium is the same as for free particles with dispersionεk in the presence of the field.
Compressibility
The compressibility κ is defined by
κ = − 1
V
∂V
∂P=
1
n2
∂n
∂µ,
where n is the total density. Since a variation in the total density of particles coincides with thevariation in the quasiparticle one, we find through (1.20) that 2
δn =1
V
∑kσ
δnkσ =1
V
∑k
δρk
=1
2√
4πN δρ00 =
1
2√
4πN δρ00
1 + FS0=
1
1 + FS0δn.
The variation δn with respect to a variation of chemical potential is the same as for free particleswith density of states N , namely
δn
δµ= N .
Therefore we finally obtain that
κ =1
n2
N1 + FS0
=1
1 + FS0
NN (0)
κ(0), (1.30)
where κ(0) is the compressibility of the original free particles which have density of states N (0).
Magnetic susceptibility
A magnetic field B, e.g. in the z-direction, introduces a Zeeman term in the Hamiltonian
δH = −g µB B1
2(N↑ −N↓) ,
2Note that
Y00(Ω) =
√1
4π.
14
whereNσ is the total number of spin-σ electrons, that acts as a difference of chemical potential forspin up and down electrons: δµ↑−δµ↓ = g µB B. Since the total magnetization is also conserved,the quasiparticles acquire the same difference in chemical potentials. The magnetization per unitvolume δm can therefore be calculated by the quasiparticles through
δm = g µB1
2V
∑k
δnk↑ − δnk↑ = g µB1
2V
∑k
δσzk = g µB1
4√
4πN δσz00
= g µB1
4√
4πN δσz00
1 + FA0=
δm
1 + FA0.
Once again, since the local equilibrium magnetization δm is equivalent to the response of freeparticles with density of states N , instead of N0 as for the original particles, we obtain that themagnetic susceptibility χ is
χ =δm
δB=
1
1 + FA0
NN (0)
χ(0), (1.31)
where χ(0) is the susceptibility the original free-particles.
1.3 Quasiparticle transport equation
Let us suppose to perturb the Landau-Fermi liquid by an external probe that varies in space ona wavelength 1/q and in time on a period 1/ω. If
q kF , ω µ,
where kF is the Fermi momentum (ε(kF ) = µ), the probe will affect only quasiparticles extremelyclose to the Fermi-sea. In this limit, we can safely adopt a semi-classical approach and introducea space- and time-dependent density of quasiparticle occupation numbers at momentum |k| ∼ kF
nkσ(x, t),
and its space and time Fourier transform
nkσ(q, ω).
Fixing both coordinate x and momentum k is allowed in spite of the Heisenberg principle∆x∆k ' 1, because
∆k ' 1
∆x' q kF ,
so that the quantum indeterminacy of momentum ∆k is much smaller than its typical value kF .
15
Since the quasiparticle occupation number is not a conserved quantity, its density shouldfollow a Liouville equation
∂nkσ(x, t)
∂t+∂nkσ(x, t)
∂x
∂x
∂t+∂nkσ(x, t)
∂k
∂k
∂t= Ikσ (x, t) , (1.32)
where Ikσ(x, t) is a collision integral and corresponds to the transition rate within a volume dxaround x of all processes in which quasiparticles have a transition from any other state into(k, σ) minus the transition rate of the inverse processes.
In the absence of the external probe, quasiparticles are at equilibrium and their density isuniform. Once the probe is applied, the density deviates from its uniform equilibrium value andacquires space and time dependence. In general the external force that acts on the real particlescan not be identified with the force felt by the quasiparticles unless the external field couples toa conserved quantity. This is the case of a scalar potential coupled to the charge density, or amagnetic field that couples to the spin density. In those cases, the deviation from equilibriumof the energy becomes non-uniform hence can be generalized into
δE(t) =
∫dx δE(x, t) =
∑kσ
∫dx
(εk + Vσ(x, t)
)δnkσ(x, t)
+1
2
∑kk′
∑σσ′
∫dxdy δnkσ(x, t) fkσ k′σ′(x− y) δnk′σ′(y, t), (1.33)
where Vσ(x, t) represents the action of the external probe, V↑ = V↓ for a scalar potential andV↑ = −V↓ for a magnetic field, and we assume that the quasiparticle interaction is instantaneousand only depends on the distance. The above expression provides also the proper definition ofthe quasiparticle excitation energy as
εkσ(x, t) = εk + Vσ(x, t) +∑k′σ′
∫dy fkσ k′σ′(x− y) δnk′σ′(y, t). (1.34)
For neutral particles we can assume that
fkσ k′σ′(x− y) = fkσ k′σ′ δ(x− y),
in which caseεkσ(x, t) = εk + Vσ(x, t) +
∑k′σ′
fkσ k′σ′ δnk′σ′(x, t). (1.35)
By means of the Hamiltonian equations for conjugate variables
∂x
∂t=∂H
∂k,
∂k
∂t= −∂H
∂x,
16
Eq. (1.32) can be finally written as
∂nkσ(x, t)
∂t+∂nkσ(x, t)
∂x
∂εkσ(x, t)
∂k− ∂nkσ(x, t)
∂k
∂εkσ(x, t)
∂x= Ikσ (x, t) , (1.36)
At linear order in the external probe Vσ(x, t), the occupation density
nkσ(x, t) = n0kσ + δnkσ(x, t),
and, consistently,εkσ(x, t) = εk + δεkσ(x, t),
where
δεkσ(x, t) =δE
δnkσ(x, t)− εk = Vσ(x, t) +
∑k′σ′
fkσ k′σ′ δnk′σ′(x, t).
Consequently, the linearized transport equation (1.36) reads
∂δnkσ(x, t)
∂t+∂δnkσ(x, t)
∂x
∂εk∂k− ∂n0
k
∂k
∂δεkσ(x, t)
∂x
=∂δnkσ(x, t)
∂t+∂δnkσ(x, t)
∂x
∂εk∂k− ∂n0
k
∂k
[∂Vσ(x, t)
∂x+∑k′σ′
fkσ k′σ′∂δnk′σ′(x, t)
∂x
]
=∂δnkσ(x, t)
∂t+∂δnkσ(x, t)
∂x
∂εk∂k
−∂n0k
∂εk
∂εk∂k
[∂Vσ(x, t)
∂x+∑k′σ′
fkσ k′σ′∂δnk′σ′(x, t)
∂x
]
=∂δnkσ(x, t)
∂t+ vk ·
∂δnkσ(x, t)
∂x
−∂n0k
∂εkvk ·
[∂Vσ(x, t)
∂x+∑k′σ′
fkσ k′σ′∂δnk′σ′(x, t)
∂x
]
≡ ∂δnkσ(x, t)
∂t+ vk ·
∂δnkσ(x, t)
∂x+∂n0
k
∂εkvk · Fkσ(x, t)
= Ikσ (x, t) , (1.37)
where
vk =∂εk∂k
is the quasiparticle group velocity. The Landau transport equation (1.37) resembles the con-ventional Boltzmann equation apart from the fact that the effective force Fkσ(x, t) dependsself-consistently on the occupation density.
17
1.3.1 Quasiparticle current density
Let us set V (x, t) = 0 in (1.37). We note that
1
V
∑kσ
δnkσ(x, t) = δρ(x, t),
where δρ(x, t) is the deviation of the density with respect to the uniform equilibrium value, andmoreover that
1
V
∑kσ
Ikσ (x, t) = 0,
by particle conservation (recall the meaning of the collision integral Ikσ). Through Eq. (1.37)and using the relation (1.29) between the deviations from global and local equilibrium, we findthe following equation for the evolution of δρ(x, t) in the absence of external probes:
∂δρ(x, t)
∂t+
∂
∂x· 1
V
∑kσ
vk
[δnkσ(x, t)− ∂n0
k
∂εk
∑k′σ′
fkσ k′σ′ δnk′σ′(x, t)
]
=∂δρ(x, t)
∂t+
∂
∂x· 1
V
∑kσ
vk δnkσ(x, t) = 0. (1.38)
Since the integral over the whole space of δρ(x, t) is the variation of the total number of realparticles, which is conserved, then δρ(x, t) should also satisfy a continuity equation
∂δρ(x, t)
∂t+∂J(x, t)
∂x= 0,
that allows to identify the current density J(x, t) (note that, at equilibrium, the current is zerohence J(x, t) = δJ(x, t))
J(x, t) =1
V
∑kσ
vk δnkσ(x, t)
=1
V
∑kσ
vk
[δnkσ(x, t)− ∂n0
k
∂εk
∑k′σ′
fkσ k′σ′ δnk′σ′(x, t)
]
=1
V
∑kσ
δnkσ(x, t)
[vk −
∑k′σ′
fkσ k′σ′∂n0
k′σ′
∂εk′vk′
]
≡ 1
V
∑kσ
δnkσ(x, t) Jkσ. (1.39)
This equation shows that the actual current matrix element Jkσ is not the group velocity vk – themoving quasiparticle induces a flow of other quasiparticles because of their mutual interaction.
18
We observe that at very low temperature
∂n0k′σ′
∂εk′=∂f (εk′ − µ)
∂εk′' −δ (εk′ − µ) ,
namely k′ lies on the Fermi surface. If the system is isotropic, then on the Fermi sphere
vk =∂ε(|k|)∂k
= vFk
|k|=kFm∗
k
|k|,
with vF and m∗ the Fermi velocity and effective mass, respectively, of the quasiparticles. Us-ing an expansion like (1.16) in terms of spherical harmonics we find that, for instance thez-component of the current density, Jz, is
Jz(x, t) =1
V
∑kσ
(vk)z δnkσ(x, t)
=1
V
∑kσ
vFkz|k|
δ (εk − µ)∑lm
Ylm (Ωk) δnlmσ(x, t)
=vF2N∑lm
δρlm(x, t)
∫dΩ
4πcos θ Ylm (Ω)
=vF
2√
12πN δρ10(x, t) =
vF
2√
12πN
(1 +
FS13
)δρ10(x, t)
=vF
v(0)F
(1 +
FS13
) [v
(0)F
2√
12πN δρ10(x, t)
]
=vF
v(0)F
(1 +
FS13
)1
V
∑kσ
(v
(0)k
)zδnkσ(x, t), (1.40)
where v(0)k is the group velocity of the free particles on the Fermi surface
v(0)k =
∂ε(0)(|k|)∂k
= v(0)F
k
|k|=kFm
k
|k|. (1.41)
The x and y component can be found analogously and involve combinations of δn1 +1(x, t)and δn1−1(x, t). Therefore in general it holds that
J(x, t) =vF
v(0)F
(1 +
FS13
)1
V
∑kσ
v(0)k δnkσ(x, t)
=m
m∗
(1 +
FS13
)1
V
∑kσ
v(0)k δnkσ(x, t). (1.42)
19
As a concluding remark we notice that we could also follow the same analysis in connection withthe spin-current density, the only difference being that FS1 would be substituted by FA1 .
Translationally invariant systems
In a system that is not only isotropic but also translationally invariant, the current matrixelement is
Jk =k
m,
with m the mass of the real particles. Since the current is conserved, the total current per unitvolume must be given in terms of quasiparticles by
Jz =1
V
∑kσ
k
mδnkσ =
kF
2m√
12πN δρ10. (1.43)
Comparing (1.43) with (1.40) in the case in which the current density, hence the occupationdensity, are constant in time and space, i.e. Jz(x, t) = Jz and δρ10(x, t) = δρ10, we concludethat, when translational invariance is unbroken, like in 3He, the following relation holds
m∗m
= 1 +FS13, (1.44)
namely the effective mass and the Landau parameter FS1 are not independent.
1.4 Collective excitations in the collisionless regime
The collision integral in the transport equation determines a typical collision time τ . If oneis interested in phenomena that occur on a time scale smaller than τ , namely on frequenciesω 1/τ , then the collision integral can be safely neglected and the transport equation (1.37)in frequency and momentum space becomes
(ω − q · vk) δnkσ(q, ω) + q · vk∂n0
k
∂εk
[Vσ(q, ω) +
∑k′σ′
fkσ k′σ′ δnk′σ′(q, ω)
]= 0. (1.45)
The solution of this equation in the absence of external field gives information about the collectiveexcitations of the quasiparticle gas. These excitations can also be thought as collective vibrationsof the Fermi sphere. From this point of view, let us suppose that, along the direction identified bythe Euler angles (θ, φ), the z-axis being directed along the momentum q, the Fermi momentumkF = kF (sin θ cosφ, sin θ sinφ, cos θ) for a spin σ quasiparticle changes into
kFσ =(kF + uσ(θ, φ)
)(sin θ cosφ, sin θ sinφ, cos θ) ,
20
as ifkFσ(θ, φ) = kF → kFσ(θ, φ) = kF + δkF (θ, σ) = kF + uσ(θ, φ).
The occupation number for a spin σ quasiparticle with momentum k = k (sin θ cosφ, sin θ sinφ, cos θ)changes accordingly as
nkσ = n0kσ ' θ (k − kFσ(θ, φ))→ nkσ = n0
kσ −∂n0
k
∂kδkF = n0
kσ + δnkσ.
We note that
δnkσ = −∂n0k
∂kδkF
= −∂n0k
∂εk
∂εk∂k
δkF = −∂n0k
∂εkvF uσ(θ, φ). (1.46)
Inserting (1.46) into (1.45) with V = 0 we get
(ω − vF q cos θ) uσ(θ, φ) + vF q cos θ∑k′σ′
fkσ k′σ′∂n0
k′σ′
∂εk′uσ′(θ
′, φ′) = 0. (1.47)
For small vibrations of the Fermi sphere, we can assume that fkσ k′σ′ only depends on the anglebetween k and k′, fkσ k′σ′ = fσσ′ (θkk′), their modula being equal to the equilibrium value ofthe Fermi momentum kF . We note that, since
∂n0k′σ′
∂εk′' −δ
(ε(|k′|)− µ
),
it follows that∑k′σ′
fkσ k′σ′∂n0
k′σ′
∂εk′uσ′(θ
′, φ′) = −2V
∫4π |k′|2 d|k′|
(2π)3 δ(ε(|k′|)− µ
)×1
2
∫dΩk′
4πfσσ′ (θkk′) uσ′ (Ωk′)
= −V N 1
2
∫dΩk′
4πfσσ′ (θkk′) uσ′ (Ωk′) = −1
2
∫dΩk′
4πFσσ′ (θkk′) uσ′ (Ωk′) ,
where, generalizing Eq. (1.22), the F -parameters are defined through
Fσσ′ (θkk′) = V N fσσ′ (θkk′) .
If furthermore we define, for spin-isotropic models,
uS = u↑ + u↓, uA = u↑ − u↓,
21
F↑↑ = F↓↓ = FS + FA, F↑↓ = F↓↑ = FS − FA,
we find, defining λ = ω/vF q, the following equation
(λ− cos θk) uS(A)(θk, φk)− cos θ
∫dΩk′
4πFS(A) (θkk′) u
S(A)(θk′ , φk′) = 0. (1.48)
We write, dropping the indices S and A,
u(θ, φ) =∑lm
Ylm(θ, φ)ulm,
F (θkk′) =∑l
Pl (θkk′) Fl,
so that Eq. (1.48) can be re-written as∑lm
Ylm (θk, φk) ulm
[(λ− cos θk)− cos θk
2l + 1Fl
]= 0. (1.49)
Since
cos θ =
√4π
3Y10(θ, φ),
l is not a good quantum number of Eq. (1.49), while m is. In particular m = 0 correspondsto longitudinal vibrations of the Fermi sphere, m = 1 to dipolar vibrations and m = 2 toquadrupolar ones. We observe that a solution of (1.49) at ω = 0 would imply that a deformationof the Fermi sphere would be cost-free, namely that the Fermi sphere is unstable. At λ = 0, l isa good quantum number, so that a λ = 0 solution of (1.49) occurs whenever there is a value ofl such that, see Eq. (1.23),
1 +FS(A)l
2l + 1= 0 =
FS(A)l
AS(A)l
, (1.50)
which is the condition for the instability of the Landau-Fermi liquid. We note that, since FS(A)l
is generally non zero, the instability implies a singularity of the Landau A-parameters.
1.4.1 Zero sound
Let us assume, as commonly done, that only FS(A)0 and F
(S(A)1 are not negligible, and study
Eq. (1.49) for longitudinal vibrations, m = 0. In this case, dropping for simplicity the labels Sand A, we obtain
(λ− cos θ)u(θ) =∑lm
(λ− cos θ) Ylm (θk, φk) ulm = cos θ F0 Y00 u00 +cos θ
3F1 Y10 u10. (1.51)
22
We define,
C =
√1
4πF0 u00, (1.52)
D =1
3
√3
4πF1 u10, (1.53)
and, by noting that
Y00 =
√1
4π, cos θ =
√4π
3Y10,
we get the formal solution
u(θ) =C cos θ +D cos2 θ
λ− cos θ. (1.54)
In terms of this solution √1
4πu00 =
∫dΩ
4π
C cos θ +D cos2 θ
λ− cos θ
= C I1 +D I2, (1.55)
1
3
√3
4πu10 =
∫dΩ
4πcos θ
C cos θ +D cos2 θ
λ− cos θ
= C I2 +D I3, (1.56)
where
In =
∫ 1
−1
dz
2
zn
λ− z. (1.57)
Therefore the self-consistency condition that has to be satisfied is
C = F0 (C I1 +D I2) , (1.58)
D = F1 (C I2 +D I3) , (1.59)
which have a solution if
1 = F0
(I1 + F1
I22
1− F1 I3
)(1.60)
By realizing that
I1 = −1 +λ
2ln
(λ+ 1
λ− 1
),
I2 = λ I1,
I3 = −1
3+ λ2 I1,
23
we finally get that (1.60) is satisfied if
1 =(F0 + λ2A1
)I1, (1.61)
where, by definition,
A1 =F1
1 + F1/3.
We note that, for λ ≤ 1,
I1 ' −1 + λ2 − iπ2λ,
has a finite imaginary part. This implies that the frequency ω that solves (1.61) is necessarilycomplex, hence that it does not correspond to a stable collective mode but rather to a dampedone. Therefore the only stable solutions of (1.61) must correspond to λ ≥ 1.
For λ ' 1 + δλ, with 0 < δλ 1,
I1 ' −1
2ln
e2 δλ
2,
leading to
δλ ' 2 exp
[−2
(1 +
1
F0 +A1
)]. (1.62)
which is consistent with the assumption δλ 1 only if
0 ≤ F0 +A1 1.
In other words, when the Landau parameters are positive and very small, the stable collectivemodes propagate like sound modes with a velocity
v ' vF(
1 + 2 exp
[−2
(1 +
1
F0 +A1
)]),
slightly larger than the Fermi velocity. These modes correspond to a deformation of the Fermisphere of the form
u(θ) ' C cos θ +D cos2 θ
1− cos θ + δλ,
that are strongly peaked to θ = 0, namely in the direction of the propagating wave-vector q.
In the opposite case of λ 1,
I1 '1
3λ2+
1
5λ4,
24
and the solution of (1.60) is readily found to be
λ2 =ω2
v2F q
2 =
(F0
3+A1
5
) (1 +
F1
3
), (1.63)
that requires the right hand side to be positive and much greater than one. Again this solutiondescribes collective modes that propagate acoustically with a velocity this time much biggerthan vF and that correspond to a deformation of the form
u(θ) ' C cos θ +D cos2 θ
λ,
that involves the whole Fermi sphere.
These collective modes that may emerge in the collisionless regime and propagate like soundmodes were called by Landau “zero sounds”, as opposed to the first sound that exists in thecollision regime. The existence of these new modes of a Landau-Fermi liquid depends on thequasiparticle interaction parameters F , that act as a restoring force absent for non-interacting
electrons. Usually, for repulsive interactions, FS0 is positive and FA0 negative, while AS(A)1 is
negligible.3 In this case only charge zero-sound, i.e. a Fermi-sphere deformation in the channel
uS = u↑ + u↓,
is a stable collective mode, while a spin zero-sound in the channel
uA = u↑ − u↓,
is not.
1.5 Collective excitations in the collision regime: first sound
For frequencies much smaller than the typical collision rate, the collision integral can not beneglected anymore and, as a result, any deformation of the Fermi sphere is destined to decay.The only exceptions are those excitations that correspond in the limit q → 0 to conserved
3 For repulsive interaction, the effective mass m∗ increases with respect to the non-interacting system – thequasiparticles move more slowly than free ones. Furthermore, the magnetic susceptibility increases even more,since the slowed down quasiparticles with opposite spin repel each other, hence can react quite efficiently to amagnetic field. On the contrary, the compressibility κ decreases, as adding particles costs more energy. Since
κ
κ(0)=m∗m
1
1 + FS0,
χ
χ(0)=m∗m
1
1 + FA0,
if follows that FS0 > −1 +m∗/m > 0 and FA0 ≤ 0.
25
quantities, in particular to charge and momentum densities, in which case it is guaranteedthat
∑kσ Ikσ =
∑kσ k Ikσ = 0. Charge and momentum density excitations correspond to
deformations of the Fermi sphere in the symmetric S-channel and with angular momenta l = 0and l = 1, respectively, and m = 0. Through Eq. (1.49), the excitations in this regime satisfythe equation
0 = (cos θ − λ) Y00(Ω)u00 + (cos θ − λ) Y10(Ω)u10
+FS0 cos θ Y00(Ω)u00 +FS13
cos θ Y10(Ω)u10. (1.64)
Note that this equation differs from Eq. (1.51) for the zero sound because in that case F0 andF1 are assumed to be non-zero, but the u(θ) solving the equation has all l-components, unlikein this case. To solve (1.64), we multiply both sides once by Y00(Ω) and integrate over the solidangle Ω, and do the same multiplying by Y10(Ω), thus obtaining the following set of equations:
−λu00 +
√1
3
(1 +
FS13
)u10 = 0,√
1
3
(1 + FS0
)u00 − λu10 = 0.
This set of equations has solution if
λ2 =ω2
v2F q
2=
1
3
(1 + FS0
) (1 +
FS13
), (1.65)
which corresponds to acoustic waves propagating with velocity s, where
s2 =v2F
3
(1 + FS0
) (1 +
FS13
). (1.66)
We note that, for an isotropic system as that we are assuming,
vF =kFm∗
, N =m∗kFπ2
, n =k3F
3π2,
so thatv2F
3=
n
m∗Nhence, through Eq. (1.30),
s2 =1
nm∗κ
(1 +
FS13
). (1.67)
One can readily realize that this is the ordinary sound velocity compatible with thermody-namics. When local thermodynamic equilibrium is enforced by collisions – the hydrodynamic
26
regime – the equations of motion for the charge density, ρ(x, t), through the continuity equation,and for the velocity field through the pressure P (x, t),4
∂ρ(x, t)
∂t= −n m
m∗
(1 +
FS13
)∇ · v(x, t)
mn∂v(x, t)
∂t= −∇P (x, t),
where n is the average density, can be solved using the local thermodynamic relation
δP (x, t) =∂P
∂ρδρ(x, t) =
1
nκδρ(x, t),
where κ is the thermodynamic compressibility. The solution gives ordinary sound with a velocitythat coincides with (1.67).
1.6 Linear response functions
An external field induces deviations of the quasiparticle occupation numbers, and the linearrelation between these deviations and the field responsible for them are the so-called “linearresponse functions”. Let us go back to the transport equation (1.37), and assume in what followsthat the collision integral is negligible. After space and time Fourier transform, Eq. (1.37) is5
(q · vk − ω) δnkσ(q, ω)− q · vk∂n0
k
∂εk
∑k′σ′
fkσ k′σ′ δnk′σ′(q, ω)
4We assume that the particle velocity field is related to the current as in Eq. (1.42), where by definition
nv(x, t) =1
V
∑kσ
v(0)k δnkσ(x, t).
5 In deriving the transport equation we assume implicitly the system to be initially, for instance at timet→ −∞, at equilibrium. This implies that the external field that moves away from equilibrium must initially beabsent, i.e. V (x, t→ −∞)→ 0. A simple and convenient choice is to represent, for all times before the measure,performed e.g. at t = 0, an external field that oscillates with frequency ω as
V (x, t) = V (x, ω) e−i ω t+η t,
with η an infinitesimal positive number. Consequently, the response of the system must follow the same timebehavior, i.e.
δnkσ(x, t) = δnkσ(x, ω) e−i ω t+η t,
so that the time dependence cancels out from (1.37). When taking the time derivative,
∂δnkσ(x, t)
∂t= −i (ω + iη) δnkσ(x, ω) e−i ω t+η t,
which implies that the frequency ω has to be interpreted in whatever follows as ω + iη.
27
= q · vk∂n0
k
∂εkVσ(q, ω). (1.68)
We stress once more that the external field that enters this equation can be identified withthe actual field acting on the true particles only if it refers to conserved quantities, hence thecharge, in which case
Vσ(q, ω) = V (q, ω), (1.69)
and the spin,V↑(q, ω) = −V↓(q, ω). (1.70)
Let us write
δnkσ(q, ω) = − q · vk
q · vk − ω∂n0
k
∂εkXkσ(q, ω), (1.71)
through which Eq. (1.68) transforms into
Xkσ(q, ω)−∑k′σ′
fkσ k′σ′q · vk′
q · vk′ − ω∂n0
k′
∂εk′Xk′σ′
= −Vσ(q, ω). (1.72)
1.6.1 Formal solution
Let us write Eq. (1.72) in matricial way like[I − f K
]~X = −~V , (1.73)
where the matrices have matrix elements
Ikσ k′σ′ = δkk′ δσσ′ ,
fkσ k′σ′ = fkσ k′σ′ ,
Kkσ k′σ′ =q · vk
q · vk − ω∂n0
k
∂εkδkk′ δσσ′ ,
and the vectors ~X and ~V (~X)kσ
= Xkσ(q, ω),(~V)kσ
= Vσ(q, ω).
The formal solution of (1.73) is
~X = −[I − f K
]−1~V . (1.74)
28
Let us suppose to introduce another matrix A, with elements(A)kσ k′σ′
= Akσ k′σ′(q, ω), (1.75)
that satisfies the set of equations [I − f K
]A = f . (1.76)
One readily derives from this equation that
A =[I − f K
]−1f ,
hence that
A K =[I − f K
]−1f K =
[I − f K
]−1 [f K − I + I
]= −I +
[I − f K
]−1.
In other words [I − f K
]−1= A K + I , (1.77)
so that~X = −
[I + A K
]~V ,
that explicitly means
Xkσ(q, ω) = −∑k′σ′
[δkk′ δσσ′ +Akσ k′σ′(q, ω)
q · vk′
q · vk′ − ω∂n0
k′
∂εk′
]Vσ′(q, ω), (1.78)
hence that
δnkσ(q, ω) =q · vk
q · vk − ω∂n0
k
∂εk
∑σ′
[δσσ′ +
∑k′
Akσ k′σ′(q, ω)q · vk′
q · vk′ − ω∂n0
k′
∂εk′
]Vσ′(q, ω), (1.79)
which is the formal solution of (1.68). The functions Abkσ k′σ′(q, ω) are the so-called quasiparticlescattering amplitudes. We define as usual
Akσ k′σ(q, ω) = ASkk′(q, ω) +AAkk′(q, ω),
Akσ k′σ′(q, ω) = ASkk′(q, ω)−AAkk′(q, ω),
with σ 6= σ′. We also introduce a vertex λσ that, in case of a scalar potential Vσ(q, ω) = V (q, ω),is defined as
λσ = 1,
29
while it isλσ = δσ↑ − δσ↓,
for a Zeeman-splitting magnetic field
Vσ(q, ω) = V (q, ω)(δσ↑ − δσ↓
)= V (q, ω)λσ.
Upon multiplying both sides of (1.79) by the corresponding vertex and summing over k andσ and dividing by the volume, we find that the charge, δρ(q, ω), and spin, δσ(q, ω), densitydeviations at linear order in the corresponding fields are given by
δρ(q, ω) = χS(q, ω)V (q, ω)
δσ(q, ω) = χA(q, ω)V (q, ω),
where
χS(A)(q, ω) =2
V
∑k
q · vk
q · vk − ω∂n0
k
∂εk
[1 + 2
∑k′
A(S(A)kk′ (q, ω)
q · vk′
q · vk′ − ω∂n0
k′
∂εk′
]. (1.80)
Thus −χS(q, ω) is n2 times the dynamical compressibility κ(q, ω), while −χA(q, ω) is the dy-namical spin-susceptibility.
Limiting cases
We readily realize that, if q = 0 and ω 6= 0, then the matrix K = 0. In this case A = f , namely
Akσ k′σ′(0, ω) = fkσ k′σ′ . (1.81)
In the opposite limit, q 6= 0 and ω = 0, Eq. (1.76) reads explicitly
Akσ k′σ′(q, 0)−∑pα
fkσ pα
∂n0p
∂εpApαk′σ′(q, 0) = fpαk′σ′ ,
showing that Akσ k′σ′(q, 0) is independent on q. By writing
V N Akσ k′σ′(q, 0) =∑l
Pl (θkk′)[ASl ±AAl
],
where the + sign refers to σ = σ′ and the − one otherwise, one readily recognize that ASl andAAl coincide with the Landau A-parameters of Eq. (1.23).
30
1.6.2 Low frequency limit of the response functions
For frequencies ω vF q, we can expand Akσ k′σ′(q, ω) as
Akσ k′σ′(q, ω) = Akσ k′σ′(q, 0) + δAkσ k′σ′(q, ω), (1.82)
where δA vanishes linearly at ω = 0. We note that
q · vk
q · vk − ω= 1 +
ω
q · vk,
so that the linear variation δA has to satisfy the equation
δAkσ k′σ′(q, ω)−∑pα
fkσ pα
∂n0p
∂εpδApαk′σ′(q, ω) =
∑pα
fkσ pα
∂n0p
∂εp
ω
q · vpApαk′σ′(q, 0). (1.83)
The integral over Ωp of the function (q · vp)−1 is singular, hence we need some regularization.Indeed, as discussed in the footnote [5], the frequency ω above is actually ω + iη, with η aninfinitesimally small positive number. Therefore
q · vk
q · vk − ω→ q · vk
q · vk − ω − iη= 1 +
ω
q · vk − iη' 1 + i π ω δ (q · vk) + P ω
q · vk,
where P indicates the principal value. Therefore, within the S and A channels, Eq. (1.83)becomes (the principal value averages to zero hence does not contribute)
δAS(A)kk′ (q, ω)− 2
∑p
fS(A)kp
∂n0p
∂εpδA
S(A)pk′ (q, ω) = 2π i ω
∑p
fS(A)kp
∂n0p
∂εpδ (q · vp) A
S(A)pk′ (q, 0).
(1.84)At first order in ω, Eq. (1.80) becomes (we drop the indices S and A)
χ(q, ω) ' 2
V
∑k
∂n0k
∂εk
[1 + 2
∑k′
Akk′(q, 0)∂n0
k′
∂εk′
]+2π i ω
1
V
∑k
δ (q · vk)∂n0
k
∂εk
[1 + 2
∑k′
Akk′(q, 0)∂n0
k′
∂εk′
]+4π i ω
1
V
∑kk′
∂n0k
∂εk
∂n0k′
∂εk′Akk′(q, 0) δ (q · vk′)
+41
V
∑kk′
∂n0k
∂εk
∂n0k′
∂εk′δAkk′(q, ω). (1.85)
Since ∫dΩ
4πPl(θ) = δl0,
31
∫dΩ
4πδ (q vF cos θ) =
1
2 q vF
it follows from Eq. (1.84) that∫dΩk
4π
dΩk′
4πδAkk′(q, ω) + F0
∫dΩp
4π
dΩk′
4πδApk′(q, ω) = −π i ω 1
VN1
2 q vFF0A0,
namely ∫dΩk
4π
dΩk′
4πδAkk′(q, ω) = −π i ω 1
VN1
2 q vF
F0
1 + F0A0 = −π i ω 1
VN1
2 q vFA2
0.
Therefore Eq. (1.85) is found to be
χ(q, ω) ' −N (1−A0)
−i π ω
2 q vFN (1−A0)
+i πω
2 q vFN A0
−i π ω
2 q vFN A2
0
= − N1 + F0
− i π ω
2 q vFN (1−A0)2
= − N1 + F0
− i π ω
2 q vFN (1 + F0)−2 . (1.86)
1.6.3 High frequency limit of the response functions
Let us now consider the case ω vF q → 0 when (again the S and A indices are not explicitlyshown)
Apk′(q, ω) = fkk′ .
In this caseq · vk
q · vk − ω' −q · vk
ω− (q · vk)2
ω2 ,
and, since the angular average of q · vk vanishes, the response functions are
χ(q→ 0, ω) = − 2
ω2 V
∑k
∂n0k
∂εk(q · vk)2
+4
ω2 V
∑kk′
(q · vk) (q · vk′)∂n0
k
∂εk
∂n0k′
∂εk′fkk′ . (1.87)
32
We note that
2
V
∑k
∂n0k
∂εk(q · vk)2 = −N
∫dΩ
4π(vF q cos θ)2
= −1
3N v2
F q2,
4
V
∑kk′
∂n0k
∂εk
∂n0k′
∂εk′fkk′ (q · vk) (q · vk′) = N v2
F q2
×∑l
Fl
∫dΩk
4π
dΩk′
4πcos θk cos θk′ Pl (θkk′)
= N v2F q
2,∑lm
Fl
∫dΩk
4π
dΩk′
4π
×√
4π
3Y10 (θk)
√4π
3Y10 (θk′)
4π
2l + 1Y ∗lm (θk) Ylm (θk′)
= N v2F q
2 F11
9.
Therefore Eq. (1.87) is
χ(q→ 0, ω) = N v2F q
2
3ω2
(1 +
F1
3
). (1.88)
In a system that is also translationally invariant, besides being isotropic, in the S channelthe following relations hold
N =m∗kFπ2
=mkFπ2
(1 +
F1
3
),
vF =kFm∗
=kFm
(1 +
F1
3
)−1
,
n =k3F
3π2,
hence
κ(q→ 0, ω) =n
m
q2
ω2, (1.89)
in agreement with the f -sum rule.
33
1.7 Charged Fermi-liquids: the Landau-Silin theory
Now, instead of a neutral Fermi system, let us consider a charged one and discuss how all previousresults change. The first thing to note is that the quasiparticle interaction fkσ k′σ′(x − y) inEq. (1.33) will now contain, besides a short range contribution, also a long range Coulomb one,so that can be assumed to be of the form
fkσ k′σ′(x− y) = fkσ k′σ′ δ(x− y) +e2
|x− y|. (1.90)
Another novel feature that we have to take into account is the possible presence of a transverseelectromagnetic field. If the system is translationally invariant or if inter-band matrix elementsof the current density are negligible, the role of a transverse field is that the conjugate momentumis not anymore k but
K = k− e
cA(x, t),
where A is the transverse vector potential felt by the real electrons. Thus the quasiparticle oc-cupation density in the semi-classical limit can be still parametrized by nKσ(x, t). Nevertheless,it is more convenient to define the occupation density in the equivalent way
nKσ(x, t) ≡ nkσ(x, t) = nK+ ecA(x,t)σ(x, t), (1.91)
since, in the presence of A, the quasiparticle excitation energy, Eq. (1.34), changes simply into
εkσ(x, t)→ εK+eA/c σ(x, t) = εkσ(x, t).
From the above equation it follows that
∂εK+eA/c σ(x, t)
∂K=∂εkσ(x, t)
∂k= vkσ(x, t)
is the proper quasiparticle group velocity, and
∂εK+eA/c(x, t)
∂x=
∑i
e
cvkσ i(x, t)
∂Ai(x, t)
∂x
+∂Vσ(x, t)
∂x+∑k′σ′
∫dy fkσ k′σ′(x− y)
∂nk′σ′(y, t)
∂y.
Through the definition Eq. (1.91), it follows that(∂nKσ(x, t)
∂t
)Kx
=
(∂nkσ(x, t)
∂t
)kx
+e
c
∂nkσ(x, t)
∂k· ∂A(x, t)
∂t,(
∂nKσ(x, t)
∂x
)K
=
(∂nkσ(x, t)
∂x
)k
+∑i
e
c
∂nkσ(x, t)
∂ki
∂Ai(x, t)
∂x,(
∂nKσ(x, t)
∂K
)x
=
(∂nkσ(x, t)
∂k
)x
.
34
Putting everything together we find the following transport equation for charged electrons:
Ikσ(x, t) =∂nkσ(x, t)
∂t+e
c
∂nkσ(x, t)
∂k· ∂A(x, t)
∂t
+vkσ(x, t) · ∂nkσ(x, t)
∂x+∑ij
vkσ j(x, t)e
c
∂nkσ(x, t)
∂ki
∂Ai(x, t)
∂xj
−∂nkσ(x, t)
∂k·[∂Vσ(x, t)
∂x+∑k′σ′
∫dy fkσ k′σ′(x− y)
∂nk′σ′(y, t)
∂y
]−∑ij
e
c
∂nkσ(x, t)
∂kjvkσ i(x, t)
∂Ai(x, t)
∂xj
We note that the two terms with the∑
ij can be written as∑ij
e
c
∂nkσ(x, t)
∂kivkσ j(x, t)
(∂Ai(x, t)∂xj
− ∂Aj(x, t)
∂xi
)= −e
c
(vkσ(x, t)×H(x, t)
)· ∂nkσ(x, t)
∂k,
whereH(x, t) =∇×A(x, t)
is the magnetic field. Since the external transverse electric field is
E(x, t) = −1
c
∂A(x, t)
∂t,
the final expression of the transport equation reads
Ikσ(x, t) =∂nkσ(x, t)
∂t+ vkσ(x, t) · ∂nkσ(x, t)
∂x
−∂nkσ(x, t)
∂k·[∂Vσ(x, t)
∂x+∑k′σ′
∫dy fkσ k′σ′(x− y)
∂nk′σ′(y, t)
∂y
]−e ∂nkσ(x, t)
∂k·E(x, t)− e
c
(vkσ(x, t)×H(x, t)
)· ∂nkσ(x, t)
∂k. (1.92)
Let us now expand the transport equation (1.92) at linear order in the deviation from equi-librium
nkσ(x, t) = n0kσ + δnkσ(x, t).
First we assume an ac electromagnetic field acting as a perturbation, in which case
−ec
(vkσ(x, t)×H(x, t)
)· ∂nkσ(x, t)
∂k' −e
c
(vk ×H(x, t)
)· ∂n
0k
∂k
35
= −ec
(vk ×H(x, t)
)· ∂n
0k
∂εkvk = 0.
This shows that an ac magnetic field does not contribute to the linearized transport equation,that becomes
Ikσ(x, t) =∂δnkσ(x, t)
∂t+ vk ·
∂δnkσ(x, t)
∂x
+∂n0
k
∂εkvk ·
[− ∂Vσ(x, t)
∂x− eE(x, t)−
∑k′σ′
∫dy fkσ k′σ′(x− y)
∂δnk′σ′(y, t)
∂y
]=
∂δnkσ(x, t)
∂t+ vk ·
∂δnkσ(x, t)
∂x
+∂n0
k
∂εkvk ·
[− ∂Vσ(x, t)
∂x− eE(x, t)
], (1.93)
where we recall that
δnkσ(x, t) = δnkσ(x, t)− ∂n0k
∂εk
∑k′σ′
∫dy fkσ k′σ′(x− y) δnk′σ′(y, t),
is the deviation from local equilibrium.In the presence of a dc magnetic field H(x), a subtle issue arises in connection with the
Lorenz’s force term
−ec
(vkσ(x, t)×H(x, t)
)· ∂nkσ(x, t)
∂k.
Indeed, a dc field, unlike an ac one, can be assumed as integral part of the unperturbed Hamilto-nian and, if large, can be taken as a zeroth order term, H(x) = H0(x). This requires to expandat linear order vkσ(x, t) and ∂nkσ(x, t)/∂k,
vkσ(x, t) ' vkσ +∑k′σ′
∫dy
∂
∂k
[fkσ k′σ′(x− y) δnk′σ′(y, t)
],
∂nkσ(x, t)
∂k' ∂n0
k
∂k+∂δnkσ(x, t)
∂k,
leading to the transport equation
Ikσ(x, t) =∂δnkσ(x, t)
∂t+ vk ·
∂δnkσ(x, t)
∂x+∂n0
k
∂εkvk ·
[− ∂Vσ(x, t)
∂x− eE(x, t)
]−ec
(vk ×H0(x, t)
)· ∂δnkσ(x, t)
∂k, (1.94)
which is the appropriate one to discuss properties like magnetoresistance, cyclotron resonances,etc. However, in what follows we will not consider such a physical situation of a large dc field,hence we will just focus on Eq. (1.93).
36
1.7.1 Formal solution of the transport equation
We note that, because of (1.90),
fSkk′(x− y) = fSkk′δ (x− y) +e2
|x− y|, fAkk′(x− y) = fAkk′δ (x− y) , (1.95)
showing that only the response in the S charge-channel is going to be modified by the Coulombrepulsion with respect to the previously studied case of neutral particles. Hence let us assumethat
Vσ(x, t) = −e φ(x, t),
is the conventional scalar potential and, neglecting the collision integral, let us write the trans-port equation for the charge deviation
δnk(x, t) = δnk↑(x, t) + δnk↓(x, t),
which is found to be
∂δnk(x, t)
∂t+ vk ·
∂δnk(x, t)
∂x− 2
∂n0k
∂εkvk ·
[− e∂φ(x, t)
∂x− eE(x, t)
]= 0, (1.96)
with
δnk(x, t) = δnk(x, t)− 2∂n0
k
∂εk
∑k′
∫dy fSkk′(x− y) δnk′(y, t).
In the Fourier space
δnk(q, ω) = δnk(q, ω)− 2∂n0
k
∂εk
∑k′
fSkk′ δnk′(q, ω)− 2∂n0
k
∂εk
4πe2
q2δρ(q, ω),
where
δρ(q, ω) =1
V
∑k′
δnk′(q, ω),
is the deviation of the total charge density. We then find
(ω − vk · q) δnk(q, ω) + 2 vk · q∂n0
k
∂εk
∑k′
fSkk′ δnk′(q, ω)
−i e 2∂n0
k
∂εkvk · E(q, ω) = 0, (1.97)
where E(q, ω) is the internal electric field felt by the quasiparticles,
E(q, ω) = E(q, ω)− iqφ(q, ω) + i 4π eq
q2δρ(q, ω), (1.98)
that contains the external transverse and longitudinal fields as well as the longitudinal fieldcreated by the same quasiparticles.
37
Longitudinal response
Let us assume a zero transverse field, and define as
V (q, ω) = −e φ(q, ω) +4π e2
q2δρ(q, ω), (1.99)
the internal scalar potential. One easily recognizes that the response to the internal scalarpotential is formally the same as for neutral particles, so that, by means of Eq. (1.80), we findthat
δρ(q, ω) = χS(q, ω)V (q, ω) ≡ χ∗(q, ω)V (q, ω), (1.100)
that also define the so-called “proper” charge response-function χ∗. Inserting the expression ofV (q, ω), Eq. (1.99), and solving for δρ(q, ω) we finally obtain
δρ(q, ω) =χ∗(q, ω)
1− 4π e2
q2χ∗(q, ω)
V (q, ω) ≡ χ(q, ω)V (q, ω), , (1.101)
that defines the “improper” response function χ, which is the response to the external field, asopposed to the proper one, i.e. the response to the internal field.
Eq.(1.101) provides the definition of the longitudinal dielectric constant ε||(q, ω) as
ε||(q, ω) = 1− 4π e2
q2χ∗(q, ω). (1.102)
In the limit ω → 0 first and then q → 0, we can make use of Eq. (1.86) and find that
limq→0
limω→0
ε||(q, ω) = limq→0
1 +4π e2
q2
N1 + FS0
→∞, (1.103)
which defines the effective Thomas-Fermi screening-length
λTF =
√π(1 + FS0
)e2N
.
In the limit q → 0, we can use Eq. (1.88) to get
limq→0
ε||(q, ω) = 1− 4π e2
q2N v2
F q2
3ω2
(1 +
F1
3
)≡ 1−
ω2p
ω2, (1.104)
that provides the definition of the conduction-band plasma-frequency
ω2p =
4π e2N v2F
3
(1 +
F1
3
), (1.105)
38
as the pole of 1/ε||(0, ω). Note that, for a translationally invariant system
ω2p =
4π n e2
m,
with n the electron density and m the real electron mass, that coincides with the conventionalexpression for the plasma oscillations.
1.8 Dirty quasiparticle gas
So far we have not taken into account the collision integral, that is however important at frequen-cies smaller than the typical collision rate. At very low temperature T , the main contributionto the collision rate comes from impurity scattering. Indeed, collisions due to phonons vanish atlow temperature, as phonon occupations go exponentially to zero as T → 0, as well as collisionsdue to scattering off other quasiparticles. 6 Taking into account only the scattering off (non-magnetic) impurities, and moreover assuming a short-range impurity potential, we can write thecollision integral as
Ikσ(x, t) = −2π∑p
Wkp
[nkσ(x, t) (1− npσ(x, t))− npσ(x, t) (1− nkσ(x, t))
]×δ(εkσ(x, t)− εpσ(x, t)
), (1.106)
6An excited quasiparticle at εk > µ can in principle decay into a quasiparticle at momentum k′ with εk′ > µplus a quasiparticle-quasihole pair, namely a quasiparticle with momentum p′ with εp′ > µ and a quasihole withmomentum p and energy εp < µ. Because of momentum conservation k = k′+p′−p, and of energy conservation
εk − µ = (εk′ − µ) + (εp′ − µ)− (εp − µ) .
The collision integral expressed through the Fermi golden-rule will have the general form (we discard for simplicitythe spin label)
Ik =1
V 2
∑k′pp′
Wkpp′k′ δ(k + p− k′ − p′
)δ (εk + εp − εk′ − εp′) nk nk′ np′ (1− np) ,
where W is the modulus square of the scattering amplitude. Since one momentum is fixed by the momentumconservation, say p, the sum becomes a double sum over k′ and p′ with the condition that
|εk′ − µ|+ |εp′ − µ|+ |εk′+p′−k − µ| = |εk − µ| ' T,
the last almost equivalence deriving from the fact that the energy variation with respect to the chemical potentialof a quasiparticle excitation at temperature T is of the order of T itself. This implies that both (εk′ − µ) and(εp′ − µ) are positive but smaller that T , so that, by simple phase space arguments, the collision integral is foundto decay approximately as T 2. Extending this analysis, one readily realizes that any decay into a quasiparticleplus m quasiparticle-quasihole pairs occurs with probability T 2m. Therefore the collision integral due to scatteringoff quasiparticles is negligible at low temperature.
39
where the energy conservation involves, as it should, the true quasiparticle excitation energy
εkσ(x, t) =δE
δnkσ(x, t)= εk +
∑k′σ′
∫dy fkσ k′σ′(x− y) δnk′σ′(y, t).
We note that[nkσ(x, t) (1− npσ(x, t))− npσ(x, t) (1− nkσ(x, t))
]δ(εkσ(x, t)− εpσ(x, t)
)=[nkσ(x, t)− npσ(x, t)
]δ(εkσ(x, t)− εpσ(x, t)
)=[n0kσ(x, t)− n0
pσ(x, t) + δnkσ(x, t)− δnpσ(x, t)]δ(εkσ(x, t)− εpσ(x, t)
)=[δnkσ(x, t)− δnpσ(x, t)
]δ(εkσ(x, t)− εpσ(x, t)
),
so that
Ikσ(x, t) = −2π∑p
Wkp
[δnkσ(x, t)− δnpσ(x, t)
]δ(εkσ(x, t)− εpσ(x, t)
), (1.107)
showing that the collision integral only depends upon deviations from local equilibrium. Withinlinear approximation
Ikσ(x, t) ' −2π∑p
Wkp
[δnkσ(x, t)− δnpσ(x, t)
]δ (εk − εp) . (1.108)
Let us write
Wkp =1
V
∑l
Wl Pl (θkp) ,
δnkσ(x, t) = −∂n0k
∂εk
∑lm
δnlmσ(x, t)Ylm (Ωk)
so that Eq. (1.108) becomes
Ikσ(x, t) = π∂n0
k
∂εkN∑lm
W0 δnlmσ(x, t)Ylm (Ωk)
−π ∂n0k
∂εkN∑lm
Wl
2l + 1δnlmσ(x, t)Ylm (Ωk)
=∂n0
k
∂εk
∑lm
1
τlδnlmσ(x, t)Ylm (Ωk) , (1.109)
40
where the inverse relaxation time is defined through
1
τl= πN
(W0 −
Wl
2l + 1
)=
2π
V
∑p
Wkp δ (εk − εp)[1− Pl (θkp)
]. (1.110)
We note that 1/τ0 = 0.Therefore the transport equation in Fourier space and for the charge S component reads
−ω δnk(q, ω) + q · vk δnk(q, ω)− 2∂n0
k
∂kq · vk
∑k′
fSkk′ δnk′(q, ω)
+i e 2∂n0
k
∂εkvk · E(q, ω) = −i ∂n
0k
∂εk
∑lm
1
τlδnlm(q, ω)Ylm (Ωk)
= −ω δnk(q, ω) + q · vk δnk(q, ω) + i e 2∂n0
k
∂εkvk · E(q, ω), (1.111)
where we recall thatE(q, ω) = E(q, ω) + i 4π e
q
q2δρ(q, ω), (1.112)
is the internal field felt by quasiparticles, E including both transverse and longitudinal compo-nents, and we have defined
δnk(q, ω) = δnk(q, ω)− 2∂n0
k
∂k
∑k′
fSkk′ δnk′(q, ω),
as the deviation from local equilibrium excluding the long-range interaction.
1.8.1 Conductivity
We observe that
−e δρ(q = 0, ω) =
∫dt ei ω t δN(t),
is the frequency Fourier-transform of the change in the total electron charge. If we assume thatthe system is maintained neutral at any time by a compensating positive ionic charge-density,then the q = 0 component of the field generated by the electrons must be canceled exactly by theionic field. In other words, δρ(q, ω) must not include any q = 0 component, i.e. δρ(0, ω) = 0.Thus, taking q = 0 in (1.111) we find
− ω δnk(ω) + i e 2∂n0
k
∂εkvk ·E(ω) = −i ∂n
0k
∂εk
∑lm
1
τlδnlm(ω)Ylm (Ωk) . (1.113)
41
We observe that
vk ·E(ω) =1
m∗k ·E(ω) =
k
m∗
√4π
3
[Y10 (Ωk) Ez(ω)
+Y1+1 (Ωk)1
2
(Ex(ω)− iEy(ω)
)+ Y1−1 (Ωk)
1
2
(Ex(ω) + iEy(ω)
)],
namely that only the l = 1 components are coupled to the electric field, so that the actualtransport equation reads [dropping the label (l,m)=(1,m)]
−ω δnk(ω) + i e 2∂n0
k
∂εkvk ·E(ω) = i
1
τ1δnk(ω).
Since, for l = 1 deviations
δn =δn
1 +FS13
,
we finally get
δnk(ω) = 2 e∂n0
k
∂εk
(1 +
FS13
)1
−iω +1
τ
vk ·E(ω), (1.114)
whereτ =
τ1
1 +FS13
. (1.115)
The electric current is therefore
J(ω) = − eV
∑k
vk δnk(ω)
= − eV
∑k
vk 2 e∂n0
k
∂εk
(1 +
FS13
)1
−iω +1
τ
vk ·E(ω) (1.116)
= e2Nv2F
3
(1 +
FS13
)1
−iω +1
τ
E(ω) ≡ σ(ω) E(ω), (1.117)
which defines the conductivity σ(ω). We note that
Nv2F
3=m∗ kF
π2
k2F
3m2∗
=n
m∗,
42
with n the average electron density. Thus the conductivity is
σ(ω) =n e2
m∗
(1 +
FS13
)1
−iω +1
τ
, (1.118)
with a dc value
σ = σ(0) =n e2 τ
m∗
(1 +
FS13
). (1.119)
This expression shows that the effective mass that controls dc transport is not m∗ but
m∗
1 +FS13
,
which, for a system that is translationally invariant before adding the impurities, is just thereal-electron mass m.
1.8.2 Diffusive behavior
Let us now consider a finite q such that vF q τ 1 as well as ωτ 1, and assume a longitudinalelectric field E||q, q taken to be along the z-axis. If we write
δnk(q, ω) = −∂n0k
∂εk
∑lm
δnlm(q, ω)Ylm (Ωk) ,
and analogously for δnk, we note that only the m = 0 component is coupled to the externalfield. If we substitute these expressions for δn and δn into Eq. (1.111), multiply both sides ofthis equation by Y ∗l0 (Ωk) and integrate over the solid angle, by means of
∫dΩY ∗l0(Ω)Yl1,0(Ω)Yl2,0(Ω) =
√(2l1 + 1) (2l2 + 1)
4π (2l + 1)
(C l0l10 l20
)2
,
where C l0l10 l20 are Clebsch-Gordan coefficients, we find the following set of equations (we discardthe label m = 0)
− ω δnl(q, ω) + q vF∑l′
√2l′ + 1
2l + 1
(C l0l′0 10
)2
δnl′(q, ω)− 2 i e vF
√4π
3E(q, ω) δl1 = i
δnl(q, ω)
τl.
(1.120)
43
We observe that only l = 1 is directly coupled to the field, so that, for any l > 1, we find− τlω
1 +FSl
2l + 1
− i
δnl(q, ω) + τl q vF∑l′ 6=1
√2l′ + 1
2l + 1
(C l0l′0 10
)2
δnl′(q, ω)
= −τl q vF√
3
2l + 1
(C l010 10
)2
δn1(q, ω).
This implies that all deviations with l > 1 are driven by q vF τl δn1 δn1, therefore arenegligible. What remains are just the components with l = 0, 1, satisfying the coupled equations
− ω
1 + FS0δn0(q, ω) + q vF
√1
3δn1(q, ω) = 0,
− ω
1 +FS13
δn1(q, ω) + q vF
√1
3δn0(q, ω)− 2 i e vF
√4π
3E(q, ω) = i
δn1(q, ω)
τ1,
whose solutions are readily obtainable:
δn1(q, ω) = 2 e vF
√4π
3
(1 +
FS13
)−i ω τ
iω − q2 v2F τ
3
(1 + FS0
) (1 +
FS13
)+ ω2 τ
E(q, ω),
δn0(q, ω) =
√1
3
vF q
ω
(1 + FS0
)δn1(q, ω)
= 2 e vF
√4π
3
(1 +
FS13
) (1 + FS0
)× −i q vF τ
iω − q2 v2F τ
3
(1 + FS0
) (1 +
FS13
)+ ω2 τ
E(q, ω).
Therefore the electric current is found to be
J(q, ω) = −e vF
2√
12πN δn1(q, ω)
= −e vF
2√
12πN
[ −i ω τ 2 e vF
√4π
3
(1 +
FS13
)
iω − q2 v2F τ
3
(1 + FS0
) (1 +
FS13
)+ ω2 τ
E(q, ω)
]
44
=n e2 τ
m∗
(1 +
FS13
)iω
iω − q2 v2F τ
3
(1 + FS0
) (1 +
FS13
)+ ω2 τ
E(q, ω)
=n e2 τ
m∗
(1 +
FS13
)iω
iω −D q2 + ω2 τE(q, ω)
≡ σ(q, ω) E(q, ω), (1.121)
where
D =v2F τ
3
(1 + FS0
) (1 +
FS13
), (1.122)
45
is the diffusion coefficient. 7 Seemingly, the charge density is
δρ(q, ω) =1
2√
4πN δn0(q, ω)
=1
1 + FS0N −iD q
iω −D q2 + ω2 τe E(q, ω). (1.123)
We recall that, for a longitudinal field,
−eE(q, ω) = e iqφ(q, ω)− i4π e2
q2q δρ(q, ω) ≡ −iqV (q, ω),
7Indeed this behavior is characteristic of diffusive modes. If E(q, ω) = 0, the two coupled equations reduce to
− ω
1 + FS0δn0(q, ω) + q vF
√1
3δn1(q, ω) = 0,
− ω
1 +FS13
δn1(q, ω) + q vF
√1
3δn0(q, ω) = i
δn1(q, ω)
τ1,
If we multiply the first by N/2√
4π , and the second by N vF
(1 + FS1
3
)/2√
12π, we find
−ω δρ(q, ω) + qJ(q, ω)
−e = 0
−ω J(q, ω)
−e + qD
τδρ(q, ω) = i
1
τ
J(q, ω)
−e .
The first equation is nothing but the continuity equation (the electric current is −e times the charge current j):
∂ρ(x, t)
∂t+ ∇ · j(x, t) = 0.
The second equation for ωτ 1, givesj(q, ω) = −i q D δρ(q, ω),
namely that the charge current,j(x, t) = −D∇ ρ(x, t),
is proportional to minus the space-gradient of the charge density, the proportionality constant being the diffusioncoefficient. Inserting this expression in the continuity equation, we find
∂ρ(x, t)
∂t−D∇2 ρ(x, t) = 0,
showing that charge diffuses with equation of motion(−iω +D q2
)δρ(q, ω) = 0.
46
where V is the internal scalar potential felt by the electrons. Therefore,
δρ(q, ω) = κD q2
iω −D q2 + ω2 τV (q, ω), (1.124)
with κ the compressibility. This equation provides an expression of the “proper” responsefunction in the presence of disorder
χ∗(q, ω) = κD q2
iω −D q2 + ω2 τ, (1.125)
hence of the “improper” one
χ(q, ω) =χ∗(q, ω)
1− 4πe2
q2 χ∗(q, ω)
. (1.126)
47
Chapter 2
Second Quantization
The first difficulty encountered in a many-body problem is how to deal with a many-bodywavefunction. The reason is that a many-body wavefunction has to take into account boththe indistinguishability of the particles and their statistics, while any operator, including theHamiltonian, does not have in first quantization such properties. Hence it would be desirableto have at disposal an alternative scheme where the indistinguishability principle as well as thestatistics of the particles were already built in the expression of the operators. This is actuallythe scope of second quantization.
2.1 Fock states and space
Let us take a system of N particles, either fermions or bosons. The Hilbert space spans a basis ofN -body orthonormal wavefunctions which should satisfy both the indistinguishability principleas well as the appropriate statistics of the particles. The simplest way to construct this space isas follows.
We start by choosing an orthonormal basis of single particle wavefunctions:
φa(x), a = 1, 2, · · · . (2.1)
Here x is a generalized coordinate which include both the space coordinate x as well as e.g. thez-component, σ, of the spin, which is half-integer for fermions and integer for bosons. The suffixa is a quantum label and ∫
dxφa(x)∗φb(x) = δab∑a
φa(x)∗φa(y) = δ(x− y).
[The∫dx . . . means
∑σ
∫dx . . . , while δ(x−y) ≡ δ(x−y)δσσ′ with x = (x, σ) and y = (y, σ′)].
A generic N -body wavefunction with the appropriate symmetry properties can be constructed
48
through the above single-particle states. Since the particles are not distinguishable, we do notneed to know which particle occupies a specific state. Instead, what we need to know are justthe occupation numbers na’s, i.e. the number of particles occupying each single-particle stateφa. This number is either na = 0, 1 for fermions, because of Pauli principle, or an arbitraryinteger na ≥ 0 for bosons. Apart from that, the occupation numbers should satisfy the trivialparticle-conservation constraint ∑
a
na = N.
Since the occupation numbers are the only ingredients we need in order to build up the N -bodywave-function, we can formally denote the latter as the ket
|n1, n2, . . . 〉, (2.2)
which is called a Fock state, while the space spanned by the Fock states is called Fock space.Within the Fock space, the state with no particles, the vacuum, will be denoted by |0〉.
For instance, if the N fermions with coordinates xi, i = 1, · · · , N , occupy the states aj ,j = 1, · · · , N , with a1 < a2 < · · · < aN , namely na = 1 for a ∈ aj, otherwise na = 0, then theappropriate wave-function is the Slater determinant
Ψna(x1, . . . , xN ) =
√1
N !
∣∣∣∣∣∣∣∣∣φa1(x1) · · · φa1(xN )φa2(x1) · · · φa2(xN )
............
...φaN (x1) · · · φaN (xN )
∣∣∣∣∣∣∣∣∣ , (2.3)
which is therefore the first quantization expression of the Fock state |na〉 with the sameoccupation numbers. The above wave-function satisfies the condition of being anti-symmetricif two coordinates are interchanged, namely two columns in the determinant. Analogously, it isantisymmetric by interchanging two rows, i.e. two quantum labels.
On the contrary, if we have N bosons with coordinates xi, i = 1, · · · , N , which occupy thestates aj , j = 1, . . . ,M , M ≤ N and a1 < a2 < · · · < aM , with occupation numbers naj , thenthe appropriate wave-function is the permanent
Φna(x1, . . . , xN ) =
√∏j nj !
N !
∑p
φa1(xp1) . . . φa1(xpna1 )
φa2(xpna1+1) . . . φa2(xpna1+na2) . . .
. . . φaM (xpN−naM+1) . . . φaM (xpN ), (2.4)
where the sum is over all non-equivalent permutations p’s of the N coordinates 1. Indeed thewave-function is even by interchanging two coordinates or two quantum labels.
1Non-equivalent means for instance that φi(x)φi(y) is equivalent to φi(y)φi(x)
49
In conclusion, the space spanned by all possible Slater determinants built with the samebasis set of single-particle wavefunctions constitues an appropriate Hilbert (Fock) space formany-body fermionic wavefunctions. Analogously the space spanned by all possible permanentsis an appropriate Hilbert space for many-body bosonic wavefunctions.
In the following we will introduce operators acting in the Fock space. We will considerseparately the fermionic and bosonic cases.
2.2 Fermionic operators
Let us introduce the creation, c†a, and annihilation, ca, operators which add or remove, respec-tively, one fermion in state a. The operator c†aca first annihilates then creates a particle in a,which can be done as many times as many particles occupy that state. Therefore
c†aca|na〉 = na|na〉, (2.5)
so it acts like the occupation number operator c†aca ≡ na. Since by Pauli principle na = 0, 1,then
c†aca|0〉 = 0, c†aca|na = 1〉 = |na = 1〉. (2.6)
Analogously, the operator cac†a first creates then destroys a fermion in state a, which can not do
if a is occupied, while it can do once if empty. Therefore
cac†a|0〉 = |0〉, cac
†a|na = 1〉 = 0. (2.7)
Thus, either a is empty or occupied, the following equation holds(cac†a + c†aca
)|na = 0, 1〉 = |na = 0, 1〉,
which leads to the operator identity(cac†a + c†aca
)=ca, c
†a
= 1, (2.8)
where the symbol . . . means the anti-commutator. Moreover, since one can not create nordestroy two fermions in the same state, it also holds
ca, ca
=c†a, c
†a
= 0. (2.9)
Eqs. (2.8) and (2.9) are the anti-commutation relations satisfied by the fermion operators withthe same quantum label. Going back to (2.6) and (2.7), we readily see that they are all satifiedif
c†a|0〉 = |na = 1〉, c†a|na = 1〉 = 0
50
ca|0〉 = 0, ca|na = 1〉 = |0〉,
also showing that c†a is the hermitean conjugate of ca.Let us consider now the action of the above operators on a Fock state. First of all we need
to provide a prescription to build a Fock state by the creation operators. We shall assume that,if
| na = 1〉 = c†a | 0〉,then by definition and for b 6= a
c†b | na = 1〉 = c†bc†a | 0〉 ≡| nb = 1, na = 1〉. (2.10)
Since the Slater determinant, hence the corresponding Fock state, is odd by interchanging tworows it follows that
| nb = 1, na = 1〉 = − | na = 1, nb = 1〉 ≡ −c†ac†b | 0〉. (2.11)
Comparing (2.11) with (2.10) we conclude that
c†ac†b = −c†bc
†a →
c†a, c
†b
= 0, (2.12)
The hermitean conjugate thus implies that alsoca, cb
= 0. (2.13)
We note that because of (2.9), both Eqs. (2.12) and (2.13) remain valid even if a = b.
Finally we need to extract the reciprocal properties of c†b and ca for a 6= b. We assume thefollowing result:
ca | na = 1, nb = 1〉 = cac†ac†b | 0〉 ≡| nb = 1〉 = c†b | 0〉. (2.14)
Since | 0〉 = cac†a | 0〉, and c†ac
†b = −c†bc
†a, it follows that
−cac†bc†a | 0〉 = c†bcac
†a | 0〉,
namely that, for a 6= b, ca, c
†b
= 0. (2.15)
All the above anti-commutation relations can be cast in the general formulasca, c
†b
= δab,
c†a, c
†b
= 0,
ca, cb
= 0. (2.16)
If we extend our prescription (2.10) to more than two electrons, we find that any Fock statehas the simple expression
| n1, n2, . . . 〉 =∏i≥1
(c†i
)ni| 0〉, (2.17)
where the occupation numbers ni = 0, 1.
51
2.2.1 Second quantization of multifermion-operators
Let us consider the single-particle operator in first quantization
V =∑i
V (xi), (2.18)
where the sum runs over all particles and V (xi) is an operator acting both on space coordinatesand spins. We want to calculate the matrix element
〈n′1, n′2, . . . |V |n1, n2, . . . 〉, (2.19)
among two Slater determinats, equivalently Fock states, with occupation numbers n′i’s and ni’s,respectively. Since the operator V conserves the number of particles, then∑
i
ni =∑j
n′j = N.
Being the particles not distinguishable, it derives that
〈n′1, n′2, . . . |V |n1, n2, . . . 〉 = N〈n′1, n′2, . . . |V (x1)|n1, n2, . . . 〉
=
∫dx1 dx2 . . . dxN Ψn′(x1, x2, . . . , xN )∗ V (x1) Ψn(x1, x2, . . . , xN )
We assume that in the Fock ket |n1, n2, . . . 〉, the states ai, i = 1, . . . , N are occupied, witha1 < a2 < · · · < aN , while in |n′1, n′2, . . . 〉 are occupied the bj ’s, j = 1, . . . , N . We can expandthe Slater determinant in the first column, corresponding to particle x1 and get
Ψn(x1, x2, . . . , xN ) =1√N
N∑i=1
(−1)i−1φai(x1)Ψn:nai=0(x2, . . . , xN ),
where Ψn:nai=0(x2, . . . , xN ) is the Slater determinant of the N − 1 particles x2, x3, . . . , xNwith the same occupation numbers as Ψn(x1, x2, . . . , xN ) apart from state ai which is empty.Analogously
Ψn′(x1, x2, . . . , xN )∗ =1√N
N∑j=1
(−1)j−1φbj (x1)∗Ψn′:n′bj=0(x2, . . . , xN )∗.
Therefore
〈n′1, n′2, . . . |V |n1, n2, . . . 〉 =∑i,j
(−1)i+j∫dx1φbj (x1)∗V (x1)φai(x1) 〈n′1, n′2, . . . , nbj − 1, . . . |n1, n2, . . . , nai − 1, . . . 〉
52
=∑i,j
Vbj ,ai〈n′1, n′2, · · · | c
†bjcai | n1, n2, . . . 〉, (2.20)
the last expression following from (2.17) and (2.16). Therefore the second quantization expres-sion of the single particle operator (2.18) is
V =∑i,j
Vi,j c†icj , (2.21)
where
Vi,j =
∫dxφi(x)∗ V (x)φj(x). 2
Let us continue and consider a two particle operator
U =1
2
∑i 6=j
U(xi, xj). (2.22)
The matrix element between two Fock states is now
〈n′1, n′2, . . . |U |n1, n2, . . . 〉 =N(N − 1)
2〈n′1, n′2, . . . |U(x1, x2)|n1, n2, . . . 〉 (2.23)
In this case we need to expand the Slater determinants in the first two columns:
Ψn(x1, x2, . . . , xN ) =1√
N(N − 1)
N∑i 6=j=1
(−1)i−1(−1)j−1−θj,iφaj (x2)φai(x1)
Ψn:nai=0,naj=0(x3, . . . , xN ), (2.24)
where now Ψn:nai=0,naj=0(x3, . . . , xN ) is the Slater determinant of theN−2 particles x3, x4, . . . , xN
in which, differently from Ψn(x1, x2, . . . , xN ), the states ai and aj are empty. The θj,i functionis = 1 if j > i and = 0 if j < i. Analogously
Ψn′(x1, x2, . . . , xN )∗ =1√
N(N − 1)
N∑i 6=j=1
(−1)i−1(−1)j−1−θj,iφbi(x1)∗ φbj (x2)∗
Ψn′:n′bi=0,n′bj=0(x3, . . . , xN )∗. (2.25)
2Notice that if the operator V (x) acts on the spins, namely V (x)→ Vσσ′(x), then
Vi,j =∑σσ′
∫dx φi(x, σ)∗ Vσσ′(x)φj(x, σ
′).
53
Therefore
N(N − 1)
2〈n′1, n′2, . . . |U(x1, x2)|n1, n2, . . . 〉 =
1
2
N∑i 6=j=1
N∑m 6=n=1
(−1)i+j+θj,i(−1)m+n+θn,m
∫dx1 dx2 φbm(x1)∗ φbn(x2)∗ U(x1, x2)φaj (x2)φai(x1)∫dx3 . . . dxNΨn′:n′bm=0,n′bn=0(x3, . . . , xN )∗Ψn:nai=0,naj=0(x3, . . . , xN )
=1
2
∑i,j,m,n
Ubm,bn;aj ,ai 〈n′1, n′2, · · · | c
†bmc†bncajcai | n1, n2, . . . 〉, (2.26)
hence the second quantization expression of the two-body interaction
U =1
2
∑ijkl
Ui,j;k,l c†ic†jckcl , (2.27)
where
Ui,j;k,l =
∫dx dy φi(x)∗φj(y)∗ U(x, y)φk(y)φl(x).
Analogously, one can introduce m-particle operators
Um =1
m!
∑i1 6=i2 6=···6=im
U(x1, x2, . . . , xm),
which translate in second quantization language into
Um =1
m!
∑i1,j1,i2,j2,...,im,jm
Ui1,i2,...,im;jm,jm−1,...,j1
c†i1c†i2. . . c†imcjmcjm−1
. . . cj1 , (2.28)
with
Ui1,i2,...,im;jm,jm−1,...,j1 =
∫dx1 dx2 . . . dxm φi1(x1)∗φi2(x2)∗ . . . φim(xm)∗
U(x1, x2, . . . , xm)φjm(xm)φjm−1(xm−1) . . . φj1(x1).
We conclude by emphasizing the advantages of second quantization with respect to first quan-tization. In the former, any multiparticle operator depends explicitly on the particle coordinates.It is only the wave-function which contains information about either the indistiguishability ofthe particles as well as their statistics. On the contrary, in second quantization those proper-ties are hidden in the definition of creation and annihilation operators, hence the multiparticle
54
operators do not depend anymore on the particle coordinates. Moreover in second quantizationwe can also introduce operators which have no first-quantization counterpart. For instance wecan define particle-non-conserving operators which connect subspaces with different numbers ofparticles of the whole Hilbert space. For instance∑
i,j
∆i,j c†ic†j + ∆∗i,j cjci ,
is an operator which connects Fock states with particle numbers differing by two. As we shallsee such operators are useful for discussing superconductivity.
2.2.2 Fermi fields
Till now we have defined fermionic operators for a given set of single-particle wavefunctions. Letus now introduce new operators which are independent of this choice. We define annihilationand creation Fermi fields by
Ψσ(x) ≡ Ψ(x) =∑i
φi(x) ci , Ψ(x)† =∑i
φi(x)∗ c†i . (2.29)
They satisfy the following properties
Ψ(x),Ψ(y)† =∑ij
φi(x)φj(y)∗ci , c†j
=∑i
φi(x)φi(y)∗ = δ(x− y), (2.30)
as well asΨ(x),Ψ(y) = Ψ(x)†,Ψ(y)† = 0. (2.31)
which are indeed independent upon the basis. If we change the basis via the unitary transfor-mation U acting on the basis set
φi(x) =∑α
Uα,i φα(x),
with U U † = I, the identity matrix, then
Ψ(x) =∑i
φi(x) ci =∑i,α
φα(x)Uα,i ci
=∑α
φα(x) cα, (2.32)
55
showing the proper transformation properties of the fermionic operators
cα =∑i
Uα,i ci . (2.33)
The multiparticle operators in second quantization have a very simple expression in termsof the Fermi fields. For instance it is easy to show that Eq. (2.21) can also be written as
V =∑ij
Vi,j c†icj =
∫dxΨ(x)† V (x) Ψ(x), (2.34)
and analogously (2.27) as
U =1
2
∑ijkl
Ui,j;k,l c†ic†jckcl =
1
2
∫dx dyΨ(x)†Ψ(y)† U(x, y) Ψ(y)Ψ(x), (2.35)
and finally (2.28) as
Um =1
m!
∑i1,j1,i2,j2,...,im,jm
Ui1,i2,...,im;jm,jm−1,...,j1
c†i1c†i2. . . c†imcjmcjm−1
. . . cj1
=1
m!
∫dx1 . . . dxm Ψ(x1)† . . .Ψ(xm)† U(x1, . . . , xm) Ψ(xm) . . .Ψ(x1). (2.36)
Let us for instance consider the density operator in first quantization
ρσ(x) =N∑i=1
δ(x− xi) δσ,σi .
In second quantization, through Eq. (2.34), it reads
ρσ(x) =∑σ′
∫dy Ψσ′(y)† δ(x− y) δσ,σ′ Ψσ′(y)
= Ψσ(x)†Ψσ(x) = Ψ(x)†Ψ(x), (2.37)
which also ahows that Ψ(x)† is nothing but the operator which creates a particle in coordinatex, while Ψ(x) destroys it.
56
2.3 Bosonic operators
As previoulsy done for the fermionic case, we introduce the creation, d†a, and its hermiteanconjugate, the annihilation da, operators which respectively create and destroy a boson in statea. Again the operators d†ada counts how many times we can destroy and create back a boson instate a, hence it is just the occupation number na. However, since Pauli principle does not holdfor bosons, the operator dad
†a first adds one boson in state a, hence increases na → na + 1, then
destroys one in the same state. This latter process can be done na + 1 times, being na + 1 theactual occupation number once one more boson has been added. Therefore
d†ada|na〉 = na|na〉, dad†a|na〉 = (1 + na)|na〉, (2.38)
hence the following commutation relation holds[da, d
†a
]= dad
†a − d†ada = 1, (2.39)
where [. . . ] denotes the commutator. Eq. (2.38) is trivially satisfied by
da|na〉 =√na|na − 1〉, d†a|na〉 =
√na + 1|na + 1〉. (2.40)
The permanent, contrary to the Slater determinant, is invariant upon interchanging twoquantum labels. As a consequence, bosonic operators corresponding to different states commuteinstead of anti-commuting as the fermionic ones. Namely, for a 6= b,[
d†a, db
]=[d†a, d
†b
]= 0.
Therefore, in general, [da, d
†b
]= δab,
[d†a, d
†b
]=[da, db
]= 0. (2.41)
Moreover, through (2.40) and (2.41) we can write a generic Fock state as
|n1, n2, . . . 〉 =∏i
(d†i
)ni√ni!|0〉. (2.42)
2.3.1 Bose fields and multiparticle operators
The analogous role of the Fermi fields is now played by the Bose fields defined through
Φ(x) =∑a
φa(x) da, Φ(x)† =∑a
φa(x)∗ d†a, (2.43)
57
which satisfy the commutation relations[Φ(x),Φ(y)†
]= δ(x− y), [Φ(x),Φ(y)] =
[Φ(x)†,Φ(y)†
]= 0, (2.44)
Let us consider the expansion of the permanent (2.4) over e.g. the coordinate x1. One caneasily show that it is given by
|n1, n2, . . . 〉 = Φn(x1, x2, . . . , xN ) =∑a
√naNφa(x1) Φn:na→na−1(x2, . . . , xN )
=∑a
√naNφa(x1) |n1, n2, . . . , na − 1, . . . 〉, (2.45)
where the√na also enforces that the sum does not include empty states. By means of (2.45)
we can write the matrix element of a single-particle operator like (2.18) as
〈n′1, n′2, . . . |V |n1, n2, . . . 〉 = N〈n′1, n′2, . . . |V (x1)|n1, n2, . . . 〉
=∑ab
√na n′b
∫dx1 φb(x1)∗ V (x1)φa(x1) 〈n′1, n′2, . . . , n′b − 1, . . . |n1, n2, . . . , na − 1, . . . 〉
= 〈n′1, n′2, . . . |∫dxΦ(x)† V (x) Φ(x) |n1, n2, . . . 〉,
thus showing that
V =
∫dxΦ(x)† V (x) Φ(x). (2.46)
Analogously we can expand the permanent in two coordinates, e.g x1 and x2,
|n1, n2, . . . 〉 = Φn(x1, x2, . . . , xN ) =∑ab
√naN
√nb − δabN − 1
φb(x2)φa(x1) Φn:na→na−1:nb→nb−1(x3, . . . , xN ),
and show that the average of a two-body interaction like (2.22) can be written as
〈n′1, n′2, . . . |U |n1, n2, . . . 〉 =N(N − 1)
2〈n′1, n′2, . . . |U(x1, x2)|n1, n2, . . . 〉
=1
2
∑abcd
√n′a(n
′b − δab)nd(nc − δcd)
∫dx1 dx2 φa(x1)∗φb(x2)∗ U(x1, x1)φc(x2)φd(x1)
〈n′1, n′2, . . . , n′a − 1, . . . , n′b − 1, . . . |n1, n2, . . . , nc − 1, . . . , nd − 1, . . . 〉.
Therefore, like in the fermionic case, we can formally identify
U =1
2
∫dx dyΦ(x)†Φ(y)† U(x, y) Φ(y)Φ(x), (2.47)
58
as well as for the m-particle operators
Um =1
m!
∫dx1 . . . dxm Φ(x1)† . . .Φ(xm)† U(x1, . . . , xm) Φ(xm) . . .Φ(x1). (2.48)
2.4 Canonical transformations
In general an interacting Hamiltonian, which contains besides bilinear also quartic and higherorder terms in creation and annihilation operators, can not be diagonalized. On the contrary, abilinear Hamiltonian is diagonalizable by a canonical transformation.
A canonical transformation preserves the commutation/anti-commutationproperties of the operators.
Since the interaction is commonly analysed perturbatively starting from an appropriate non-interacting theory, it is useful to begin with bilinear Hamiltonians and introduce the canonicaltransformations which diagonalize them.
The simplest bilinear Hamiltonian is the second quantized expression of a non-interactingfirst-quantization Hamiltonian, which has the general form
H =∑ab
tab c†acb, (2.49)
both for fermions and bosons. Since H is hermitean, then
tab = t∗ba.
If t is the hermitean matrix with elements tab, there exists a unitary transformation U , U U † = I,with I the identity matrix, such that∑
ab
U †αa tab Ubβ = εα δαβ. (2.50)
Therefore, if we define
ca =∑α
Uaα cα, (2.51)
then through (2.50) we find
H =∑αβ
∑ab
c†αU†αa tab Ubβcβ =
∑α
εαc†αcα. (2.52)
59
We have now to check whether the above is a canonical transformation.If the cα’s are fermionic/bosonic operators then
ca, c†b± =
∑αβ
Uaα U†βb cα, c
†β± =
∑α
Uaα U†αb = δab,
where . . . ± stands for the anticommutator (+) and commutator (-), respectively. Therefore
also the ca’s are fermions/bosons, hence U is indeed canonical.Once the Hamiltonian has been transformed into the diagonal form (2.52), the problem is
solved. Indeed any Fock state constructed through the new basis set with operators cα, namelya wave-function |nα〉 where each state α is occupied by nα particles, is an eigenstate of theHamiltonian
H |nα〉 =
[∑α
εα nα
]|nα〉.
2.4.1 More general canonical transformations
In second quantization we have the opportunity to introduce bilinear Hamiltonians which doesnot conserve the number of particles:
H =∑ab
tab c†acb + ∆ab cacb + ∆∗ab c
†bc†a. (2.53)
These Hamiltonians are relevant for a wide class of physical problems. Again tab = t∗ba while
∆ab = −∆ba
for fermions, and for bosons∆ab = ∆ba.
The most general transformation which may diagonalize the Hamiltonian is of the form
ca =∑α
Uaαcα + Vaαc†α, (2.54)
c†a =∑α
U∗aαc†α + V ∗aαcα. (2.55)
Let us derive the conditions under which the above is a canonical transformation:
ca, c†b± =
∑αβ
Uaα U†βbcα, c
†β± + Vaα V
†βbc
†α, cβ±
=∑α
Uaα U†αb ± Vaα V
†αb ≡ δab
60
ca, cb± =∑αβ
Uaα Vbβcα, c†β± + Vaα Ubβc†α, cβ±
=∑α
Uaα Vbα ± Vaα Ubα ≡ 0.
This implies that, for fermions
U U † + V V † = I , U V T + V UT = 0, (2.56)
while, for bosons,U U † − V V † = I , U V T − V UT = 0. (2.57)
In addition, U and V should diagonalize (2.53), namely, once we write the original ca’s and c†a’sin terms of the new operators, we should obtain
H =∑α
εα c†αcα.
We conclude by pointing out that, due to the particle-non-conserving terms, the vacuum ofthe original particles, |0〉, is nomore the vacuum of the new ones, |0〉. Indeed
ca|0〉 =∑α
Vaα c†α|0〉 6= 0.
61
2.5 Examples and Exercises
Let us consider electrons in a square box of linear length L with periodic boundary conditions.As a basis of single-particle wave-functions we use simple plane waves and spinors, namely thequantum label a = (k, s) with
k =2π
L(nx, ny, nz) ,
being ni’s integers, s = ±1/2 =↑, ↓. Hence the single-particle wavefunctions of the basis set are
φa(x) =1√L3
eik·x δs σ.
The annihilation and creation operators are defined as
ckσ, c†kσ,
hence the Fermi fields are
Ψσ(x) =1√L3
∑k
eik·x ckσ,
Ψ†σ(x) =1√L3
∑k
e−ik·x c†kσ.
Let us start from the kinetic energy, which is in first quantization
Hkin =N∑i=1
− ~2
2m∇2i ,
diagonal in the spin coordinate. Since
1
L3
∫dx e−ik·x
(− ~2
2m∇2
)eip·x =
~2 k2
2mδkp,
then, in the plane-wave basis, the kinetic energy in second quantization becomes
Hkin =∑kσ
~2 k2
2mc†kσckσ ≡
∑kσ
εk c†kσckσ. (2.58)
The ground state with N , assumed to be even, electrons is the Fermi sea |FS〉 obtained by fillingwith a spin-up and spin-down electrons the lowest energy levels up to the Fermi momentum kFdefined through
N = 2∑|k|≤kF
,
62
namely
|FS〉 =∏
k:|k|≤kF
c†k ↑c†k ↓|0〉, (2.59)
hence the occupation number in momentum space is
nk ↑ = nk ↓ = θ (kF − |k|) ,
where the θ-function is defined through θ(x) = 1 if x ≥ 0, otherwise θ(x) = 0.Let us add an electron-electron interaction of the general form
Hint =1
2
∑i 6=j
V (xi − xj).
In second quantization it becomes
Hint =1
2
∑σσ′
∫dx dy Ψ†σ(x)Ψ†σ′(y)V (x− y) Ψσ′(y)Ψσ(x)
=1
2L3
∑σσ′
∑ki,i=1,...,4
c†k1 σc†k2 σ′
ck3 σ′ck4 σ
1
L3
∫dx dy e−ik1·xe−ik2·yeik3·yeik4·x V (x− y).
We define the Fourier transform of the potential as
V (q) =
∫dr e−iq·r V (r),
so that
V (x− y) =1
L3
∑q
eiq·(x−y) V (q),
and the interaction is finally found to be
Hint =1
L3
∑σσ′
∑kpq
V (q) c†kσc†p+qσ′cpσ′ck+qσ. (2.60)
We showed that the electron density for spin σ in second quantization is
ρσ(x) = Ψ†σ(x) Ψσ(x).
Its Fourier transform is
ρσ(q) =
∫dx e−iq·x Ψ†σ(x) Ψσ(x) =
∑k
c†kσck+qσ.
63
Let us consider interacting electrons also in the presence of a single-particle potential, providede.g. by the ions,
Hpot =∑σ
∫dxU(x) ρσ(x),
so that the total Hamiltonian reads
H = Hkin +Hint +Hpot.
Since the Hamiltonian conserves the total number of electrons, the electron density summedover the spins, i.e. ρ = ρ↑ + ρ↓ must satisfy a continuity equation. Let us define the Heisenbergevolution of the density through
ρ(x, t) = eiHt ρ(x) e−iHt.
Since the integral of the density over the whole volume is the total number of electrons, N ,which is conserved, it follows that∫
dx∂ρ(x, t)
∂t=
∂
∂t
∫dx ρ(x, t) =
∂N
∂t≡ 0.
This condition is satisfied if∂ρ(x, t)
∂t= −∇J(x, t),
where J(x, t) is the current density operator. In fact, the integral over the volume of the lefthand side is also equal to minus the flux of the current through the surface of the sample. Ifthe number of electrons is conserved, it means that the flux of the current through the surfacevanishes, which is the desired result.Exercises:
• Calculate the formal expression of the average value of
H = Hkin +Hint,
over the Fermi sea wave-function (2.59);
• Using the Heisenberg equation of motion of the operators, according to which
i~∂ρ(x, t)
∂t= [ρ(x, t),H] ,
calculate the expression of the Fourier transform of the current J(q);
• Calculate the following commutator
[ρ(q, t),J(−q, t)] , (2.61)
which is commonly known as the f -sum rule.
64
2.6 Application: fermionic lattice models and the emergence ofmagnetism
Let us consider the electron Hamiltonian in the presence of the periodic potential provided bythe ions in a lattice:
H =
N∑i=1
p 2i
2m+∑i
∑R
V (xi −R) +1
2
∑i 6=j
U(xi − xj)
= Hkin +Hel−ion +Hel−el = H0 +Hel−el, (2.62)
where R are lattice vectors. We start by rewriting the Hamiltonian in second quantization. Forthat purpose we need to introduce a basis set of single-particle wavefunctions. Since the Hamilto-nian is spin independent, it is convenient to work with factorized single-particle wavefunctions:φ(x, σ) = φ(x)χσ. As a basis for the space-dependent φ(x) we use Wannier wave-functionsφ(x)n,R satisfying the usual conditions∫
dxφ(x)∗n,R1φ(x)m,R2 = δnm δR1R2 ,
∑n,R
φ(x)∗n,Rφ(y)n,R = δ(x− y),
as well asφ(x)n,R+R0 = φ(x−R0)n,R . (2.63)
Consequently we associate to any wavefunction φ(x)n,R χσ creation, c†n,R,σ, and annihilation,
cn,R,σ, operators, and introduce the Fermi fields
Ψσ(x) =∑n,R
φ(x)n,R cn,R,σ, Ψ†σ(x) = (Ψσ(x))† .
Let us start by second quantization of the non interacting part of the Hamiltonian, H0 inEq. (2.62):
H0 =∑σ
∫dx Ψ†σ(x)
[−~2∇2
2m+∑R
V (x−R)
]Ψσ(x)
=∑σ
∑nm
∑R1,R2
tnmR1,R2c†n,R1,σ
cm,R2,σ. (2.64)
The matrix elements are
tnmR1,R2=
∫dxφ(x)∗n,R1
[−~2∇2
2m+∑R
V (x−R)
]φ(x)m,R2 , (2.65)
65
and satisfytnmR1,R2
=(tmnR2,R1
)∗. (2.66)
By the property (2.63) it derives that
tnmR1+R0,R2+R0=
∫dxφ(x)∗n,R1+R0
[−~2∇2
2m+∑R
V (x−R)
]φ(x)m,R2+R0
=
∫dxφ(x−R0)∗n,R1
[−~2∇2
2m+∑R
V (x−R)
]φ(x−R0)m,R2
=
∫dxφ(x)∗n,R1
[−~2∇2
2m+∑R
V (x + R0 −R)
]φ(x)m,R2 = tnmR1,R2
,
since ∑R
V (x + R0 −R) =∑R
V (x−R).
Let us now introduce the operators in the reciprocal lattice through the canonical transformation
cn,k,σ =1√V
∑R
e−ik·Rcn,R,σ, (2.67)
and its inverse
cn,R,σ =1√V
∑k
eik·Rcn,k,σ, (2.68)
where V is the number of lattice sites and k belongs to the reciprocal lattice, namely∑R
eik·R = V δk0,∑k
e−ik·R = V δR0.
One can check that the above transformation is indeed canonical:
cn,k1,σ1, c†m,k2,σ2
=1
V
∑R1,R2
e−ik1·R1eik2·R2cn,R1,σ1, c†m,R2,σ2
= δnmδσ1σ21
V
∑R1
e−i(k1−k2)·R1 = δnmδσ1σ2δk1k2 .
Let us substitute (2.68) into (2.64):
H0 =1
V
∑σ
∑nm
∑R1,R2
∑k1,k2
e−ik1·R1eik2·R2 tnmR1,R2c†n,k1,σ
cm,k2,σ.
66
We notice that ∑R1,R2
e−ik1·R1eik2·R2 tnmR1,R2
=1
V
∑R1,R2,R0
e−ik1·R1eik2·R2 tnmR1−R0,R2−R0
=1
V
∑R1,R2,R0
e−ik1·(R1+R0)eik2·(R2+R0) tnmR1,R2
=∑
R1,R2
e−ik1·R1eik2·R2 tnmR1,R2
1
V
∑R0
e−i(k1−k2)·R0
= δk1k2
∑R1,R2
e−ik1·(R1−R2) tnmR1,R2= V δk1k2 t
nmk1,
where we define
tnmk ≡ 1
V
∑R1,R2
e−ik·(R1−R2) tnmR1,R2=∑R
e−ik·R tnmR,0. (2.69)
Since (2.66) holds it also follows that
(tnmk )∗ =∑R
eik·R(tnmR,0
)∗=
∑R
eik·R tmn0,R =∑R
eik·R tmn−R,0
=∑R
e−ik·R tmnR,0 = tmnk ,
namely the matrix tk, with elements tmnk , is hermitean. The Hamiltonian is therefore
H0 =∑σ
∑nm
∑k
tnmk c†n,k,σcm,k,σ. (2.70)
As we said tk is hermitean, hence it is possible to write
tk = U(k)† εk U(k)→ tnmk = U †(k)ni εi,k U(k)im,
with U(k)† U(k) = I. Therefore, upon the canonical transformation
ci,k,σ =∑n
U(k)incn,k,σ,
67
the non interacting Hamiltonian gets a diagonal form
H0 =∑σ
∑i
∑k
εi,k c†i,k,σci,k,σ. (2.71)
The index i identifies the bands, and εi,k is the energy dispersion in the reciprocal lattice: wehave thus obtained the band structure.
We can formally write
ci,k,σ =1√V
∑R
e−iR·kci,R,σ,
thus introducing a new basis of Wannier functions. Since the Fermi field is invariant upon thebasis choice, then
Ψσ(x) =∑R,n
φ(x)n,R χσ cn,R,σ =1√V
∑R,n,k
φ(x)n,R χσ eik·R cn,k,σ
=1√V
∑R,n,i,k
φ(x)n,R χσ eik·R U †(k)ni ci,k,σ
=1
V
∑R,R′,n,i,k
φ(x)n,R χσ eik·(R−R′) U †(k)ni ci,R′,σ ≡
∑R′
φ(x)i,R′ χσ ci,R′,σ ,
thus implying the following expression of the new Wannier functions
φ(x)i,R′ =1
V
∑n,R,k
eik·(R−R′) U †(k)ni φ(x)n,R. (2.72)
Going back to the Hamiltonian in the diagonal basis, we can also rewrite it as
H0 =∑σ
∑i
∑k
εi,k c†i,k,σci,k,σ
=1
V
∑σ
∑i
∑k
∑R1,R2
εi,k eik·(R1−R2)c†i,R1,σci,R2,σ
≡∑σ
∑i
∑R1,R2
tiR1,R2c†i,R1,σ
ci,R2,σ, (2.73)
namely like a tight-binding Hamiltonian, diagonal in the band index. Once we know the bandstructure, the ground state of the non interacting Hamiltonian is simply obtained by filling allthe lowest bands with the available electrons. If the highest occupied band is full, the modelis a band insulator, otherwise is a metal. In particular, since each band can accomodate 2Velectrons, V of spin up and V of spin down, a necessary condition for a band insulator is to have
68
an even number of available electrons per unit cell. This is not sufficient since the bands mayoverlap.
Exercises:
• Calculate the formal expression of the Fourier transform of the density ρ(q) in the basis
of eigenoperators of H0, c†i,k,σ and ci,k,σ;
• Do the same calculation for the current density J(x), and discuss the meaning of thef -sume rule, Eq. (2.61) in this representation.
2.6.1 Hubbard models
In many physical situations, the Wannier orbitals φ(x)i,R’s in Eq. (2.72) for the highest occupiedbands (valence bands) are quite delocalized, hence the lattice vector label R looses its physicalmeaning. In those cases, although formally exact, the tight-binding way (2.73) of rewritingthe non interacting Hamiltonian is of little use. 3 However there is a wide class of materials,commonly called strongly correlated systems, where the tight-binding formalism is meaningful.There the valence bands derive from d or f orbitals of transition metals, rare earth or actinides,and the Wannier orbitals keep noticeable atomic character, thus leading to short range hoppingelements tiR1,R2
’s. Let us consider just the above situation and write down the left-over electron-electron interaction in the Wannier basis. We further assume that there is only one valence bandwell separated from lower and higher ones, so that we are allowed to neglect interband transitionprocesses induced by the interaction. Since we take into account only one band, we are goingto drop the band index i, so that the Wannier orbitals for the valence band are φR(x) and thetight-binding term projected on the same band is
H0 =∑σ
∑R1,R2
tR1,R2 c†R1σ
cR2σ. (2.74)
The interaction term within the valence band is given by
Hint =1
2
∑σ1σ2
∑R1,R2,R3,R4
UR1,R2;R3,R4 c†R1σ1
c†R2σ2cR3σ2
cR4σ1, (2.75)
where
UR1,R2;R3,R4 =
∫dxdy φR1(x)∗φR2(y)∗ U(x− y)φR3(y)φR4(x). (2.76)
Let us make use of our assumption of well localized Wannier orbitals, namely that φR(x) decayssufficiently fast with |x −R|. Within this assumption, the leading matrix element is when alllattice sites are the same:
UR,R;R,R ≡ U. (2.77)
3 For instance the hopping matrix elements tiR1,R2turns out to be long ranged.
69
This term gives rise to an interaction
HU =U
2
∑R
nR nR, (2.78)
where the operator
nR =∑σ
c†RσcRσ,
counts the number of particles at site R. The interaction term (2.78), which simply describesan on-site Coulomb repulsion, plus the hopping (2.74) represent the so-called Hubbard model,which is the prototype of the strongly-correlated lattice models.
Let us continue and consider in (2.76) two other cases: either (1) R1 = R4, R2 = R3 withR2 nearest neighbor of R1 or (2) R1 = R3, R2 = R4 still with R2 nearest neighbor of R1. Incase (1) we obtain
U1
2
∑<RR′>
nR nR′ ,
where < RR′ > stands for nearest neighbor sites and
U1 = UR,R′;R′,R.
In case (2) we find
U2
2
∑<RR′>
∑σσ′
c†Rσc†R′σ′cRσ′cR′σ = −U2
2
∑<RR′>
∑σσ′
c†RσcRσ′c†R′σ′cR′σ,
whereU2 = UR,R′;R,R′ .
One can easily show that∑σσ′
c†RσcRσ′c†R′σ′cR′σ =
1
2[nR nR′ + 4SR · SR′ ] , (2.79)
where the spin density operator
SR =1
2
∑αβ
c†Rα σαβ cRβ,
being σ = (σ1, σ2, σ3), with σi’s the Pauli matrices. Therefore, upon defining V = U1 − U2/2and Jex = −2U2, the whole nearest neighbor interaction reads
Hn.n. =V
2
∑<RR′>
nR nR′ +Jex2
∑<RR′>
SR · SR′ . (2.80)
70
Since U1 > U2 > 0, the first term describes a nearest-neighbor repulsion, while the second a spinexchange which tends to aligns the spin ferromagnetically since J < 0, so called direct exchange.Therefore, although we have started from a spin independent interaction, in the Wannier basiswe have been able to identify a spin interaction, thus showing in a simple way how magnetismemerges out of the Coulomb repulsion.
2.6.2 The Mott insulator within the Hubbard model
Let us summarize the approximate Hubbard Hamiltonian which we have so far derived, byfurther assuming a nearest neighbor hopping:
H = −t∑σ
∑<RR′>
c†RσcR′σ +U
2
∑R
nR nR +V
2
∑<RR′>
nR nR′ +Jex2
∑<RR′>
SR · SR′
= H0 + Hint.
By construction U > V > 0 and Jex < 0. We consider the case in which the number of valenceelectrons is equal to the number of sites N . In the absence of interaction, the hopping formsa band which can accomodate 2N electrons: therefore the band is half-filled and the system ismetallic.
Exercise: Consider the nearest neighbor tight-binding model
H = −t∑σ
∑<RR′>
c†RσcR′σ,
with t > 0 in a two-dimensional square lattice. The number of electrons is equal to the numberof sites. Find the shape of the Fermi surface.
Let us analyse the opposite case of a very large U t, V, |Jex|. In this case we shouldstart from the configuration which minimizes the Coulomb repulsion U and treat what is left byperturbation theory. This lowest energy electronic configuration is the one in which each site issingly occupied. Indeed the energy cost in having just an empty site and a doubly occupied oneinstead of two singly occupied sites is given by
E(2) + E(0)− 2E(1) = 2U + 0− 2U
2= U,
and is much larger then the energy gain in letting the electrons move, which is of order t. Inthis situation the model describes an insulator, but of a particular kind. Namely the insulatingstate is driven by the strong correlation, while the conventional band structure argument wouldpredict always a metal. This correlation-induced insulator is called a Mott insulator. Therefore,as the strength of U increases with respect to the bandwidth proportional to t, an interaction
71
driven metal-to-insulator transition, commonly referred as a Mott transition, should occur ata critical Uc. However, the configuration with one electron per site is hugely degenerate, sincethe electron can have either spin up or down. There are 2N degenerate states with one electronper site, which are going to be split by the left-over terms in the Hamiltonian. The nearestneighbor interaction V is not effective in splitting the degeneracy. On the contrary, the directexchange does play a role and tends to prefer the configuration in which all spins are aligned –a ferromagnetic ordering. Yet, this is not the only source of spin correlations. Indeed also thehopping term is able to split the degeneracy within second order in perturbation theory. Let usfocus on two states which only differs in the spin configurations of two nearest neighbor sites:|σR1 , σR2〉 and |σ′R1
, σ′R2〉. Within second order in t we obtain a matrix element bewteen these
two states given by
t2∑n
∑σσ′
〈σR1 , σR2 |(c†R1σ
cR2σ+H.c.
)|n〉 1
E0 − En〈n|(c†R1σ′
cR2σ′+H.c.
)|σ′R1
, σ′R2〉,
where E0 = NU/2 is the energy of the degenerate ground states with all singly occupied sites,and |n〉 an excited state with energy En. Since the hopping creates one empty and one doublyoccupied site out of two singly occupied ones, E0 − En = −U . Moreover the sum over theintermediate states act as a complete sum, so that we can write also
− t2
U
∑σσ′
〈σR1 , σR2 |(c†R1σ
cR2σc†R2σ′
cR1σ′+ c†R2σ
cR1σc†R1σ′
cR2σ′
)|σ′R1
, σ′R2〉.
By means of (2.79) we therefore find that the operator responsable of the additional splitting is
− t2
U
∑σσ′
(c†R1σ
cR2σc†R2σ′
cR1σ′+ c†R2σ
cR1σc†R1σ′
cR2σ′
)=t2
U
∑σσ′
2 c†R1σcR1σ′
c†R2σ′cR2σ
− (nR1σ + nR2σ) δσσ′
=4t2
U
[SR1 · SR2 +
1
4(nR1nR2 − nR1 − nR2)
].
Since the last term is a constant over states where each site is singly occupied, the relevant partis just
2t2
U
∑<RR′>
SR · SR′ , (2.81)
the missing factor two coming from the fact that in the sum over nearest neighbors each bondis counted twice. Therefore the second order perturbation theory in the hopping gives rise to anovel spin exchange which is antiferromagnetic, usually called super-exchange. All together, the
72
large U effective Hamiltonian describes localized spin-1/2’s (each site occupied by one electron)coupled by the spin exchange
HHeis =J
2
∑<RR′>
SR · SR′ , (2.82)
where
J = Jex +4t2
U= −2U2 +
4t2
U.
The spin exchange may be either ferromagnetic or antiferromagnetic depending on the strengthof the direct- with respect to the super-exchange couplings. The effective Hamiltonian (2.82)describes an Heisenberg model.
2.7 Spin wave theory in the Heisenberg model
We have previoulsy shown how magnetism may emerge in a single band Mott insulator wherestrong Coulomb repulsion localizes the electrons which become effectively local moments. Wehave also shown that both the Coulomb repulsion itself, via the direct exchange, and the covalentbinding, via the super-exchange proportional to the square of the hopping, induce a couplingamong the local moments which has the form
HHeis =∑R,R′
JR,R′ SR · SR′ . (2.83)
Notice that, by lattice translational symmetry,
JR,R′ = JR′,R = JR′−R−R′,R−R−R′ = J−R,−R′ ,
namely the exchange is inversion symmetric. In the single band case we discussed, the spinshave magnitude S = 1/2, but, in what follows, we will consider the general case of arbitrary S.As a result, the Mott insulator will likely be magnetically ordered according to the propertiesof the exchange terms. If the spins were classical vectors, then the classical ground state wouldshow a magnetic order of the form
〈SR〉 = S [u cos(Q ·R) + v sin(Q ·R)] , (2.84)
with u and v two orthogonal unit vectors. Upon inserting this expression into (2.83), we find aclassical energy
E(Q) =∑R,R′
JR,R′ cos(Q · (R−R′)
)= N
∑R
JR,0 cos (Q ·R) =N
2(JQ + J−Q) = NJQ,
4
4 Since, as we showed, the exchange is inversion symmetric, namely JR,0 = J−R,0, then JQ = J∗Q = J−Q
73
which is minimized by the value of Q for which the JQ is minimum. The classical ground stateenergy does not depend on u and v, reflecting the global spin-rotational symmetry. In thereciprocal lattice, (2.84) implies an order parameter
〈Sq〉 =∑R
e−iq·R 〈SR〉 =SN
2(u− iv) δq,Q +
SN
2(u + iv) δq,−Q. (2.85)
In reality the spins are quantum operators satisfying the usual commutation relations[SaR, S
bR′
]= iεabc S
cR δRR′ ,
which, in the reciprocal lattice, transform into[Saq, S
bq′
]= iεabc S
cq+q′ .
Therefore, through the Heisenberg equation of motion
i~dSR
dt=[SR, HHeis
]= −i
∑R′
JR,R′ SR × SR′ ,
we find that
i~dSQ
dt= −i
∑q
JqN
Sq+Q × S−q.
Since the right hand side is non zero, then the classical order parameter does not correspondquantum mechanically to a conserved quantity. In turns this implies that quantum fluctuationsare going to reduce the magnitude of the classical order parameter. There is just one exceptionwhen the classical ordering wave-vector Q = 0, which corresponds to a ferromagnetic order.In this case the order parameter is just the total magnetization, which is indeed a conservedquantity. Therefore, if all J ’s are negative so that the min(Jq) is at Q = 0, the quantum groundstate remains the classical fully polarized ferromagnet. In all other cases when Q 6= 0, we donot know apriori if the classical ordered phase is going to survive quantum fluctuations. A quitesimple but effective way to analyse the role of quantum fluctuations is by the so-called spin-wavetheory.
We now show how this method works for the simple example of an antiferromagnetic Heisen-berg model with nearest neighbor exchange on an hypercubic lattice. The Hamiltonian is givenby
HHeis = J∑
<RR′>
SR · SR′ , (2.86)
with J > 0. The classical ground state is the Neel antiferromagnet with order parameter (wechoose to break the SU(2) symmetry along the z-axis)
〈SzR〉 = S cos(Q ·R) = S (−1)R,
74
where Q = π(1, 1, 1, . . . , 1)5 and, if R = (n1, n2, . . . , nd) with d the dimension of the space, thenR =
∑di=1 ni. Let us go back to the commutation relation[
Saq, Sbq′
]= iεabc S
cq+q′ ,
and assume that the left hand side is substituted by the classical average value, i.e.[Saq, S
bq′
]= iεabz NSδQ,q+q′ .
The meaning of this approximation is the following. If we want to analyse the role of quantumfluctuations, we need to study the dynamics of the spin operators which, by the equations ofmotion, is related to commutators. If the true ground state is a slight modification of theclassical one, which has to be checked aposteriori, then the commutators can be approximatedwith their classical averages. Within this approximation[
Sxq, Sy−q+Q
]' iNS.
Yet we have to further impose that [Sxq, S
zQ
]= −iSyq+Q,[
Syq, SzQ
]= iSyq+Q,
Both conditions can be fulfilled if we write
Sxq =√
SN2
(dq + d†−q
),
Syq = −i√
SN2
(dq−Q − d
†−q+Q
),
SzQ = SN −∑
q d†qdq.
(2.87)
The operators dq’s and d†q’s are bosonic and satisfy[dq, d
†q′
]= δq,q′ ,
[dq, dq′
]= 0.
The expression for the order parameter operator SzQ clarifies also the meaning of our startingassumption of a quantum ground state slightly different from the classical one. Indeed thisapproximation holds if
〈SzQ〉 ∼ SN,5Notice that Q = −Q, being the two connected by a reciprocal lattice vector
75
namely if1
N
∑q
〈d†qdq〉 S.
Therefore the larger is the spin magnitude S the more correct is the approximation, i.e. theweaker are the effects of quantum fluctuations. The Hamiltonian in the reciprocal lattice is
HHeis =1
N
∑q
J(q) Sq · S−q,
where
J(q) = 2J
d∑i=1
cos(qi) ≡ 2 J γq,
d being the dimension of the space and q = (q1, q2, . . . , qd). Notice that
J(Q) = −2 J d.
Since
J(q + Q) = 2J
d∑i=1
cos(qi + π) = −J(q),
we can also rewrite the Hamiltonian, compatibly with our assumption of weak quantum fluctu-ations, as
HHeis =1
2N
∑q
J(q) (Sq · S−q − Sq+Q · S−q+Q)
' 1
2N
∑q
J(q)(SxqS
x−q + SyqS
y−q − Sxq+QS
x−q+Q − S
yq+QS
y−q+Q
)+
1
NJ(Q)SzQS
zQ, (2.88)
At this stage we should substitute (2.87) into (2.88), keep only bilinear terms and diagonalize theHamiltonian by a canonical transformation. However it is much better to follow an equivalentroute where calculations are simpler. We introduce conjugate variables through
xq =
√1
2
(dq + d†−q
), (2.89)
pq = i
√1
2
(d†−q − dq
), , (2.90)
which satisfy the canonical commutation relation[xq, pq′
]= iδq,−q′ .
76
In terms of these variables the expressions for the spin operators read
Sxq =√SN xq,
Syq =√SN pq−Q,
SzQ =
(S +
1
2
)N − 1
2
∑q
xqx−q + pqp−q,
which substituted into (2.88) and keeping up to order S terms, lead to
HHeis =S
2
∑q
J(q) (xqx−q + pq−Qp−q+Q − xq−Qx−q+Q − pqp−q)
+1
NJ(Q)
(S2N2 + SN2 − SN
∑q
xqx−q + pqp−q
)= N J(Q)S (S + 1) + S
∑q
J(q) (xqx−q − pqp−q)
−S J(Q)∑q
(xqx−q + pqp−q)
= N J(Q)S (S + 1)
+S∑q
(J(q)− J(Q)) xqx−q + (−J(q)− J(Q)) pqp−q. (2.91)
The constant term N J(Q)S2 represents the classical Neel energy, the rest being the correctiondue to quantum mechanics. In order to diagonalize this Hamiltonian, let us consider the followingcanonical transformation
xq =√KqXq, (2.92)
pq =
√1
KqPq, (2.93)
which preserves the commutation relations provided Kq = K−q. Upon substitution, the quan-tum fluctuation term transforms into
S∑q
(J(q)− J(Q)) xqx−q + (−J(q)− J(Q)) pqp−q →
→ S∑q
Kq (J(q)− J(Q)) XqX−q +K−1q (−J(q)− J(Q)) PqP−q.
If we impose that
K2q =−J(Q)− J(q)
−J(Q) + J(q), (2.94)
77
the Hamiltonian becomes that of independent harmonic oscillators, i.e.
HHeis = N J(Q)S (S + 1) +1
2
∑q
ωq (XqX−q + PqP−q) , (2.95)
with the dispersion relation defined through
ωq = 2S√
(−J(Q)− J(q)) (−J(Q) + J(q)) = 4S J√
(d− γq) (d+ γq). (2.96)
Notice that, since d ≥ |γq|, the frequencies are positive real numbers. It is clear that theHamiltonian has a diagonal form. Indeed if we introduce new bosonic operators through
Xq =
√1
2
(bq + b†−q
),
Pq = −i√
1
2
(bq − b
†−q
),
the Hamiltonian transforms into
HHeis = N J(Q)S (S + 1) +∑q
ωq
(b†qbq +
1
2
), (2.97)
thus showing that the eigenstates are Fock states in the b-basis |nq〉 with energies
HHeis |nq〉 =
[N J(Q)S (S + 1) +
∑q
ωq
(nq +
1
2
)]|nq〉 ≡ E [nq] |nq〉.
The elementary bosonic excitations are called spin-waves. The vacuum is the ground state,whose energy per site including quantum fluctuations is
E0 = −2JdS2 − 1
N
∑q
(2SJd− 1
2ωq
)= Eclass −
2JS
N
∑q
[d−
√(d− γq) (d+ γq)
]< Eclass,
which, as expected, is lower than the classical counterpart Eclass.
Let us analyse the energy spectrum of the spin-waves. First we notice that
γq = −γq−Q,
78
hence that ωq = ωq+Q, showing that the true Brillouin zone is half of the original one. This isbecause the Neel state breaks translational symmetry and the new unit cell contains two sites.Secondly, by expanding (2.96) close to q = 0, which as we said is equivalent to q = Q, and since
γq ' d−1
2q · q,
we find that
ωq = 4S J√
(d− γq) (d+ γq) ' 4S J√d |q| ≡ v |q|,
namely a linearly vanishing dispersion with velocity v = 4S J√d. This was actually predictable
by the Goldstone theorem which states that when a continuos symmetry is broken, in our casethe spin SU(2), and in the absence of long range interaction, there should exist a mode, theGoldstone mode, with vanishing energy as q→ 0.
Let us go back to the canonical transformation (2.92) and (2.93). One can prove that thistransformation is accomplished by the unitary operator
U = exp
[− i
2
∑q
lnKq xq p−q
], (2.98)
namely thatU † xq U =
√Kq xq. (2.99)
This result can be obtained through to the following expression
e−AB eA =∞∑n=0
(−1)n
n!Cn,
where C0 = B and Cn = [A,Cn−1].Exercise: Prove the above statement, namely that (2.98) gives rise to (2.99).
Therefore the unitary transformation (2.98) applied to the Hamiltonian (2.91) gives
U † HHeis U = E0 +1
2
∑q
ωq (xqx−q + pqp−q − 1) ,
namely the desired diagonal form. This way of rewriting the canonical transformation is quiteuseful since it allows to identify the eigenstates in a simple way. Let us consider the state
|nq〉 = U |nq〉0,
where |nq〉0 is a Fock state in the old basis with the occupation number configuration nq.Then
HHeis |nq〉 = HHeis U |nq〉0
79
= U U †HHeis U |nq〉0 = U
[E0 +
∑q
ωq nq
]|nq〉0
=
[E0 +
∑q
ωq nq
]U |nq〉0 = E [nq] |nq〉,
namely it is an eigenstate with energy
E [nq] =
[E0 +
∑q
ωq nq
].
Analogously, the new vacuum state can be obtained by the old one through
|0〉 = U |0〉0.
Finally we have to check whether our approximation is valid. Let us evaluate the groundstate average value of the order parameter
M =1
N〈0|SzQ|0〉 = S +
1
2− 1
2N
∑q
〈0|xqx−q + pqp−q|0〉
= S +1
2− 1
2N
∑q
〈0|KqXqX−q +K−1q PqP−q|0〉
= S +1
2− 1
4N
∑q
Kq +K−1q
= S +1
2− 1
2N
∑q
√d+ γqd− γq
, (2.100)
where we used the fact that on the new vacuum
〈0|XqX−q|0〉 = 〈0|PqP−q|0〉 =1
2,
and that K−1q = Kq+Q. If S−M 1 then the approximation indeed holds. The only dangerous
region in the sum is when d ∼ γq, i.e. when the spin-wave energy vanishes. Here the sum behavesas
1
2N
∑q
√d+ γqd− γq
∼∫dq
|q|.
This integral is convergent for d ≥ 2, but is singular in one dimension. Therefore the spin waveapproximation is always wrong in one dimension, in agreement with Mermin-Wagner theorem
80
which states that a continuos symmetry can not be broken in 1+1 dimensions (A quantumproblem in d dimensions corresponds to a classical one in d+ 1).
To complete our analysis, let us evaluate the contribution of thermal fluctuations, namelylet us move to finite temperature T . Instead of ground state averages, we need the thermalaverages
〈XqX−q〉 = 〈PqP−q〉 =1
2(1 + 2b(ωq)) =
1
2coth
ωq
2T,
where b(ωq) is the Bose distribution function
b(x) =1
eβ x + 1.
(Exercise: Derive the above result). Therefore, at finite temperature, the order parameter is
M(T ) = S +1
2− 1
2N
∑q
√d+ γqd− γq
cothωq
2T. (2.101)
We notice that in two-dimensions the contribution of thermal fluctuations is diverging since, forωq T ,
cothωq
2T∼ 2T
ωq,
and the sum behaves likeT
v
∫dq
|q|2,
which is singular in two-dimensions. This implies that for d = only at T strictly zero there is atrue Neel order. For dimensions d > 2, Eq. (2.101) allows to estimate the value of the criticaltemperature Tc above which the order parameter vanishes through the equation M(Tc) = 0.Since J is the only dimensional coupling, one easily realize that the Neel critical temperatureTc ∼ J .
2.7.1 More rigorous derivation: the Holstein-Primakoff transformation
Let us conclude by showing more rigorously how one can derive the spin wave theory. One canreadily demonstrate that the following way of writing spin operators in terms of bosonic one(Holstein-Primakoff transformation) preserves proper spin commutation relations:
SzR = (−1)R[S − d†RdR
], (2.102)
S+R =
1
2
[1− (−1)R
]d†R√
2S − nR +1
2
[1 + (−1)R
]√2S − nR dR, (2.103)
81
S−R =1
2
[1 + (−1)R
]d†R√
2S − nR +1
2
[1− (−1)R
]√2S − nR dR, (2.104)
where nR = d†RdR. Notice that the bosonic vacuum has the properties
〈0|SzR|0〉 = (−1)R S, 〈0|S+R|0〉 = 〈0|S−R|0〉 = 0,
namely it corresponds to the classical Neel state. The square root operators in (2.103) and (2.104)assure that it is not possible to create more than 2S bosons at any given site, so that 〈SzR〉 =−S, . . . , S as it should. If one inserts the above expressions in the Heisenberg Hamiltonian andexpands it up to order S, the effective Hamiltonian (2.91) is recovered, after moving from bosonicoperators to conjugate variables.
Exercises:
• Determine how the specific heat behaves at low energy for the antiferromagnetic HeisenbergHamiltonian (2.86) in an hypercubic lattice in d ≥ 3 dimensions.
• Add to the antiferromagnetic Heisenberg Hamiltonian (2.86) with symmetry breakingalong the z-direction a magnetic field along the x-direction with Fourier components Bq,i.e. a term
δH = −∑q
B−q Sxq ' −
√SN
∑q
B−q xq.
Diagonalize the Hamiltonian in the presence of this term and calculate the ground stateenergy.
• Calculate the spin wave spectrum of the Heisenberg ferromagnetic Hamiltonian
H = −J∑
<RR′>
SR · SR′ ,
with J > 0 and where < RR′ > means sum over nearest neighbor sites on an hypercubiclattice in d dimensions.
82
Chapter 3
Linear Response Theory
In order to access the physical properties of a system, one has to act on it with some externalprobe. This amounts to add to the unperturbed Hamiltonian H0 a time-dependent perturbationof the general form
V (t) =
∫dx A(x) v(x, t), (3.1)
where v(x, t) represents the external probe which couples to the hermitean operator A(x). Ourscope is to study the effects of V (t) on some measurable quantity described e.g. by an operatorB(x), namely to calculate
B(x, t) ≡ Tr[ρ(t) B(x)
],
being ρ(t) the time-dependent density matrix in the presence of the perturbation.
3.1 Linear Response Functions
We assume that the perturbation is switched on at time t → −∞. Initially the system is inthermal equilibrium, so that the density matrix
limt→−∞
ρ(t) = ρ0, (3.2)
with
ρ0 =1
Z0e−βH0 =
1
Z0
∑n
e−βEn |φn〉〈φn|. (3.3)
Here H0 |φn〉 = En |φn〉 and Z0 =∑
n exp (−βEn). Therefore the time evolution of the density
matrix in the presence of V (t) is given by
ρ(t) =1
Z0
∑n
e−βEn |φn(t)〉〈φn(t)|, (3.4)
83
where
i~∂
∂t|φn(t)〉 =
[H0 + V (t)
]|φn(t)〉, (3.5)
is the Shrœdinger equation which determines the evolution of the eigenstates of the unperturbedHamiltonian in the presence of the perturbation. The meaning of (3.4) is that initially thesystem is described by a statistical ensamble of sub-systems, each in a given eigenstate of H0
and weigthed by the Boltzmann factor. After we switch on the perturbation, |φn〉 ceases to bean eigenstate of the perturbed Hamiltonian, so it acquires a non-trivial time evolution.
Through Eq. (3.5) one readily finds the equation of motion for the density matrix
i~∂
∂tρ(t) =
[H0 + V (t), ρ(t)
]. (3.6)
We introduce the Dirac, also called interaction, representation of the density matrix as
ρD(t) = eiH0t/~ ρ(t) e−iH0t/~,
which satisfies
i~∂
∂tρD(t) = −
[H0, ρD(t)
]+ eiH0t/~
[H0 + V (t), ρ(t)
]e−iH0t/~
=[VD(t), ρD(t)
], (3.7)
where
VD(t) =
∫dx eiH0t/~ A(x) e−iH0t/~ v(x, t) =
∫dx A(x, t) v(x, t),
being A(x, t) the Heisenberg evolution of A(x) with the unperturbed Hamiltonian. We solve
(3.7) perturbatively in v, i.e. ρD(t) = ρ(0)D (t) + ρ
(1)D (t) + . . . , where ρ
(n)D (t) contains n-powers of
the perturbation. Obviously
limt→−∞
ρD(t) = ρ0 = ρ(0)D .
We will limit our analysis to the linear response, hence we just need the first order term whichsatisfies
i~∂
∂tρ
(1)D (t) =
[VD(t), ρ
(0)D (t)
],
with solution
ρ(1)D (t) = − i
~
∫ t
−∞dt′[VD(t′), ρ0
]. (3.8)
Therefore, at linear order,
B(x, t) = Tr[ρ(t) B(x)
]= Tr
[ρD(t) eiH0t/~ B(x) eiH0t/~
]84
= Tr[ρD(t) B(x, t)
]= Tr
[ρ0 B(x, t)
]+Tr
[ρ
(1)D (t) B(x, t)
]= B(x)0 + Tr
[ρ
(1)D (t) B(x, t)
]where we used the fact that
Tr[ρ0 B(x, t)
]= Tr
[ρ0 B(x)
]= B(x)0,
is the unperturbed average value. We then find that the variation of the average value is givenby
B(x, t)−B(x)0 = − i~
∫ t
−∞dt′Tr
[VD(t′), ρ0
]B(x, t)
= − i
~
∫ t
−∞dt′∫dy Tr
ρ0
[B(x, t), A(y, t′)
]v(y, t′)
= − i~
∫ ∞−∞
dt′∫dy θ(t− t′)Tr
ρ0
[B(x, t), A(y, t′)
]v(y, t′)
≡∫ ∞−∞
dt′∫dyχBA(x,y; t− t′) v(y, t′), (3.9)
with the linear response function defined through
χBA(x,y; t− t′) = − i~θ(t− t′) 〈
[B(x, t), A(y, t′)
]〉, (3.10)
where 〈. . . 〉 means a thermal and quantum average with the unperturbed density matrix andwe recall that the operators evolve in time with H0.
Eq. (3.9) shows that, at linear order, the variation of any measurable quantity is obtainedthrough the linear response function (3.10) which is only related to averages on the unperturbedsystem.
We conclude by noticing that any average of pairs of time-evolved operators in the Heisenbergrepresentation only depends on the time-difference, since the Schrœdinger equation is time-translationally invariant. In fact
〈B(t) A(t′)〉 =1
ZTr(
e−β H eiH t/~ B e−iH t/~ eiH t′/~ A e−iH t′/~)
=1
ZTr(
e−β H eiH (t−t′)/~ B e−iH (t−t′)/~ A)
= 〈B(t− t′) A(0)〉
=1
ZTr(
e−β H B eiH (t′−t)/~ A e−iH (t′−t)/~)
= 〈B(0) A(t′ − t)〉.
85
3.2 Kramers-Kronig relations
Let us now study the analytical properties of the response function in the frequency domain. Inthe following we drop the space-coordinate dependence of the response function, which is notrelevant for what we are going to demonstrate. The response of an operator A in the presenceof an external probe which couples to B is therefore
χAB(t) = − i~θ(t) 〈
[A(t), B
]〉, (3.11)
where
A(t) = eiH0t~ A e−i
H0t~ .
The response function (3.11) vanishes for t < 0, which is a consequence of causality. Weintroduce the Fourier transform through
χAB(t) =
∫ ∞−∞
dω
2πe−iωtχAB(ω), (3.12)
as well as its analytical continuation in the complex frequency plane χAB(z). If we assume, asit is always the case, that χAB(z) does not diverge exponentially for |z| → ∞, we can regard(3.12) as the result of a contour integral
χAB(t) =
∮dz
2πe−iztχAB(z),
where the contour is in the upper half plane for t < 0 and in the lower for t > 0. The integralcatches all poles lying inside the contour. Since χAB(t) = 0 for t < 0, it follows that
• as consequence of causality χAB(z) is analytic in the upper half plane. •
Let us now consider the contour drawn in Fig. 3.1. Since there are no poles enclosed by thecontour, it is trivially zero the integral∮
Cdz
χAB(z)
ω − z= 0. (3.13)
On the other hand the above integral is also equal to the line integral along the lower edge,hence
0 =
∫ ω−ε
−∞+
∫ ∞ω+ε
dω′
χAB(ω′)
ω − ω′+
∮z=ω+ε exp(iθ);θ∈[π,0]
dzχAB(z)
ω − z
= P∫ ∞−∞
dω′χAB(ω′)
ω − ω′− i∫ 0
πdθ χAB
(ω + ε eiθ
),
86
ω
Re(z)
Im(z)
2ε
Figure 3.1: Integration contour in Eq. (3.13).
the symbol P denoting the principal value of the integral. In the limit ε → 0, the aboveexpression simplifies into
P∫ ∞−∞
dω′χAB(ω′)
ω − ω′+ i π χAB(ω) = 0, (3.14)
which implies that
ImχAB(ω) = P∫ ∞−∞
dω′
π
Re χAB(ω′)
ω − ω′, (3.15)
Re χAB(ω) = −P∫ ∞−∞
dω′
π
ImχAB(ω′)
ω − ω′, (3.16)
known as the Kramers-Kronig relations. Therefore, because of causality, the real and imaginaryparts of the response function are not independent. It is possible to rewrite both expressions as
χAB(ω) =
∫ ∞−∞
dω′
π
ImχAB(ω′)
ω′ − ω − iη, (3.17)
with η an infinitesimal positive number.
3.2.1 Symmetries
Let us introduce back the space dependence, so that
χAB(x,y; t− t′) = − i~θ(t− t′) 〈
[A(x, t), B(y, t′)
]〉. (3.18)
87
Since both operators are hermitean, it follows that
χAB(x,y; t− t′)∗ =i
~θ(t− t′) 〈
[B(y, t′), A(x, t)
]〉 = χAB(x,y; t− t′). (3.19)
By definition
χAB(x,y;ω) =
∫dt eiωt χAB(x,y; t),
therefore, through (3.19), we find that
χAB(x,y;ω)∗ =
∫dt e−iωt χAB(x,y; t)∗ = χAB(x,y;−ω).
This implies that
Re χAB(x,y;ω) =1
2[χAB(x,y;ω) + χAB(x,y;−ω)] ,
is even in frequency, while
ImχAB(x,y;ω) =1
2i[χAB(x,y;ω)− χAB(x,y;−ω)] ,
is odd.
3.3 Fluctuation-Dissipation Theorem
Let us introduce other types of correlation functions. The first are the so-called structure factorsdefined through
SAB(x,y; t) =1
~〈A(x, t) B(y)〉. (3.20)
In addition we introduce the dissipation response function
χ′′AB(x,y; t) =
1
2~〈[A(x, t), B(y)
]〉 =
1
2[SAB(x,y; t)− SBA(y,x;−t)] , (3.21)
whose meaning will be explained in the following section, as well as the fluctuation one
FAB(x,y; t) =1
2〈A(x, t), B(y)
〉 =
~2
[SAB(x,y; t) + SBA(y,x;−t)] . (3.22)
One readily verifies that the former is related to the response function through
χ′′AB(x,y; t) =
i
2[χAB(x,y; t)− χBA(y,x;−t)] ,
88
which in the frequency domain reads
χ′′AB(x,y;ω) =
i
2[χAB(x,y;ω)− χBA(y,x;−ω)] . (3.23)
In particularχ′′AA(x,x;ω) = −ImχAA(x,x;ω). (3.24)
Through the definition (3.20) we find that
SBA(y,x;−t) =1
Z0Tr(
e−βH0 e−iH0t~ B(y) eiH0
t~ A(x)
)=
1
Z0Tr(
e−βH0 eiH0(t−iβ~)
~ A(x) e−iH0(t−iβ~)
~ B(y))
= SAB(x,y; t− i~β).
Therefore
SBA(y,x;−ω) =
∫dt e−iωtSBA(y,x; t) =
∫dt eiωtSBA(y,x;−t)
=
∫dt eiωtSAB(x,y; t− i~β) = e−β~ω SAB(x,y;ω), (3.25)
namely
χ′′AB(x,y;ω) =
1
2SAB(x,y;ω)
(1− e−β~ω
), (3.26)
FAB(x,y;ω) =~2SAB(x,y;ω)
(1 + e−β~ω
). (3.27)
In other words the following relation holds
FAB(x,y;ω) = ~ coth
(β~ω
2
)χ′′AB(x,y;ω), (3.28)
which is the so-called fluctuation-dissipation theorem. Indeed, if A = B and x = y, FAA(x,x; t =0) is an estimate of the fluctuations of A. On the other hand
FAA(x,x; t = 0) =
∫dω
2πFAA(x,x;ω) (3.29)
= ~∫dω
2πcoth
(β~ω
2
)χ′′AA(x,x;ω), (3.30)
which relates the fluctuations to the dissipation.
89
3.4 Spectral Representation
The spectral representation of the response functions gives instructive information about theirphysical meaning. Let us start from the structure factor (3.20) which can be written as (in thefollowing we do not explicitly indicate the space dependence)
SAB(t) =1
~Z∑n
e−βEn 〈n|eiHt/~ A e−iHt/~ B|n〉
=1
~Z∑nm
e−βEn ei(En−Em)t/~〈n|A|m〉〈m|B|n〉.
After Fourier transformation we get
SAB(ω) =2π
Z
∑nm
e−βEn 〈n|A|m〉〈m|B|n〉 δ (~ω + En − Em) . (3.31)
The meaning is now self-evident. The matrix element 〈m|B|n〉 is the transition amplitudefor the excitation of the initial state |n〉 into the final one |m〉 induced by the operator B,while 〈n|A|m〉 decribes the reverse process but now induced by A. The excitation followed byrelaxation process is weigthed by the Boltzmann factor for the initial state and contributes toSAB(ω) only if the energy difference Em − En is ~ω. Thus SAB(ω) is a spectral function whichmeasures the transition amplitude for excitations induced by B and de-excitation induced by Awith a given energy ~ω.
Through (3.25) we also find that
χ′′AB(ω) =
π
Z
∑nm
e−βEn(
1− e−β~ω)〈n|A|m〉〈m|B|n〉 δ (~ω + En − Em)
=π
Z
∑nm
(e−βEn − e−βEm
)〈n|A|m〉〈m|B|n〉 δ (~ω + En − Em) , (3.32)
which means that χ′′AB(ω) is the transition amplitude for |n〉 → |m〉 induced by B and |m〉 → |n〉
induced by A weighted by the occupation probability
pn =e−βEn
Z,
of the initial state |n〉 minus that one
pm =e−βEm
Z,
of the final state |m〉. We notice that
pn − pm = pn(1− pm)− pm(1− pn),
90
namely it is the probability of n being occupied and m empty, minus the opposite. In otherwords, χ
′′AB(ω) measures the absorption minus the emission probability of an energy ~ω, namely
the total absorption probability.Finally, one can analogously derive the spectral representation of the response function χAB,
χAB(t) = − i~θ(t)
1
Z
∑nm
e−βEn[ei(En−Em)t/~ 〈n|A|m〉〈m|B|n〉 − e−i(En−Em)t/~ 〈n|B|m〉〈m|A|n〉
]= − i
~θ(t)
1
Z
∑nm
〈n|A|m〉〈m|B|n〉 ei(En−Em)t/~(
e−βEn − e−βEm).
For the Fourier transform one has to evaluate the integral
−i∫ ∞
0dt eiωt ei(En−Em)t/~.
Since the perturbation has been assumed to be switched on at very early times, a meaningfulregularization of the above integral is
−i∫ ∞
0dt eiωt ei(En−Em)t/~ e−ηt/~ =
~~ω − (Em − En) + iη
,
where η/~ is the switching rate of the perturbation, and is taken to be an infinitesimal positivenumber. As a result we find that
χAB(ω) =1
Z
∑nm
〈n|A|m〉〈m|B|n〉 e−βEn − e−βEm
~ω − (Em − En) + iη. (3.33)
3.5 Power dissipation
Till now we have formally introduced several response functions. In this section and in thefollowing ones we are going to show how those functions emerge in real experiments.
Let us first analyse the power dissipated in the presence of the perturbation. Given ourstarting assumption about the time-evolution of the density matrix (3.4), it is clear that theentropy defined through the phase space occupied by the statistical ensamble remains constantand equal to the thermal equilibrium one, S0. Therefore the system free energy is
F (t) = U(t)− TS0 = (U(t)− U0) + F0,
so that∂F (t)
∂t=∂U(t)
∂t.
91
On the other hand, since
U(t) = Tr(ρ(t) H0
),
it follows that
∂U(t)
∂t= − i
~Tr([H0 + V (t), ρ(t)
]H0
)= − i
~Tr(ρ(t)
[H0, H0 + V (t)
])= − i
~Tr(ρ(t)
[H0, V (t)
]). (3.34)
We assume that the perturbation has the general form
V (t) =∑J
AJ vJ(t),
so that∂U(t)
∂t= − i
~∑J
vJ(t) Tr(ρ(t)
[H0, AJ
])=i
~∑J
vJ(t) 〈[AJ , H0
]〉(t), (3.35)
where the last term denotes the average value of the operator within the bracket in the presenceof the time-dependent perturbation. We further notice that
AJ(t) = Tr(ρ(t) AJ
)=∑J ′
∫dt′ χJJ ′(t− t′) vJ ′(t′),
where
χJJ ′(t− t′) = − i~θ(t− t′) 〈
[AJ(t), AJ ′(t
′)]〉.
Therefore, by recalling that the operators evolve with the unperturbed Hamiltonian, we findthat
i~∑J ′
∫dt′
∂
∂tχJJ ′(t− t′) vJ ′(t′) =
∑J ′
∫dt′ δ(t− t′)〈
[AJ(t), AJ ′(t)
]〉 vJ ′(t′)
−∑J ′
∫dt′
i
~θ(t− t′) 〈
[[AJ(t), H0
], AJ ′(t
′)]〉 vJ ′(t′)
=∑J ′
〈[AJ(t), AJ ′(t)
]〉 vJ ′(t) + 〈
[AJ(t), H0
]〉.
After inserting into (3.35) we finally get
∂U(t)
∂t= −
∑J,J ′
∫dt′
∂
∂tχJJ ′(t− t′) vJ(t) vJ ′(t
′)
92
− i~∑J,J ′
〈[AJ(t), AJ ′(t)
]〉 vJ(t) vJ ′(t).
The last term vanishes since the commutator is odd by interchanging J with J ′, while vJ(t) vJ ′(t)is even. Hence
∂U(t)
∂t= −
∑J,J ′
∫dt′
∂
∂tχJJ ′(t− t′) vJ(t) vJ ′(t
′). (3.36)
Let us write
vJ(t) =1
2
(vJ e−iωt + v∗J eiωt
), (3.37)
and define the power dissipated within a cycle, W , through
W =ω
2π
∫ 2π/ω
0dt∂U(t)
∂t. (3.38)
By performing the integral and by means of (3.36) and (3.37) we obtain for W the expression:
W = iω
4
∑J,J ′
v∗JvJ ′ [χJJ ′(ω)− χJ ′J(−ω)] =ω
2
∑J,J ′
v∗J χ′′JJ ′(ω) vJ ′ , (3.39)
where Eq. (3.23) has been used. The power dissipated during a cycle is proportional to what wedefined as the dissipation response function, thus explaining its name. Indeed, as we showed inthe previous section, χ
′′JJ ′(ω) measures the probability of energy absorption during the process,
hence its appearance in Eq. (3.39) it is not unexpected. We notice that, since W > 0, it derivesthat ω χ
′′JJ ′(ω) is a positive-definite quadratic form. In particular
ω χ′′JJ(ω) = −ω ImχJJ(ω) > 0,
namely the imaginary part of χJJ(ω) is positive for ω < 0 and negative otherwise.
3.5.1 Absorption/Emission Processes
The power dissipation is related to the absorption minus emission probability. However, thereare other measurements where only absorption or emission is revealed. For instance one can shoton a sample with a beam of particles, either photons, neutrons, electrons etc..., and measurethe absorption probability of an energy ~ω. If the coupling between the beam and the sampleis represented by an operator A, the Fermi golden rule tells us that the absorption rate per unittime of an ensamble at thermal equilibrium is
PA(ω) =2π
Z
∑if
e−βEi∣∣∣〈f |A|i〉∣∣∣2 δ (~ω + Ei − Ef ) = SAA(ω), (3.40)
which enlights the meaning of the structure factors.
93
3.5.2 Thermodynamic Susceptibilities
Let us consider again a perturbation of the form
V (t) =∑J
∫dx AJ(x) vJ(x, t).
Let us further assume that the only time-dependence of the external probes vJ(x, t) comes froma very slow switching rate (adiabatic switching) which just sets the proper regularization oftime-integrals as in Eq. (3.33). Hence, for times far away from the time at which the pertur-bation is switched on, the external probes become constant in time, vJ(x, t) = vJ(x), and thethermodynamic averages lose any time-dependence. In this limit
AJ(x)−AJ(x)0 =∑J ′
∫ ∞−∞
dt′∫dyχJ J ′(x,y; t− t′) vJ ′(y)
=∑J ′
∫dyχJ J ′(x,y;ω = 0) vJ ′(y). (3.41)
Let us now consider a generically perturbed Hamiltonian of the form
H = H0 +∑J
∫dy AJ(y) vJ(y),
which therefore admits stationary eigenstates. The perturbed free-energy turns out to be afunctional of the external fields vJ(y), F = F [vJ(y)]. Standard thermodynamics tells us that
〈AJ(x)〉 =δF
δvJ(x),
which is in general different from its average value, 〈AJ(x)〉0, in the absence of external fields.For very small external fields one finds at linear order that
〈AJ(x)〉 − 〈AJ(x)〉0 =∑J ′
∫dy
(δ2F
δvJ(x) δvJ ′(y)
)v=0
vJ ′(y), (3.42)
where the second derivatives of the unperturbed free-energy are the so-called thermodynamicsusceptibilities. Comparing (3.42) with (3.41) one obtains that(
δ2F
δvJ(x) δvJ ′(y)
)v=0
= χJ J ′(x,y;ω = 0), (3.43)
which relates thermodynamic susceptibilities to the response functions at zero frequency. Noticethat thermodynamic stability implies that −χJ J ′(x,y;ω = 0) is positive definite.
94
Exercises
(1) Consider a free electron Hamiltonian
H0 =∑kσ
εk c†kσckσ,
in the presence of a Zeeman splitting due to a slowly varying magnetic field oriented alongthe z-direction B(q, t). The perturbation is
V (q, t) =1
VµB g B(q, t)σ−q,
where V is the volume, µB the Bohr magneton, g ' 2 the electron gyromagnetic ratio andσq the spin density operator at momentum q defined through
σq =1
2
∑k
(c†k↑ck+q↑ − c
†k↓ck+q↓
).
By linear response theory, the average of the spin-density operator at momentum q is
〈σq〉(t) =
∫dt′ χ(q, t− t′)
(gµB B(q, t′)
),
where
χ(q, t− t′) = − i~
1
Vθ(t− t′)〈[σq(t), σ−q]〉,
is the magnetic response function per unit volume. (Prove that for any p 6= q 〈[σp(t), σ−q]〉 =0, so that only the Fourier component q is affected by the magnetic field.)
• Find the formal expression of
χ(q, ω) =
∫dt eiωt χ(q, t).
• Calculate the magnetic susceptibility, namely
χ = limq→0
χ(q, ω = 0),
at zero temperature in terms of the density of states
N(ε) =1
V
∑k
δ(ε− εk).
95
µ
L
R
IT
Figure 3.2: Geometry of the tunneling problem (2).
• Prove and discuss thatχ(q = 0, ω) = 0.
(2) Consider two disconnected metallic leads, a right one (R) and a left one (L), see Fig. 3.2,described by the unperturbed Hamiltonian
H0 =∑kσ
εk
(c†RkσcRkσ + c†LkσcLkσ
).
At some time, a voltage bias V is applied, implying that the two leads are kept at differentchemical potential, i.e. that the following perturbation is added to the Hamiltonian
Vµ =µ
2
∑kσ
(c†RkσcRkσ − c
†LkσcLkσ
)=µ
2(NR −NL) ,
with µ = eV . At the same time a tunnelling between the two leads is switched on, whichis described by the additional perturbation
VT = −∑kσ
T (εk, εp) c†RkσcLpσ +H.c.,
where we further assume that the tunneling amplitudes T (εk, εp) depends only on theenergy. Therefore the fully perturbed Hamiltonian is
H = H0 + Vµ + VT . (3.44)
• Using the Heisenberg equation of motion with the Hamiltonian (3.44), calculate theexpression of the operator of the current flowing from the L to the R lead, which isdefined through
I =∂
∂t(NR −NL) = − i
~[(NR −NL) ,H] .
96
• Extending the derivation of the linear response up to second order in the perturbation,calculate the average value of the current I = 〈I〉. (One needs the second orderdensity matrix ρ(2) in the perturbation Vµ + VT , and in particular the mixed termwhich derives from Vµ VT .) For the explicit calculation it is convenient to introducethe density of states N(ε), which is by definition equal for the L and R leads.
97
Chapter 4
Hartree-Fock Approximation
In this chapter we describe the simplest approximate technique to study a model of interactingfermions: the Hartree-Fock approximation. This technique is variational and can be appliedboth at zero and at finite temperature. Essentially within the Hartree-Fock approximation theeffects of the particle-particle interaction are simulated by an effective external field acting onthe particles which is self-consistently generated by the same particles. This is also the reasonwhy the Hartree-Fock approximation is a Mean Field theory.
The main body is devoted to the Hartree-Fock approximation for interacting fermions. Inthe last Section the case of bosons is briefly discussed in connection with superfluidity.
4.1 Hartree-Fock Approximation for Fermions at Zero Temper-ature
The Hartree-Fock (HF) approximation at zero temperature consists in
• searching for a Slater determinant which minimizes the total energy.
Since in general the ground state wavefunction is not a single Slater determinant, the HF ap-proach is variational hence the HF energy is an upper bound to the true ground state energy.
The trial HF wavefunction is a Slater determinant, namely
ΦHF (x1, . . . , xN ) =1√N !
∣∣∣∣∣∣∣∣∣φ1(x1) · · · φ1(xN )φ2(x1) · · · φ2(xN )
............
...φN (x1) · · · φN (xN )
∣∣∣∣∣∣∣∣∣ , (4.1)
where N is the electron number and the sigle particle wavefunctions φi(x) are for the meanwhile
98
unknown. They are assumed to belong to a complete set of orthonormal wavefunctions∫dxφi(x)∗φj(x) = δij . (4.2)
The Hamiltonian in first quantization is given by
H =N∑i=1
T (xi, pi) +1
2
∑i 6=j
U(xi, xj),
where
T (x, p) = − ~2
2m∇2 + V (x)
is a single particle contribution including the kinetic energy as well as a potential term.The average value of the Hamiltonian over the wavefunction (4.1) is
EHF =〈ΦHF |H|ΦHF 〉〈ΦHF |ΦHF 〉
. (4.3)
One has to impose that the variation of EHF vanishes upon varying the trial wavefunction,keeping it still of the form of a Slater determinant. Most generally, this amounts to change oneof the single particle wavefunctions, namely
φi(x)→ Ni (φi(x) + δφi(x)) . (4.4)
Clearly, if the variation δφi(x) has a finite overlap with anyone of the φk(x)’s already presentin (4.1), the Slater determinant does not change, hence such a variation is irrelevant. Thereforethe only meaningful possibility is that
δφi(x) = η φj(x), (4.5)
where η is an infinitesimal quantity and φj belongs to the set of wavefunctions (4.2) but doesnot appear in (4.1), namely j > N . The normalization Ni in Eq. (4.4) is given by
N−2i =
∫dx (φi(x)∗ + η∗φj(x)∗) (φi(x) + ηφj(x)) = 1 + |η|2.
Therefore Ni ' 1 at linear order in η. This implies that also the normalization of the Slaterdeterminant ΦHF + ηδΦHF remains one at linear order in η, hence
δEHF = η∗〈δΦHF |H|ΦHF 〉+ η〈ΦHF |H|δΦHF 〉= Re η Re〈δΦHF |H|ΦHF 〉+ Imη Im〈δΦHF |H|ΦHF 〉 = 0.
99
Since η is an arbitrary infinitesimal number, this implies that
Re〈δΦHF |H|ΦHF 〉 = Im〈δΦHF |H|ΦHF 〉 = 0,
namely〈δΦHF |H|ΦHF 〉 = 0. (4.6)
Let us rephrase everything in second quantization. We associate to any of the wavefunctionsφi(x) an annihilation and a creation operator, ci and c†i , respectively. The wave-function (4.1)is given by
|ΦHF 〉 =N∏i=1
c†i |0〉,
while the variation|δΦHF 〉 = c†jci |ΦHF 〉,
with j > N while i ≤ N . The Hamiltonian in second quantized form is given by
H =∑ij
tij c†icj +
1
2
∑ijkl
Uijkl c†ic†jckcl , (4.7)
where the parameters are the matrix elements over the basis set (4.2). What we need to solveis therefore the equation
〈ΦHF |c†icjH|ΦHF 〉 = 0,
where i ≤ N and j > N . For that we need the following two equalities, which can be easilyderived: ∑
kl
tkl〈ΦHF |c†icjc†kcl |ΦHF 〉 = tji
1
2
∑klmn
Uklmn〈ΦHF |c†icjc†kc†l cmcn|ΦHF 〉
=1
2
N∑m=1
Ujmmi + Umjim − Ujmim − Umjmi
=N∑m=1
Ujmmi − Ujmim,
where the last expression comes from the symmetry relation
Uijkl = Ujilk.
100
Eq. (4.6) implies that
tji +N∑m=1
Ujmmi − Ujmim = 0, (4.8)
if j > N and i ≤ N . In other words, the Slater determinant which minimizes the total energyis constructed by single particle wavefunctions φi’s which have matrix elements obeying (4.8).Let us suppose that we have found instead a set of wavefunctions φi’s satisfying
tji +
N∑m=1
Ujmmi − Ujmim = εi δij . (4.9)
This set authomatically satisfies also (4.8), hence it does solve our variational problem. In firstquantization Eq.(4.9) reads
T (x, p)φi(x) +N∑m=1
∫dy U(x, y) (φm(y)∗φm(y)φi(x)− φm(y)∗φm(x)φi(y)) = εiφi(x), (4.10)
which is the standard Hartree-Fock set of equations.Notice that there might me several Slater determinants built up using N of the wavefunctions
solving (4.9) which would satisfy the HF variational principle. Among them one has to find theSlater determinant which makes minimum the total energy for N electrons, which can be easilyfound to be
EHF (N) = 〈ΦHF |H|ΦHF 〉 =
N∑i=1
tii +1
2
N∑i,m=1
Uimmi − Uimim
=
N∑i=1
εi −1
2
N∑i,m=1
Uimmi − Uimim. (4.11)
Notice that, if for N − 1 particles the Hartree-Fock single-particle wavefunctions stay approxi-mately invariant, then
EHF (N − 1) 'N−1∑i=1
tii +1
2
N−1∑i,m=1
Uimmi − Uimim
=
N∑i=1
tii (1− δiN ) +1
2
N∑i,m=1
(Uimmi − Uimim) (1− δiN ) (1− δmN )
= EHF (N)− εN , (4.12)
showing that the Hartree-Fock single particle energies correspond approximately to the ioniza-tion energies.
101
4.1.1 Alternative approach
The Hartree-Fock equations (4.10) are complicated non-linear integral-differential equations.This is especially true if one does not impose any constraint on the form of the variationalSlater determinant dictated for instance by the symmetry properties of the Hamiltonian, whatis called unrestricted Hartree-Fock approximation. However, very often one expects that thetrue ground state has well defined properties under symmetry transformations which leave theHamiltonian invariant. For instance, if the Hamiltonian is translationally and spin-rotationallyinvariant, one may expect that the true ground state is an eigenstate of the total spin and of thetotal momentum. For this reason one would like to search for a variational wavefunction withinthe subspace of Slater determinants which are eigenstates of the total spin and momentum.This amounts to impose symmetry constraints on the general form (4.1) of the variationalwavefunction. Yet, this is not a simple task if one keeps working within first quantization.The second quantization approach to the Hartree-Fock approximation that we describe in thefollowing has the big advantage to allow an easy implementation of such symmetry constraints.
Let us suppose we have our Hamiltonian written in a basis of single particle wavefunctionsφα(x) which is more convenient to work with (for instance Block waves)
H =∑αβ
tαβ c†αcβ +
1
2
∑αβγδ
Uαβγδ c†αc†βcγcδ. (4.13)
Our scope is to find the basis φi(x) which solves the Hartree-Fock equations (4.9). TheHartree-Fock wavefunction (4.1), being a Slater determinant, should be the ground state of asingle-particle Hamiltonian, which we define as HHF . We write such an Hamiltonian in thefollowing general form
HHF =∑αβ
hαβ c†αcβ, (4.14)
where we have introduced a set of unknown variational parameters satisfying hαβ = h∗βα, forthe Hamiltonian to be hermitean. The ignorance about the basis set φi(x) is reflected in theignorance about the h’s. The Hartree-Fock wavefunction satisfies:
HHF | ΦHF 〉 = E | ΦHF 〉, (4.15)
namely is the lowest energy eigenstate of HHF . We define as
∆αβ = 〈ΦHF | c†αcβ | ΦHF 〉, (4.16)
the average values of all one-body operators on the wavefunction, which are therefore functionalof the variational parameters h. Since
E = 〈ΦHF | HHF | ΦHF 〉 =∑αβ
hαβ ∆αβ, (4.17)
102
it follows that∂E
∂hµν= ∆µν +
∑αβ
hαβ∂∆αβ
∂hµν. (4.18)
On the other hand, since | ΦHF 〉 is an eigestate of the Hamiltonian HHF , the Hellmann-Feynmantheorem holds1 that states
∂E
∂hµν= 〈ΦHF |
∂HHF
∂hµν=| ΦHF 〉 = ∆µν . (4.19)
By comparing (4.18) with (4.19) we therefore obtain that∑αβ
hαβ∂∆αβ
∂hµν= 0, (4.20)
for any hµν .On the other hand, the average value of the original Hamiltonian (4.13) on the Hartee-Fock
wavefunction is readily found to be
EHF = 〈ΦHF | H | ΦHF 〉
=∑αβ
tαβ ∆αβ +1
2
∑αβγδ
∆αβ∆γδ
(Uαγδβ − Uαγβδ
), (4.21)
which is also a functional of the h’s. Minimization of EHF requires as a necessary condition that
∂EHF∂hµν
=∑αβ
∂∆αβ
∂hµν
[tαβ +
∑γδ
∆γδ
(Uαγδβ − Uαγβδ
)]= 0. (4.22)
1 The Hellmann-Feynman theorem states that, if | ψ〉 is the normalized eigenstate with eigenvalue E of aHamiltonian H that depends on some parameter λ, then
∂E
∂λ= 〈ψ | ∂H
∂λ| ψ〉.
This theorem can be easily proved. Indeed
∂E
∂λ= 〈ψ | ∂H
∂λ| ψ〉+ 〈∂ψ
∂λ| H | ψ〉+ 〈ψ | H | ∂ψ
∂λ〉
= 〈ψ | ∂H∂λ| ψ〉+ E
(〈∂ψ∂λ| ψ〉+ 〈ψ | ∂ψ
∂λ〉)
= 〈ψ | ∂H∂λ| ψ〉+ E
∂〈ψ | ψ〉∂λ
= 〈ψ | ∂H∂λ| ψ〉,
since |ψ〉 has norm equal to one for any λ.
103
The Hartree-Fock wavefunction, hence the variational parameters hµν , must therefore satisfyboth (4.20) and (4.22), which implies that
hαβ = tαβ +∑γδ
∆γδ[h](Uαγδβ − Uαγβδ
). (4.23)
Since the average values ∆γδ[h] are themselves functional of the h’s, the above set of equationsis actually a self-consistency condition for determining the latter variational parameters. Theseequations are fully equivalent to solving the Hartree-Fock non-linear differential equations thatare found in first quantization with one major advantage. Indeed, the symmetry properties of| ΦHF 〉 are completely determined by the parameters hαβ. In turns, this implies that we can
implement any desired symmetry operation O by imposing that[O, HHF
]= 0,
which leads to simple conditions to be imposed on the hαβ. For instance, if the quantum numberα = (a, σ) include an orbital index a and spin σ, and we would like to enforce full spin-rotationalsymmetry, then we must assume that
h(aσ)(bσ′) = hab,
does not depend on the spin indices. If we expect that spin SU(2) is lowered down to U(1),namely the rotation around the magnetization axis, which we can always assume to be also thequantization axis, then we have to impose that, while h(aσ)(bσ) are finite, h(a↑)(b↓) and h(a↓)(b↑)are identically zero.
We note that (4.23) could be also regarded as the definition of hαβ in terms of new variationalparameters ∆γβ that, only at the end, must be determined in such a way that
∆αβ = 〈ΦHF | c†αcβ | ΦHF 〉, (4.24)
the average being done on a choosen eigenstate of
HHF =∑αβ
c†αcβ
[tαβ +
∑γδ
∆γδ
(Uαγδβ − Uαγβδ
)], (4.25)
which depends parametrically on the ∆’s. Moreover it follows that the variational energy
EHF
[∆]
= 〈ΦHF | H | ΦHF 〉
= 〈ΦHF | HHF | ΦHF 〉 −1
2
∑αβγδ
∆αβ ∆γδ
(Uαγδβ − Uαγβδ
)
104
= E[h[∆]]− 1
2
∑αβγδ
∆αβ ∆γδ
(Uαγδβ − Uαγβδ
), (4.26)
also becomes functional of the ∆’s. Through Eqs. (4.19) and (4.23) it follows that
∂E
∂∆γδ=
∑αβ
∂EHF∂hαβ
∂hαβ∂∆γδ
=∑αβ
∆αβ
(Uαγδβ − Uαγβδ
). (4.27)
The equations (4.23), (4.25), (4.26) and (4.27) suggest two alternative ways of stating theHartee-Fock variational problem, which are very convenient to implement:
Given the interacting Hamiltonian
H =∑αβ
tαβ c†αcβ +
1
2
∑αβγδ
Uαβγδ c†αc
†βcγcδ
then the Hartree-Fock variational wavefunction |ΦHF 〉 is the ground state withenergy E of the single particle Hamiltonian
HHF =∑αβ
tαβ c†αcβ +
∑αβγδ
c†αcβ∆γδ (Uαγδβ − Uαγβδ)
where the parameters ∆αβ have to be determined self-consistently by imposingeither of the following two conditions
Condition 1.:
∆αβ = 〈ΦHF |c†αcβ|ΦHF 〉
Condition 2.:δ
δ∆µν
(E[∆]−−
1
2
∑αβγδ
∆αβ ∆γδ
(Uαγδβ − Uαγβδ
))= 0
105
4.2 Hartree-Fock approximation for fermions at finite temper-ature
A variational approach of the Hartree-Fock type can also be implemented at finite temperatureas a variational minimization of the free energy. Before presenting the method, it is necessaryto introduce some preliminary results.
4.2.1 Preliminaries
Let us consider a generic Hamiltonian
H = H0 + V ,
where H0 is the unperturbed Hamiltonian while V is the perturbation, e.g. the interaction,which does not commute with H0. The partition function is defined through
Z = e−βF = Tr(
e−βH). (4.28)
We assume that the free energy can be expanded in powers of the perturbation, namely
F =∑n=0
F (n),
where F (n) contains n-powers of V .Let us define for real τ ’s
e−Hτ = e−H0τ S(τ), (4.29)
implying that
S(τ) = eH0τe−Hτ .
From this expression it follows that
∂S
∂τ= eH0τ
(H0 − H
)e−Hτ
= −eH0τ V e−H0τeH0τe−Hτ = −V (τ)S(τ), (4.30)
where we define the unperturbed evolution (Dirac or interaction representation) in imaginarytime as
V (τ) = eH0τ V e−H0τ .
The equation of motion satisfied by the S-operator allows a very simple perturbative expan-sion. The 0-th order S(0) = 1 since at this order H = H0. Morover, the boundary condition at
106
τ = 0 is S(0) = 1, implying that all S(n)(τ = 0) = 0 for n > 0, since already S(0) = 1 and, forany n and τ , S(n)(τ) > 0. The first order term satisfies
∂S(1)
∂τ= −V (τ) S(0) = −V (τ).
The solution with the appropriate boundary condition is
S(1)(τ) = −∫ τ
0dτ1 V (τ1). (4.31)
For a generic n, the equation of motion reads
∂S(n)
∂τ= −V (τ) S(n−1),
whose solution can be iteratively found starting from the expression of S(1), and reads
S(n)(τ) = (−1)n∫ τ
0dτn
∫ τn
0dτn−1 . . .
∫ τ2
0dτ1 V (τn) V (τn−1) . . . V (τ1). (4.32)
We can rewrite this term as 1/n! times the sum of all the terms which are obtained by permutingthe n-indices. This allows us to formally write
S(n)(τ) =(−1)n
n!
∫ τ
0
n∏i=1
dτi Tτ
(V (τ1)V (τ2) . . . V (τn)
), (4.33)
where we have introduced the so-called time-ordered product of operators, Tτ , which is definedas follows:
• the time-ordered product Tτ of several operators at different (imaginary) times is theproduct where the operators are ordered in such a way that those at later times appearon the left of those at earlier times. If some of those operators are fermionic-like, namelycontain odd number of fermionic operators, the result has to be multiplied by (−1)P ,where P is the number of permutations needed to bring the original sequence of only thefermionic operators into the time-ordered one.
For instance the time-ordered product of two operators A1(τ1) and A2(τ2) is
Tτ
(A(τ1)A(τ2)
)= θ (τ1 − τ2) A(τ1)A(τ2)± θ (τ2 − τ1) A(τ2)A(τ1),
where the − sign has to be used if both operators are fermionic-like.
107
Having introduced the time-ordered product, we can write the general expression for theS-operator as
S(τ) = Tτ
(e−∫ τ0 dτ ′ V (τ ′)
). (4.34)
Coming back to the partition function, it can be rewritten as
Z = Tr(
e−βH)
= Tr(
e−βH0S(β))
= Z01
Z0Tr(
e−βH0S(β))
= Z0〈S(β)〉0, (4.35)
whereZ0 = Tr
(e−βH0
)= e−βF
(0),
is the unperturbed partition function and 〈. . .〉0 means a quantum and thermal average with theuncorrelated Boltzmann weight, namely
〈. . .〉0 ≡1
Z0Tr(
e−βH0 . . .).
In a perturbation expansion
〈S(β)〉0 =∑n=0
〈S(n)(β)〉0 =∑n=0
S(n) = 1 + S(1) + S(2) + . . . .
Therefore
Z = e−βF(0)
[1− βF (1) +
1
2β2(F (1)
)2− βF (2) + . . .
]= e−βF
(0)[1 + S(1) + S(2) + . . .
],
which leads to
F (1) = − 1
βS(1) = − 1
β〈S(1)(β)〉0 (4.36)
F (2) = − 1
βS(2) +
1
2β(F (1)
)2= − 1
β〈S(2)(β)〉0 +
1
2β
(〈S(1)(β)〉0
)2. (4.37)
Let us first consider (4.36). We need to calculate
〈S(1)(β)〉0 = − 1
Z0Tr
[e−βH0
∫ β
0dτ eτH0 V e−τH0
]= − 1
Z0Tr
[e−βH0
∫ β
0dτ V
]= −β〈V 〉0. (4.38)
108
HenceF (1) = 〈V 〉0, (4.39)
is the thermal average of the perturbation.From the expression of S(2)(β) one can easily show that
F (2) = − 1
2β
∫ β
0dτ1 dτ2〈Tτ
[(V (τ1)− 〈V 〉0
)(V (τ2)− 〈V 〉0
)]〉0. (4.40)
Going up in perturbation theory, one finds that the generic F (n) is obtained by a cumulantexpansion in powers of V , namely an expansion in
V (τ)− 〈V 〉0.
We want now to show that F (2) < 0, which is the finite temperature analogoue of the knownresult that second-order perturbation theory always decreases the ground state energy. Let usconsider an hermitean operator A(τ) such that 〈A〉0 = 0, a property shared also by V (τ)−〈V 〉0.Then ∫ β
0dτ1
∫ τ1
0dτ2 Tr
[e−βH0A(τ1)A(τ2)
]=∑n,m
∫ β
0dτ1
∫ τ1
0dτ2 e−βεne(εn−εm)(τ1−τ2)
∣∣∣〈n|A|m〉∣∣∣2=∑n,m
∣∣∣〈n|A|m〉∣∣∣2 [e−βεm − e−βεn
(εn − εm)2− β e−βεn
εn − εm
].
Since the matrix element∣∣∣〈n|A|m〉∣∣∣2 is even by interchanging n ↔ m, the first term in the
square brackets, being odd, vanishes. The second term gives, after symmetrizing the sum∑n,m f(n,m) = 1/2
∑n,m f(n,m) + f(m,n),
1
2β∑n,m
∣∣∣〈n|A|m〉∣∣∣2 e−βεm − e−βεn
εn − εm,
which is always greater than zero.By this result, one can write
F (2) = − 1
2Z0
∑n,m
∣∣∣〈n|(V − 〈V 〉0) |m〉∣∣∣2 e−βεm − e−βεn
εn − εm,
hence the desired result F (2) ≤ 0.
109
How do we use this property? Let us consider the perturbed Hamiltonian
H(λ) ≡ H0 + λV .
ThenF (λ) = F (0) + λF (1) + λ2F (2) + . . . ,
where, as we showed, the curvature F (2) ≤ 0. Therefore in the interval λ ∈ [0, 1], the functionF (λ) has a downwards curvature. This implies that in this interval
F (λ) ≤ F (0) + λF (1),
leading to the variational principle
F = F (λ = 1) ≤ F (0) + F (1) = − 1
βln Tr
(e−βH0
)+
1
Tr(
e−βH0
)Tr(
e−βH0 V). (4.41)
4.2.2 Variational approach at T 6= 0
Let us apply this finite temperature variational principle to our interacting problem.The interacting Hamiltonian is, as usual,
H =∑αβ
tαβ c†αcβ +
1
2
∑αβγδ
Uαβγδ c†αc†βcγcδ.
We define an unperturbed Hamiltonian
H0 =∑αβ
hαβ c†αcβ,
which contains variational parameters hαβ [the similarity with (4.14) is not accidental, as wewill show], so that
V ≡ H − H0 =1
2
∑αβγδ
Uαβγδ c†αc†βcγcδ +
∑αβ
(tαβ − hαβ
)c†αcβ. (4.42)
By the variational principle (4.41), we can get an upper bound to the true free energy by choosingthe variational parameters hαβ which minimize the right hand side of (4.41). This is what weintend to do in the following.
Since H0 is quadratic in the fermionic fields, it can be diagonalized by some unitary trans-formation
cα =∑i
Uαici ,
110
leading to
H0 =∑i
εi c†ici .
Any many-body eigenstate of H0, e.g. |n〉, with energy En, can be written as a single Slaterdeterminant. The following property then holds
〈c†αc†βcγcδ〉0 =
1
Z0
∑ijkl
U†iα U†jβ Uγk Uδl Tr
(e−βH0c†ic
†jckcl
)=
1
Z0
∑ijkl
U†iα U†jβ Uγk Uδl
∑n
e−βEn〈n|c†ic†jckcl |n〉
=1
Z0
∑ijkl
U†iα U†jβ Uγk Uδl
∑n
e−βEn (δilδjk − δikδjl) 〈n|c†ici |n〉〈n|c†jcj |n〉
=∑ijkl
U†iα U†jβ Uγk Uδl (δilδjk − δikδjl) fi(T )fj(T )
= 〈c†αcδ〉0〈c†βcγ〉0 − 〈c
†αcγ〉0〈c
†βcδ〉0 ≡ ∆αδ(T )∆βγ(T )−∆αγ(T )∆βδ(T ), (4.43)
whereZ0 = Tr
(e−βH0
)=∏i
(1 + e−βεi
),
is the partition function of H0,
fi(T ) =1
1 + eβεi,
is the Fermi distribution function and we have introduced the thermal averages
∆αβ(T ) = 〈c†αcβ〉0.
The previous result shows in a particular case the following general result
• the thermal average of a product of n creation and n annihilation operators with a bilinearHamiltonian H0, namely
〈c†i1c†i2. . . c†incj1cj2 . . . cjn〉0 =
1
Z0Tr(
e−βH0c†i1c†i2. . . c†incj1cj2 . . . cjn
)=∑P
(−1)P 〈c†i1cjP1 〉0〈c†i2cjP2〉0 . . . 〈c†incjPn 〉0, (4.44)
where (jP1 , jP2 , . . . , jPn) = P (j1, j2, . . . , jn) is a permutation over the n j-indices and P isthe order of the permutation. The most general case where the operators are not orderedin such a way that creation operators appear on the left can be straightforwardly derived
111
from the above result. Essentially one has to consider all possible contractions between acreation, c†i , and an annihilation, cj , operator. If the former is on the left of the latter, thecontraction means
〈c†icj〉0,
otherwise it means〈cjc
†i 〉0.
The sign of a given product of contractions is plus or minus depending on the numberof fermionic hops one has to do to bring each pair of operators to be contracted closetogether.
Coming back to our original scope, we need to solve the equation
∂F (0)
∂hαβ+∂F (1)
∂hαβ= 0.
Let us first consider the first term on the left-hand side. We note that
∂F (0)
∂hαβ= − T
Z0
∂Z0
∂hαβ. (4.45)
On the other hand, if En are the eigenvalues of the many-body eigenstates | ψn〉 of H0, then
∂Z0
∂hαβ=
∑n
∂e−βEn
∂hαβ
= −β∑n
e−βEn∂En∂hαβ
= −β∑n
e−βEn 〈ψn |∂H0
∂hαβ| ψn〉
= −β∑n
e−βEn 〈ψn | c†αcβ | ψn〉 = −β Z0 ∆αβ(T ), (4.46)
where we made use of the Hellmann-Feynman theorem previously discussed. Therefore, through(4.45) and (4.46) we find that
∂F (0)
∂hαβ= ∆αβ(T ), (4.47)
a simple extension of the Hellmann-Feynman theorem at finite temperature.
By means of (4.43) we find that F (1) defined through (4.39) and (4.42) is given by
F (1) = 〈V 〉0 =1
2
∑αβγδ
∆αβ(T )∆γδ(T ) (Uαγδβ − Uαγβδ) +∑αβ
(tαβ − hαβ
)∆αβ(T ),
112
hence
∂F (1)
∂hαβ= −∆αβ(T ) +
∑γδ
(tγδ − hγδ
)∂∆γδ
∂hαβ+∑γδµν
∂∆γδ
∂hαβ∆µν (Uγµνδ − Uγµδν) . (4.48)
The sum of (4.47) and (4.48) gives therefore
∑γδ
∂∆γδ
∂hαβ
[hγδ − tγδ −
∑µν
∆µν (Uγµνδ − Uγµδν)
]= 0, (4.49)
which should hold for every pair of indices (αβ). The solution of this equation reads
hγδ = tγδ +∑µν
∆µν(T ) (Uγµνδ − Uγµδν)
= tγδ +1
Z0
∑µν
(Uγµνδ − Uγµδν) Tr(
e−βH0c†µcν
). (4.50)
Therefore the single particle Hamiltonian H0 which minimizes F (0) + F (1), hence providing anupper bound to the exact free energy, is defined through variational parameters hαβ which haveto be determined self-consistently through Eq. (4.50). Notice that this equation is just anextension of the Hartree-Fock Hamiltonian (4.14) at finite temperature, where the variationalparameters have to correspond now to thermal averages and nomore to ground state averages.
Finally notice that the Hartree-Fock Hamiltonian H0 represents non-interacting electrons inthe presence of a fictitious external field which, through (4.50), is indeed the self-consistent fieldgenerated by the same electrons. This is the reason why the Hartree-Fock approximation is alsocalled Mean-Field approximation.
113
4.3 Mean-Field approximation for bosons and superfluidity
The naıve Hartree-Fock approximation for interacting bosons, in the spirit of what we did forfermions, might be searching for the permanent of single-particle wavefunctions which has thelowest average value of the Hamiltonian. This is however not so easy in general because apermanent, unlike a Slater determinant, does not vanish if the single particle wavefunctions,out of which it is constructed, overlap each other. In addition, bosonic Hamiltonian fall intotwo different classes. The first includes models where the number of bosons is not a conservedquantity, e.g. phonons in solids. In this case a permanent, which is by definition a wavefunctionfor a fixed number of particles, is a very poor approximation for the actual ground state. Thesecond class includes models where the number of bosons is conserved, as for instance modelsfor liquid 4He. Here however one has to face another problem, namely Bose condensation, and asingle permanent is not expected to be a good approximation, too. In other words there is nota unique prescription for a simple and accurate mean-field approximation for bosons. In whatfollows we discuss an Hamiltonian for bosonic particles in the second class, i.e. which conservesthe number of bosons, in the context of Bose superfluidity.
Let us consider the Hamiltonian for N bosonic particles
H =∑q
εq b†qbq +
1
2V
∑qpk
U(q) b†pb†k+qbkbp+q, (4.51)
where the single particle energy
εq =~2 q2
2m.
In the absence of interaction, the ground state is simply obtained by putting, i.e. condense, allN particles into the q = 0 state. The interaction is going to mix this state with excited states,by moving particles from q = 0 to q 6= 0. First of all let us rewrite the interaction term as
1
2V
∑qpk
U(q) b†pb†k+qbkbp+q =
U(0)
2V
∑pk
b†pb†kbkbp
+1
2V
∑q6=0
∑pk
U(q) b†pb†k+qbkbp+q
=U(0)
2VN (N − 1) +
1
2V
∑q6=0
∑pk
U(q) b†pb†k+qbkbp+q,
which shows that the q = 0 component of the interaction is just a constant since the number ofbosons is conserved. Therefore, discarding this constant, the Hamiltonian can be written as
H =∑q
εq b†qbq +
1
2V
∑q 6=0
∑pk
U(q) b†pb†k+qbkbp+q, (4.52)
114
We will assume that the interaction is weak enough that the actual ground state is a slightmodification of the state where all bosons condense into the state q = 0.
Let us consider the following unitary transformation acting on the q = 0 bosons:
U = exp[√
N0
(b†0 − b0
)].
It follows thatU† b0 U = b0 +
√N0, U† b†0 U = b†0 +
√N0.
Therefore, if we consider a so-called coherent state
|0〉 ≡ U |0〉,
then
〈0| b†0b0 |0〉 = N0,
〈0| b†0b†0 |0〉 = 〈0| b0b0 |0〉 = N0.
In other words |0〉 contains an average number N0 of bosons at q = 0, but, at the meantime, it
is not eigenstate of b†0b0. On the other hand, the number fluctuation behaves as
∆N2 = 〈0|(b†0b0 −N0
)2|0〉 = N0,
hence, for large N0, ∆N/N0 → 0. These properties suggest that |0〉 might be a good repre-sentation of a state in which bosons condense into the zero-momentum state but fluctuationsof charge from q = 0 to q 6= 0 are still possible. Therefore, let us assume as a variationalwavefunction
|Ψ〉 = |0〉 |Φq 6=0〉,
where |Φq 6=0〉 includes non-zero momentum bosons, supplemented by the constraint
〈Ψ| b†0b0 +∑q 6=0
b†qbq |Ψ〉 = N0 +∑q 6=0
〈 b†qbq 〉 = N. (4.53)
As we pointed out, such a variational wavefunction should be a faithful representation of theground state provided
N −N0 N, (4.54)
which we are going to assume in what follows and check a posteriori.
The average of the Hamiltonian over |0〉 leads to an effective Hamiltonian for the q 6= 0bosons which reads
Heff =∑q 6=0
εq b†qbq +
N0
V
∑q 6=0
U(q) b†qbq +N0
2V
∑q 6=0
U(q)(b†qb†−q + b−qbq
)
115
+
√N0
V
∑qpp+q 6=0
U(q)(b†pb†qbp+q + b†p+qbqbp
)+
1
2V
∑ ∑qpk 6=0
U(q) b†pb†k+qbkbp+q.
Notice that the first two interaction terms are proportional to N0 (N −N0) while the last twocan be shown to be of order (N −N0)2. Therefore, in view of the assumption (4.54), we aregoing to start from the Hamiltonian
H0 =∑q 6=0
εq b†qbq +
N0
V
∑q 6=0
U(q) b†qbq
+N0
2V
∑q 6=0
U(q)(b†qb†−q + b−qbq
), (4.55)
and eventually treat as a perturbation
Hint =
√N0
V
∑qpp+q 6=0
U(q)(b†pb†qbp+q + b†p+qbqbp
)+
1
2V
∑ ∑qpk 6=0
U(q) b†pb†k+qbkbp+q. (4.56)
Let us introduce conjugate variables through
xq =
√1
2
(bq + b†−q
),
pq = −i√
1
2
(bq − b
†−q
),
which satisfy[xp, p−k] = i δp,k.
In terms of these variables (4.55) becomes
H0 =∑q 6=0
1
2
(εq + n0 U(q)
)(xqx−q + pqp−q − 1
)+
1
2n0U(q)
(xqx−q − pqp−q
), (4.57)
where n0 = N0/V , which can be diagonalized by the canonical transformation
xq →√Kq xq, pq →
√1
Kqpq,
withK2
q =εq
εq + 2n0U(q), (4.58)
116
leading to
H0 = −1
2
∑q 6=0
(εq + n0 U(q)
)+
1
2
∑q 6=0
ωq
(xqx−q + pqp−q
), (4.59)
with the frequency
ωq =
√εq
(εq + 2n0U(q)
). (4.60)
For small q, if U(q→ 0) is finite,
ωq → ~q√n0U(0)
m= s q,
thus showing that the long wavelength excitations describe acoustic phonons, which are actually
117
the Goldstone modes since the wave-function explicitly breaks gauge invariance. 2
Let us go back to the Hamiltonian (4.59). We notice that
〈b†qbq〉 =1
2〈xqx−q + pqp−q − 1〉 =
1
4
(Kq +
1
Kq− 2
).
Therefore the constraint (4.53) implies the following self-consistency equation for n0
N
V= n0 +
1
4V
∑q 6=0
(Kq +
1
Kq− 2
).
Since Kq ∼ q for small q’s, this self-consistency equation can be always solved but in onedimension, where the summation diverges. Once more this tells us that a continuos symmetry,
2Indeed the original Hamiltonian has a gauge symmetry, namely is invariant upon the transformation
bq → eiφ bq, b†q → e−iφ b†q,
generated by the unitary operator
Tφ = exp
(iφ∑q
b†q bq
).
As usual, this implies that, given a generic normalized eigenstate |Ψn〉 with eigenvalue En, the state Tφ |Ψn〉 isalso a normalized eigenstate with the same eigenvalue. There are two possible cases: either Tφ |Ψn〉 is, for all φ′s,the same |Ψn〉 apart from a phase factor, namely
|〈Ψn| Tφ |Ψn〉| = 1,
or it is different, i.e.|〈Ψn| Tφ |Ψn〉| < 1.
The former case means that |Ψn〉 is invariant under a gauge transformation, while the latter case implies that itis not, hence that the eigenvalue En is degenerate. Let us suppose that our variational wave-function is indeed agood approximation of the actual ground state |Ψ0〉, which is therefore also characterized by a finite value of
〈Ψ0| b0 |Ψ0〉 6= 0.
On the other hand〈Ψ0| b0 |Ψ0〉 = e−iφ 〈Ψ0| T †φ b0 Tφ |Ψ0〉.
If |Ψ0〉 is not degenerate, then Tφ |Ψ0〉 = eiγ |Ψ0〉, with γ a real number, hence we would get
〈Ψ0| b0 |Ψ0〉 = e−iφ 〈Ψ0| b0 |Ψ0〉.
This equality, being true for any φ, would necessarily imply that 〈Ψ0| b0 |Ψ0〉 = 0. Therefore, in order for thisaverage value to be finite, the ground state has to be degenerate and generically not gauge invariant. Noticethat in principle, given one of the degenerate ground states, say |Ψ0〉, one can always construct a gauge invariantcombination by
|Ψ0〉 =
∫dφ Tφ |Ψ0〉.
Yet, all other orthogonal combinations will remain not gauge invariant.
118
as the gauge symmetry is, can not be broken spontaneously in one dimension. Notice in additionthat the summation vanishes when U(q) → 0, namely when Kq → 1, so that, for sufficientlysmall interaction, indeed N −N0 N , thus justifying our approximation.
4.3.1 Superfluid properties of the gauge symmetry breaking wavefunction
Let us uncover now some interesting properties of the variational wavefunction. By definitionthe Fourier transform of the current operator is
Jq =~m
∑k
(k +
q
2
)b†kbk+q. (4.61)
In the spirit of our approximation, the leading term of the current is obtained when either k = 0or k + q = 0, leading to
Jq '√N0
~q
2m
(bq − b
†−q
)= i√
2N0~q
2mpq. (4.62)
In other words the current is purely longitudinal, i.e. q ∧ Jq = 0. Let us check whether (4.62)is compatible, within the same scheme, with the continuity equation
i~∂ρq∂t
= [ρq,H0] = q · Jq.
The density operator at leading order is
ρq =∑k
b†kbk+q '√N0
(bq + b†−q
)=√
2N0 xq,
and one readily verifies that its commutation with the Hamiltonian H0 indeed reproduces q ·Jq,with the current given by (4.62).
Non-Classical Rotational Inertia
What is the consequence of a purely longitudinal current? Let us suppose that our system of Nbosons with mass m is enclosed in a cylindrical annulus, initially at rest, with internal radiousR and thickness d, with d/R 1. At a given time the cylinder is made moving around its axis,assumed to be the z-axis, with constant angular velocity ω. A normal liquid would be draggedby the walls of the cylinder and rotate at the same angular velocity ω. Neglecting the mass ofthe cylinder, the energy change at equilibrium due to rotation would simply be at leading orderin ω
E(ω)− E(0) =1
2I0ω
2,
119
where the classical moment of inertia I0 ' mNR2 if d R. Let us consider what happens inour Bose-condensed fluid. The Schrœdinger equation describing the system reads
i~dΨ(x1,x2, . . . ,xN ; t)
dt=−~2
2m
N∑i=1
∇2i Ψ(x1,x2, . . . ,xN ; t)
+1
2
∑i 6=j
U(xi − xj) Ψ(x1,x2, . . . ,xN ; t)
+N∑i=1
V (xi; t) Ψ(x1,x2, . . . ,xN ; t),
where V (x; t) represents the coupling of a boson with the rotating walls of the cylinder, hence itis explicitly time-dependent. We use cylidrical coordinates, so that the position of particle i is
xi = (r cos θ, r sin θ, z) ≡ ri + zi z,
where z is the unit vector in the z-direction. Let us consider the following transformation onthe planar coordinate
ρi(t) = cosωt ri + sinωt ri ∧ z, (4.63)
which corresponds to a reference frame in which the cylinder is at rest. In this frame
V (x; t) = V (r, z; t) = V (ρ(t), z) ,
namely the explicit time-dependence of the interaction with the wall is being transformed intoan implicit one via the time-dependence of ρ(t). We rewrite the many-body wavefunction interms of the new variables, i.e.
Ψ ((r1, z1), (r2, z2), . . . , (rN , zN ); t) = Ψ ((ρ1, z1), (ρ2, z2), . . . , (ρN , zN ); t) .
In this representation the wavefunction depends both explicitly upon time and also implicitly,via the coordinates ρi’s. Therefore
i~dΨ ((r1, z1), (r2, z2), . . . , (rN , zN ); t)
dt= i~
∂Ψ ((ρ1, z1), (ρ2, z2), . . . , (ρN , zN ); t)
∂t
+i~N∑i=1
∂ρi∂t·∇iΨ ((ρ1, z1), (ρ2, z2), . . . , (ρN , zN ); t)
= i~∂Ψ ((ρ1, z1), (ρ2, z2), . . . , (ρN , zN ); t)
∂t
+i~ωN∑i=1
ρi ∧ z · ∇iΨ ((ρ1, z1), (ρ2, z2), . . . , (ρN , zN ); t)
120
= i~∂Ψ ((ρ1, z1), (ρ2, z2), . . . , (ρN , zN ); t)
∂t
−mωN∑i=1
ρi ∧ z · J iΨ ((ρ1, z1), (ρ2, z2), . . . , (ρN , zN ); t) ,
where ∇i is assumed to act on the new coordinate basis and by definition the current operatoris
J i = − i~m∇i.
One can readily verify that the Laplacian is invariat upon the transformation (4.63), hence theSchrœdinger equation becomes
i~∂Ψ
∂t=
N∑i=1
(− ~2
2m∇2i +mωρi ∧ z · J i
)Ψ
+1
2
∑i 6j
U(ρi − ρj , zi − zi) Ψ +
N∑i=1
V (ρi, zi) Ψ. (4.64)
Unlike in the original representation, this equation is not explicitly time-dependent, hence itadmits stationary solutions
Ψ ((ρ1, z1), (ρ2, z2), . . . , (ρN , zN ); t) = eiE(w)t/~ Ψ ((ρ1, z1), (ρ2, z2), . . . , (ρN , zN )) .
The minimum energy E(ω) corresponds to the equilibrium condition.3
It is important to recognize that the actual average value of the Hamiltonian is not E(ω)but4
E(ω) =
∫ ∏i
dri dzi Ψ ((r1, z1), (r2, z2), . . . , (rN , zN ); t)∗ i~d
dtΨ ((r1, z1), (r2, z2), . . . , (rN , zN ); t)
=
∫ ∏i
dρi dzi Ψ ((ρ1, z1), (ρ2, z2), . . . , (ρN , zN ); t)∗
i~ ∂∂t−mω
∑j
ρj ∧ z · Jj
Ψ ((ρ1, z1), (ρ2, z2), . . . , (ρN , zN ); t)
=
∫ ∏i
dρi dzi Ψ ((ρ1, z1), (ρ2, z2), . . . , (ρN , zN ))∗
E(ω)−mω∑j
ρj ∧ z · Jj
3 Notice that the equilibrium condition corresponds to the minimum energy in a reference frame in which the
cylinder is at rest. Indeed, it is only in such a reference frame that the walls of the cylinder can stop transferingenergy to the liquid.
4The jacobian of the transformation (4.63) is unity.
121
Ψ ((ρ1, z1), (ρ2, z2), . . . , (ρN , zN ))
= E(ω)−mω∑i
〈ρi ∧ z · Ji〉. (4.65)
Therefore, in order to evaluate E(ω) we first need to study the model described by theHamiltonian in second quantization
H =
∫dx Ψ(x)†
[− ~
2m∇2 + A(x) · J + V (x)
]Ψ(x)
+1
2
∫dxdy Ψ(x)†Ψ(y)† U(x− y) Ψ(y) Ψ(x) ,
withA(x) = mω x ∧ z = mω (y,−x, 0), (4.66)
playing the role of a vector potential coupled to the current operator
J = −i ~m∇.
We notice that∇ ·A(x) = 0,
namely the vector potential is purely transverse. If we assume that the liquid is homogeneous,then, as we previously proved, the leading component of the current operator is purely longitu-dinal, hence the net effect of the transverse perturbation to the system is negligible. Therefore,within our approximation, E(ω) ' E(0) does not depend upon ω. Analogoulsy
〈J ·A〉 ' 0,
implying that also E(ω) ' E(0) is ω-independent. In other words, the actual moment of inertiais zero, i.e. the quantum liquid can not be made moving by rotating the cylinder, so calledNon-Classical Rotational Inertia (NCRI).
In reality the situation may be more complex. Indeed, if the frequency of rotation exceeda critical value ωc1, the lowest energy configuration ceases to be homogeneous, as we assumedso far, because line defects, so-called vortex lines, appear. It is however not our scope here todiscuss vortices.
Superfluid flow
Let us consider another hypotetical experiment in which the bosonic fluid is made moving insidea capillary with constant velocity v parallel to the walls of the capillary. A normal fluid, due tothe friction, will be slowed down by the walls. Let us consider instead what happens with our
122
Bose-condensed fluid. In the reference frame in which the fluid is at rest, hence the capillarymoves, we may expect that energy is transferred from the moving walls to the fluid leading tocreation of excitations. Let us assume that an excitation of momentum q and energy ωq ispresent. In the original reference frame in which the capillary is at rest, the actual energy is, bya Galilean tranformation,
E(q) = ωq + v · q +Nmv2
2.
As we said the equilibrium condition corresponds to the minimum of E(q), which is obtained bya momentum antiparallel to v, i.e. q · v = −q v. In this case
E(q) = ωq − v q +Nmv2
2.
It is clear that it is advantageous to create excitations only if
v ≥ ωq
q. (4.67)
This is the famous Landau criterium for superfluidity. It tells us that the fluid can flow withoutresistance through a capillary, so-called superfluidity, for velocities below a critical one vcdefined by
vc = limq→0
ωq
q.
From our previous analysis it derives that vc = s, the velocity of the acoustic phonons in theliquid. We notice that the interaction among the particles is essential for superfluidity. Indeedan ideal non-interacting Bose gas with dispersion ~2q2/2m has
vc = limq→0
1
q
~2q2
2m= 0,
hence it is not superfluid.
123
4.4 Time-dependent Hartree-Fock approximation for fermions
Let us move back to the Hartree-Fock approximation for fermions, and suppose to have solvedthe time-independent Hartree-Fock approximation, namely to have found a set of single particlewave-functions φi(x)’s which diagonalize
tij +
N∑k=1
Uikkj − Uikjk = εi δij . (4.68)
The Hartree-Fock wave-function is the Slater determinant
|ΦHF 〉 =N∏i=1
c†i |0〉, (4.69)
where c†i creates a fermion in state φi. The original Hamiltonian in the Hartree-Fock basis isgiven by
H =∑ij
tijc†icj +
1
2
∑ijkl
Uij,kl c†ic†jckcl . (4.70)
Let us suppose to perturb the Hamiltonian by a time-dependent perturbation
V =∑ij
Vij(t) c†icj , (4.71)
and let us study the response of the system to linear order. Therefore the full Hamiltonian H(t) =H + V (t) is time-dependent and the variational principle now concerns the time dependentShrœdinger equation. Namely we will search within the subspace of time-dependentSlater determinants for a |ΦHF (t)〉 which satisfies
δ
[〈ΦHF (t)| i~ ∂
∂t − H(t) |ΦHF (t)〉〈ΦHF (t)|ΦHF (t)〉
]= 0. (4.72)
We assume that the time-dependent Slater determinant is build up by N single particle nor-malized wave-functions φα(x, t). By analogy with the conventional derivation of the time-independent Hartree-Fock equations, one finds that
i~∂
∂tφα(x, t) = [T (x, p) + V (x, p, t)] φα(x, t)
+N∑β=1
∫dy U(x, y)
[|φβ(y, t)|2φα(x, t)− φβ(y, t)∗φα(y, t)φβ(x, t)
], (4.73)
124
where T (x, p) is the single-particle term of the Hamiltonian, V (x, p, t) the perturbation andU(x, y) the interaction. We assume that the φα(x, t)’s are related to the φi(x)’s by the time-dependent unitary transformation
φα(x, t) =∑i
Uiα(t)φi(x),
where
Uiα(t) =
∫dxφi(x)∗φα(x, t).
In terms of the matrix elements, the time-dependent equations (4.73) read
i~∂
∂tUiα(t) =
∑j
[tij + Vij(t)]Ujα(t)
+N∑β=1
∑jkl
Ukβ(t)∗Ulβ(t) Ujα(t) [Uiklj − Uikjl] . (4.74)
i~∂
∂tUiα(t)∗ = −
∑j
[tji + Vji(t)]Ujα(t)∗
−N∑β=1
∑jkl
Ukβ(t) Ulβ(t)∗Ujα(t)∗ [Ujlki − Uljki] . (4.75)
The time-dependent Hartree-Fock wave-function is identified by the matrix elements
∆ij(t) = 〈ΦHF (t)| c†icj |ΦHF (t)〉 =
N∑α=1
Uiα(t)∗Ujα(t) , (4.76)
which, calculated with the time-independent wave-function (4.69) are simply
∆(0)ij = δij ni, (4.77)
with ni = 1 if i ≤ N and ni = 0 otherwise. Through (4.74), (4.75) and (4.76) we obtain thefollowing equations for the ∆ij ’s:
i~∂
∂t∆ij(t) =
∑k
− [tki + Vki(t)] ∆kj(t) + [tjk + Vjk(t)] ∆ik(t)
+∑klm
−∆kl(t) ∆mj(t) [Umkli − Ukmli]
+∆il(t) ∆km(t) [Ujkml − Ujklm] . (4.78)
125
We solve the above equation up to first order in the perturbation V . One can easily checkthat the zero order time-independent term is indeed given by (4.77). The first order terms satisfythe equation
i~∂
∂t∆
(1)ij (t) =
∑k
−
[tki +
N∑m=1
(Ukmmi − Umkmi)
]∆
(1)kj (t)
+∑k
[tjk +
n∑m=1
(Ujmmk − Ujmkm)
]∆
(1)ik (t)
+ (ni − nj)∑lm
∆(1)ml (t) [Ujmli − Ujmil] + Vji(t) (ni − nj) . (4.79)
Through Eq. (4.68) we finally obtain[i~∂
∂t− εj + εi
]∆
(1)ij (t)
= (ni − nj)
[Vji(t) +
∑kl
∆(1)kl (t) (Ujkli − Ujkil)
], (4.80)
which is our desired result.
4.4.1 Bosonic representation of the low-energy excitations
We notice from (4.80) that the only expectation values which are influenced at linear order bythe perturbation are those which either ni = 1 and nj = 0 or viceversa, namely where one ofthe index refer to an occupied state within (4.69) and the other to an un-occupied one. Let usdenote by greek letters α, β, . . . the un-occupied states (> N), hereafter named “particles”, andby roman letters a, b, . . . the occupied ones (≤ N), named “holes”. We denote by
b†αa = c†αca, (4.81)
the particle-hole creation operator, which destroys one electron (create one hole) inside the Slaterdeterminant and creates it outside, and its hermitean conjugate
bαa = c†acα. (4.82)
The commutators [bαa, bβb
]=[b†αa, b
†βb
]= 0,
vanish, while [bαa, b
†βb
]= δαβ c
†acb − δab c
†βcα,
126
has the average value on (4.69) given by
〈ΦHF |[bαa, b
†βb
]|ΦHF 〉 = δαβδab (na − nα) = δαβδab,
as if the particle-hole creation and annihilation operators were bosonic particles. If we assumethat at linear order in V the Hartree-Fock wave-function is sligthly modified with respect toits time-independent value, in other words we make a similar assumption as we did in derivingspin-wave theory, then we can approximate the commutator by its average value, thus obtaining[
bαa, b†βb
]= δαβ δab,
[bαa, bβb
]=[b†αa, b
†βb
]= 0, (4.83)
which are indeed bosonic commutation relations. We notice that the bosonic vacuum is thetime-independent Hartree-Fock wave-function. Since
∆aα(t) = 〈ΦHF (t)|bαa|ΦHF (t)〉,
Eq. (4.80) implies the following equation of motion for the bosonic operators[i~∂
∂t− εα + εa
]bαa
=
Vαa(t) +∑βb
bβb (Uαbβa − Uαbaβ) + b†βb (Uαβba − Uαβab)
. (4.84)
We may now ask the following question: What would be a bosonic Hamiltonian leading tothe above equation of motion ? The answer can be readily found and is∑
αa
(εα − εa) b†αabαa +∑αβab
b†αabβb (Uαbβa − Uαbaβ)
+1
2
∑αβab
b†αab†βb (Uαβba − Uαβab) + bβbbαa (Uabβα − Uabαβ)
+∑αa
Vαa(t)b†αa + Vaα(t)bαa. (4.85)
It is now clear that the external field simply probes particle-hole excitations which have theirown dynamics provided by the Hamiltonian
HTDHF =∑αa
(εα − εa) b†αabαa +∑αβab
b†αabβb (Uαbβa − Uαbaβ)
+1
2
∑αβab
b†αab†βb (Uαβba − Uαβab) + bβbbαa (Uabβα − Uabαβ) . (4.86)
127
Due to the presence of the last two terms, the true ground state of (4.86) is not the vacuum,namely the Hartree-Fock wave-function, but another state
|ΦTDHF 〉 = eA |ΦHF 〉,
with A an anti-hermitean operator quadratic in the bosonic operators. The above wave-functionis actually an improvement of the Hartree-Fock wave-function that includes the zero-point fluc-tuations of the particle-hole excitations.
The above result can be re-derived in the spirit of the spin-wave theory, that we have al-ready encountered, without even invoking the time-dependent Hartree-Fock. If we assume, tobe checked a posteriori, that the Hartree-Fock wave-function is not strongly modified by theinclusion of quantum fluctuations, we can assume that the commutators[
c†a cα , c†β cb
]= δαβ c
†a cb − δab c
†β cα,[
c†α ca , c†b cβ
]= δab c
†α cβ − δαβ c
†b ca,[
c†a cb , c†c cd
]= δbc c
†a cd − δad c
†c cb,[
c†α cβ , c†γ cδ
]= δβγ c
†α cδ − δαδ c
†γ cβ,
can be effectively substituted by their average values on the Hartree-Fock wave-function, i.e.[c†a cα , c
†β cb
]= δαβ δab,[
c†α ca , c†b cβ
]= −δαβ δab,[
c†a cb , c†c cd
]= 0,[
c†α cβ , c†γ cδ
]= 0,
which justifies the identification with bosonic operators. The only terms of the interaction thatmay have non vanishing effect when applied either on the right or on the left of the Hartree-Fockwave-functtion are those which contain two particle and two holes, hence they are
Hint '1
2
∑abαβ
Ubαβa c†bc†αcβca + Uαbaβ c
†αc†bcacβ
+Ubαaβ c†bc†αcacβ + Uαbβa c
†αc†bcβca
+Uαβba c†αc†βcbca + Uabβα c
†ac†bcβcα
=1
2
∑abαβ
2(Uαbβa − Uαbaβ
)c†αc†bcβca
128
+Uαβba c†αc†βcbca + Uabβα c
†ac†bcβcα.
Next we have to express the interaction in terms of bosons, avoiding double counting. The firstterm becomes straightforwardly ∑
abαβ
(Uαbβa − Uαbaβ
)b†αa bβb.
The second term is more delicate to re-express, as one can at will couple α with a or b, thusgetting either b†αa b
†βb or −b†αb b
†βa. Although the two terms correspond to the same fermionic
operator, they are independent from the bosonic point of view, hence they both have to be kept,leading to
1
2
∑abαβ
(Uαβba − Uαβab
)b†αa b
†βb +H.c.,
which coincides with the term in (4.86).
129
4.5 Application: antiferromagnetism in the half-filled Hubbardmodel
Let us consider again the Hubbard model on an hypercubic lattice in d-dimensions:
H = −t∑
<RR′>
∑σ
(c†RσcR′ σ +H.c.
)+ U
∑R
nR ↑nR ↓, (4.87)
wherenRσ = c†RσcRσ,
and, as usual, < RR′ > means that R and R′ are nearest neighbor sites. We already showedthat for very large U/t and one electron per site, i.e. half-filling, this model is a Mott insulatorwith a Neel magnetic order in d > 1 at zero temperature and in d > 2 at finite temperaturebelow a critical Tc. Let us now try to recover this behavior within the previously describedHartree-Fock theory.
The first step is to identify the variational parameters. The interaction, being on-site, leadsto local variational parameters which are
∆R, σσ′ = 〈ΦHF |c†RσcRσ′ |ΦHF 〉.
We search for an Hartree-Fock Slater determinant which describes a Neel order with the an-tiferromagnetic order parameter along the z-direction, namely we assume that the spin SU(2)symmetry is lowered down to a U(1) symmetry which describes spin-rotations around the z-axis.This implies that we can choose |ΦHF 〉 as an eigenstate of the z-component of the total spin,hence that the only finite variational parameters are diagonal in the spin index, namely
nRσ = 〈ΦHF |nRσ|ΦHF 〉.
Since the average number of electrons per site is∑
σ nRσ = 1, we use the following parametriza-tion:
nR ↑ =1
2+m (−1)R, (4.88)
nR ↓ =1
2−m (−1)R, , (4.89)
where if the vector R = a(n1, n2, . . . , nd), a being the lattice spacing, then R =∑d
i=1 ni. Inother words the average magnetization is assumed to be
SzR =1
2(nR ↑ − nR ↓) = m (−1)R.
130
The Hartree-Fock Hamiltonian is therefore
HHF = −t∑
<RR′>
∑σ
c†RσcR′ σ +H.c. + U∑R
(nR ↑ nR ↓ + nR ↓ nR ↑)
= −t∑
<RR′>
∑σ
c†RσcR′ σ +H.c.− U m∑R
(−1)R (nR ↑ − nR ↓)
+U
2
∑R
(nR ↑ + nR ↓) .
The last term is proportional to the total number of electrons, UN/2, which is a conservedquantity. Therefore UN/2 is just a constant which can be dropped leading to the Hartree-FockHamiltonian
HHF = −t∑
<RR′>
∑σ
c†RσcR′ σ +H.c.− U m∑R
(−1)R (nR ↑ − nR ↓) . (4.90)
Let us write this Hamiltonian in momentum space. Since
(−1)R = eiQ·R,
where Q = π(1, 1, . . . , 1)/a, one readily finds that
HHF =∑kσ
εknkσ − U m∑k
(c†k ↑ck+Q ↑ − c
†k ↓ck+Q ↓
). (4.91)
If k = (k1, k2, . . . , kd), the bare dispersion is given by
εk = −2t
d∑i=1
cos kia.
The original Brillouin zone is an hypercube with linear size 2π/a. Since the Hartree-FockHamiltonian breaks translational symmetry – the new unit cell is twice larger – the actualBrillouin zone is half the original one. Let us define the new Brillouin zone, which we callMagnetic Brillouin zone (MBZ) as the volume which encloses all k-points such that εk ≤ 0.Since for any of these points
εk+Q = −εk ≥ 0,
the MBZ is indeed twice-smaller than the original one. Then let us define new fermionic operatorswith k ∈MBZ by
akσ = ckσ, bkσ = ck+Qσ,
and spinorial operators
ψkσ =
(akσbkσ
). (4.92)
131
In terms of the latter (4.91) can be written as
HHF =∑
k∈MBZ σ
εk ψ†kσ τ3 ψkσ − U m
∑k∈MBZ
ψ†k ↑ τ1 ψk ↑ − ψ†k ↓ τ1 ψk ↓, (4.93)
where τi’s, i = 1, 2, 3, are Pauli matrices acting in the spinorial basis. We notice that
εk τ3 ∓ U mτ1 = −Ek τ3
(− εkEk± iUm
Ekτ2
)= −Ek τ3 e±2iθk τ2 = −Ek e∓iθk τ2 τ3 e±iθk τ2 ,
where
Ek =√ε2k + U2m2, (4.94)
and
cos 2θk = − εkEk
, sin 2θk =Um
Ek.
Therefore if we define two new spinors through
φk ↑ =
(αk ↑βk ↑
)= eiθk τ2 ψk ↑, φk ↓ =
(αk ↓βk ↓
)= e−iθk τ2 ψk ↓, (4.95)
the Hamiltonian becomes
HHF = −∑
k∈MBZ σ
Ek φ†kσ τ3 φkσ = −
∑k∈MBZ σ
Ek
(α†kσαkσ − β
†kσβkσ
), (4.96)
namely it acquires a diagonal form. Since Ek > 0 the ground state with a number of electronsequal to the number of sites N is simply obtained by filling completely the α-band (notice thatthe MBZ contains N/2 k-points, hence each band can accomodate 2N/2 = N electrons). Thusthe Hartree-Fock Hamiltonian describes a band-insulator with two bands separated by a gap ofminimal value 2U m. The Hartree-Fock wave-function satisfies:
〈ΦHF |φ†kσ τi φkσ|ΦHF 〉 = δi 3,
which becomes at finite temperature
〈φ†kσ τi φkσ〉 = δi 3 tanhEk
2T. (4.97)
We still have to impose the self-consistency condition
m =1
2V
∑R
(−1)R 〈nR ↑ − nR ↓〉
132
=1
2V
∑k∈MBZ
〈ψ†k ↑ τ1 ψk ↑ − ψ†k ↓ τ1 ψk ↓〉
=1
2V
∑k∈MBZ
〈ψ†k ↑ eiθk τ2 τ1 e−iθk τ2 ψk ↑ − ψ†k ↓ e−iθk τ2 τ1 eiθk τ2 ψk ↓〉
=1
2V
∑k∈MBZ
〈φ†k ↑ (cos 2θk τ1 + sin 2θk τ3)φk ↑ − φ†k ↓ (cos 2θk τ1 − sin 2θk τ3)φk ↓〉
=1
V
∑k∈MBZ
sin 2θk tanhEk
2T.
In other words the self-consistency condition reads
U
2V
∑k
1
Ektanh
Ek
2T= 1. (4.98)
We solve this self-consistency condition by assuming a constant density of states, namely
1
V
∑k
· · · =∫dε ρ(ε) · · · → ρ0
∫ D
−Ddε . . . ,
where ρ0 = 1/2D, with D ∼ 2dt. At zero temperature we find
1 =Uρ0
2
∫ D
−Ddε
1√ε2 + U2m2
,
which has solution given by
m =D
U
(sinh
1
Uρ0
)−1
.
This equation has always a solution for any U 6= 0. In particular, for U D,
m ' D
Uexp
(− 1
Uρ0
). (4.99)
This results states that the Hubbard model at half-filling on an hypercubic lattice is always anantiferromagnetic insulator whatever is the value of the Hubbard U . Notice that in the oppositelimit of U D, m ∼ Dρ0 = 1/2, which is the expected result since, for large U , electronslocalize, one per site, hence behave like local spin-1/2 moments.
We can also determine the Neel temperature, which is the temperature Tc at which theself-consistency condition starts to be verified with m = 0, namely
1 =U
2V
∑k
1
εktanh
εk2Tc' Uρ0
2
∫ D
−D
dε
εtanh
ε
2Tc. (4.100)
133
One finds that Tc ∼ U m(T = 0). This implies that
Tc ∼ D exp
(− 1
Uρ0
),
for small U , but Tc ∼ U at large U . The latter results contradicts what we previously foundin the large U limit, by mapping the Hubbard model onto an Heisenberg model. There, withinspin-wave theory, we showed that Tc ∼ J ∼ t2/U , hence vanishes for large U . The origin of thedisagreement stems from the fact that the Hartree-Fock theory is valid only for an interactionwhich is weak compared to the electron bandwidth. Essentially the Mott phenomenon, whichdominates at large U/t, escapes any description using a single Slater determinant, which is onlyable to mimick a band-insulator.
4.5.1 Spin-wave spectrum by time dependent Hartree-Fock
Rigorously speaking, Hartree-Fock theory only allows to access ground state or, at finite temper-ature, equilibrium properties, although in an approximate variational manner, but not propertiesof excited states. Therefore it is not legitimate to interpret the spectrum of the Hartree-FockHamiltonian as an approximation of the actual spectrum. For instance we showed that theHeisenberg antiferromagnet, which corresponds to the large U -limit of the Hubbard model athalf-filling, has gapless spin-wave excitations. On the contrary the Hartree-Fock Hamiltonian,which describes a band-insulator, has a gap for all excitations. The simplest way to cure thisdefect, namely to have access to excitations, is by means of the time dependent Hartree-Fock.Here we apply this technique to recover the spin-wave spectrum.
Let us derive the bosonic Hamiltonian for the spin-flip particle-hole excitations above theHartree-Fock ground state. We consider, in terms of the spinors (4.95) which diagonalize theHartree-Fock Hamiltonian, the following spin-flip operators:
Xk;q =
√1
2φ†k ↑ τ1 φk+q ↓,
X†k;q =
√1
2φ†k+q ↓ τ1 φk ↑,
Pk;q =
√1
2φ†k ↑ τ2 φk+q ↓,
P †k;q =
√1
2φ†k+q ↓ τ2 φk ↑.
In the framework of time-dependent Hartree-Fock one can assume that[Xk;q, P
†k′;q′
]=
i
2
(δk+q,k′+q′ φ
†k ↑ τ3 φk′ ↑ + δk,k′ φ
†k′+q′ ↓ τ3 φk+q ↓
)134
' 〈ΦHF |[Xk;q, P
†k′;q′
]|ΦHF 〉 = i δk,k′ δq,q′ ,
which shows that X and P † are conjugate variables. The energy cost of such spin-flip excitationsare described by the free Hamiltonian
H0 =∑kq
ωk;q
(X†k;qXk;q + P †k;qPk;q
), (4.101)
whereωk;q = Ek+q + Ek.
This simply means that if one destroys an α particle at k and create a β one at k+q, the energycost is ωk;q.
Following the prescriptions of time-dependent Hartree-Fock, we still need to express theHubbard interaction in terms of these spin-flip excitations. We notice that
nR ↑nR ↓ = c†R ↑cR ↑ c†R ↓cR ↓
= −c†R ↑cR ↓ c†R ↓cR ↑ ≡ S
+R S−R,
so that
U∑R
nR ↑nR ↓ = U∑R
S+R S−R =
U
V
∑q
S+q S−−q,
whereS+q =
∑R
e−iq·R S+R =
∑k
c†k ↑ck+q ↓.
We recall that the MBZ is half the original one. For instance, while before the momentaq = 0 and q = Q were different, in the MBZ both coincide with momentum zero. Therefore,if we are interested in long wavelength excitations, we need to consider in the original BZ bothq ∼ 0 and q ∼ Q. For q ∼ 0, S+
q connects states inside the MBZ, i.e.
S+q '
∑k∈MBZ
a†k ↑ak+q ↓ + b†k ↑bk+q ↓ =∑
k∈MBZ
ψ†k ↑ψk+q ↓.
Near Q we can consider the operator S+q+Q with small q, which is
S+q+Q '
∑k∈MBZ
a†k ↑bk+q ↓ + b†k ↑ak+q ↓ =∑
k∈MBZ
ψ†k ↑ τ1 ψk+q ↓.
Therefore in the long wavelength limit it is legitimate to re-write
U
V
∑q
S+q S−−q =
U
V
∑kpq∈MBZ
ψ†k ↑ψk+q ↓ ψ†p+q ↓ψp ↑ + ψ†k ↑ τ1 ψk+q ↓ ψ
†p+q ↓ τ1 ψp ↑.
135
Notice that
ψ†k ↑ψk+q ↓ = φ†k ↑ eiθkτ2 eiθk+qτ2 φk+q ↓
' i sin (θk + θk+q) φ†k ↑ τ2 φk+q ↓ = i√
2 sin (θk + θk+q) Pk;q,
ψ†k ↑ τ1 ψk+q ↓ = φ†k ↑ eiθkτ2 τ1 eiθk+qτ2 φk+q ↓
' cos (θk − θk+q) φ†k ↑ τ1 φk+q ↓ =√
2 cos (θk − θk+q) Xk;q,
so that the interaction Hamiltonian becomes
Hint = −2U
V
∑kpq∈MBZ
sin (θk + θk+q) sin (θp + θp+q) P †k;q Pp;q
+ cos (θk − θk+q) cos (θp − θp+q) X†k;qXp;q. (4.102)
The complete Hamiltonian for the spin-flip excitations is therefore
H =∑
kq∈MBZ
ωk;q
(X†k;qXk;q + P †k;qPk;q
)−2U
V
∑kpq∈MBZ
sin (θk + θk+q) sin (θp + θp+q) P †k;q Pp;q
+ cos (θk − θk+q) cos (θp − θp+q) X†k;qXp;q. (4.103)
One might diagonalize this Hamiltonian to obtain the spin-flip excitation spectrum. Here weaim simply to show that (4.103) does contain the spin-waves.
First we perform the canonical transformation
Xk;q →
√1
ωk;qXk;q,
Pk;q → √ωk;q Pk;q,
after which
H =∑
kq∈MBZ
X†k;qXk;q + ω2k;q P
†k;qPk;q
−2U
V
∑kpq∈MBZ
√ωk;qωp;q sin (θk + θk+q) sin (θp + θp+q) P †k;q Pp;q
+
√1
ωk;qωp;qcos (θk − θk+q) cos (θp − θp+q) X†k;qXp;q.
136
Next we make a unitary transformation and define the new operators
Xq =
√√√√√ 1∑k∈MBZ
cos2 (θk − θk+q) /ωk;q
∑p∈MBZ
cos (θp − θp+q)√ωp;q
Xp;q,
Pq =
√√√√√ 1∑k∈MBZ
cos2 (θk − θk+q) /ωk;q
∑p∈MBZ
cos (θp − θp+q)√ωp;q
Pp;q,
as well as all the other orthogonal combinations, Xi;q and Pi;q. The part of the Hamiltonian
which involves X is simply
∑q∈MBZ
(1− 2U
V
∑k∈MBZ
cos2 (θk − θk+q)
ωk;q
)X†q Xq.
Notice that for q = 0 the square bracket vanishes because is nothing but the self-consistencyequation (4.98) at zero temperature, namely
1 =U
V
∑k∈MBZ
1
Ek.
Therefore for small q
1− 2U
V
∑k∈MBZ
cos2 (θk − θk+q)
ωk;q∼ A |q|2,
where A is a constant which can be determined and for large U/t behaves as t2/U2. All thecombinations of X’s, Xi;q’s, which are orthogonal to X remain degenerate and described by theterm ∑
i
∑q
X†i;qXi;q.
One realizes that the above |q|2-term is the leading one at small q, hence we can evaluatewhat is left putting q = 0. Then, through (4.98), we find that
Pq '√U
V
∑k∈MBZ
√1
EkPk;q,
and that the part of the Hamiltonian involving the conjugate variables is∑kq∈MBZ
4E2k P†k;qPk;q −
2U
V
∑kpq∈MBZ
2√EkEp
Um
Ek
Um
EpP †k;q Pp;q
137
=∑
kq∈MBZ
4E2k P†k;qPk;q − 4U2m2
∑q
P †qPq.
This implies that the conjugate variables are governed, at small q, by the term
∑q
[4U
V
∑k∈MBZ
Ek − 4U2m2
]P †qPq = B
∑q
P †qPq,
where B ∼ t2 for large U , and a bunch of other terms involving the orthogonal combinationsPi;q’s which are coupled together and to Pq. Let us neglect for the moment the coupling to Pq.
Then Xq and Pq are governed by the Hamiltonian∑q
A|q|2 X†qXq +B P †qPq,
which is diagonalized by the transformation Xq →√Kq Xq and Pq →
√K−1
q Pq, where
K2q =
B
A|q|2,
leading to a linear dispersionωq =
√AB |q| ∼ J |q|.
If we diagonalize the Hamiltonian which involves all other orthogonal combinations, we wouldget for them a gaped spectrum.
Notice that
Pq →√K−1
q Pq ∼√|q| Pq,
therefore the coupling between P and all other Pi’s is proportional to√|q| hence just renor-
malizes the spin-wave velocity. Moreover this coupling term between P and all Pi’s actuallyvanishes for large U since
∑kq∈MBZ
4E2k P†k;qPk;q ∼ 4U2m2
∑q
[P †qPq +
∑i
P †i;qPi;q
].
so that, for U/t 1, the spin-wave velocity v is, up to subleading terms,
v =√AB, (4.104)
with A and B defined previously.
138
4.6 Linear response of an electron gas
Let us consider now the response to an electromagnetic field of a solid. We start from theclassical macroscopic Maxwell equations of an isolated system
∇ ·E = 4πρ,
∇ ·B = 0,
∇ ∧E = −1c∂B∂t ,
∇ ∧B = 1c∂E∂t + 4π
c J ,
(4.105)
where ρ is the total charge density within the system, and J the current density. In generalthe current receives contributions from the free carriers, from the magnetic moments and frombound charges, so that its expression is
J = Jcarriers +∂P
∂t+ c∇ ∧M, (4.106)
where P the polarization density of bound charges5 and M the magnetization density. In whatfollows we will not take into account these two terms, unless explicitly stated.
We define the displacement field D through
1
c
dD
dt=
1
c
∂E
∂t+
4π
cJ . (4.107)
By applying ∇ on both sides we get
1
c
d
dt∇ ·D =
1
c
∂
∂t∇ ·E +
4π
c∇ · J =
4π
c
[∂ρ
∂t+∇ · J
]= 0, (4.108)
5The macroscopic charge density is assumed to be the average of the microscopic one over a volume elementmuch bigger then the unit cell and smaller than the coherent volume of the applied electromagnetic field. It isreasonable to assume that, at equilibrium, this reference volume can be taken to be neutral, so that the chargedensity vanishes but eventually for the presence of a finite polarization P and of higher multipolar componentswhich derive from the shape of the bound-charge density. Away from equilibrium, the charge density can changealso because free electrons can move or because external charges are added. Hence, neglecting higher multipolesthan the dipole, the generic macroscopic charge density is
ρ(x) = ρfree(x)−∇ ·P(x) + ρext(x),
where ρfree is the variation of the free-charge density with respect to equilibrium. The distinction between freeand bound charges is however a bit matter of convention. Indeed, at frequencies much bigger then the atomicexcitation ones, there is actually no distinction among them. The convention that is usually adopted, which weadopt here as well, is that the frequencies under consideration will always be much smaller then atomic ones.
139
where we made use of the continuity equation
∂ρ
∂t+∇ · J = 0.
We can therefore assume∇ ·D = 0. (4.109)
If there is an external charge with density, then (4.108) becomes
1
c
d
dt∇ ·D =
4π
c
dρextdt
,
implying∇ ·D = 4πρext. (4.110)
In other words the displacement field is only sensible to the external charge.In the following we assume that the system is translationally invariant. In this case we can
formally write a relationship between the Fourier components of D, D(ω, q), and those of E,E(ω, q):
D(ω, q) = ε(ω, q)E(ω, q), (4.111)
where ε(ω, q) is the dielectric constant, which is actually a matrix acting on the space compo-nents, i.e. Di = εij Ej with i, j = 1, 2, 3. If the system is isotropic, we can generally write
εij(ω, q) =qiqjq2
ε||(ω, q) +
(δij −
qiqjq2
)ε⊥(ω, q), (4.112)
where ε||(ω, q) is the longitudinal component of the dielectric constant and ε⊥(ω, q) the trans-verse one. Analogously, for any vector V (ω, q) one can define the longitudinal componentthrough
q · V (ω, q) = q V||(ω, q),
as well as the transverse one
V ⊥(ω, q) = V (ω, q)− qqV||(ω, q).
In the presence of the electromagnetic field the Hamiltonian H = H0 +HCoul, where H0 isthe kinetic energy while HCoul includes the electron-electron, electron-ion and ion-ion Coulombinteraction, changes according to
H0 →∑σ
∫dx
1
2mΨσ(x)†
(−i~∇+
e
cA(x)
)2Ψσ(x) − eφ(x) Ψσ(x)†Ψσ(x). (4.113)
140
Since we already includes the Coulomb interaction among the internal charges inside HCoul, thedisplacement, electric and magnetic fields are, in terms of the vector and scalar potentials,
D|| = −∇φ− 1
c
∂
∂tA||, (4.114)
E⊥ = −1
c
∂
∂tA⊥, (4.115)
B = ∇ ∧A⊥. (4.116)
In other words the scalar potential and the longitudinal component of the vector potential arerelated only to the displacement field, which is in turn determined by the external charge through
∇ ·D|| = 4πρext.
Therefore φ and A|| are actually the external fields φext and A||ext applied on the system. Noticethat, since only D|| has physical meaning there is a redumdancy in having both φ and A||. Thisis a manifestation of the so-called gauge invariance. On the contrary, since we do not includethe relativistic current-current interaction among the internal charges, the transverse electricand magnetic fields are those actually felt by the electrons inside the sample, so called internalfields.
We conclude by noticing that, in the presence of a vector potential, the charge currentoperator as obtained through the continuity equation
∂ρ
∂t+∇ · J = 0,
changes according to
J(x) = −e∑σ
[−i ~m
Ψσ(x)†∇Ψσ(x) +e
mcΨσ(x)†Ψσ(x) A(x, t)
]≡ −e j(x)− e2
mcρ(x) A(x, t), (4.117)
where we defined as −e j(x) the purely electronic current, while the last term is commonlyreferred to as the diamagnetic term.
If we take into account also the Zeeman splitting, the Hamiltonian further contains the term
gµB∑αβ
∫dx B(x, t) · Ψα(x)† Sαβ Ψβ(x) = gµB
∫dx A⊥(x, t) · ∇ ∧ σ(x), (4.118)
whereσ(x) =
∑αβ
Ψα(x)† Sαβ Ψβ(x) ,
141
is the spin density operator, being S the spin-1/2 matrices. The Zeeman splitting contributesto the current with the transverse term6
δJ(x) = −c g µB∇ ∧ σ(x). (4.119)
4.6.1 Response to an external charge
Let us start by studying the response to an external charge density e ρext(x, t). For simplicitywe will represent the ions as an infinitely massive uniform background of positive charge whichhas only the effect of neutralizing on average the electronic charge. In other words, if e n is thecharge density of the background and ρ(x) the density operator for the electrons, then∫
dx〈ρ(x)〉 ≡ ρ(q = 0) = V n,
where V is the sample volume and ρ(q) the Fourier transform of the density. Notice that theoperator ρ(q = 0) is the total number of electrons, which is conserved, hence we can alwayswrite ρ(q = 0) = V n
If we choose a gauge in which A|| = 0, the coupling to the scalar potential φ(x, t) leads tothe perturbation
δH = −∫dx dy
e2
|x− y|ρext(x, t) (ρ(y)− n)
= − 1
V
∑q
4πe2
q2ρext(q, t) (ρ(−q)− V nδq0)
= − 1
V
∑q 6=0
4πe2
q2ρext(q, t)ρ(−q) ≡ −e 1
V
∑q 6=0
φ(q, t) ρ(−q), (4.120)
since the Fourier transform of the scalar potential is
φ(q, t) =4πe
q2ρext(q, t).
In the absence of the external charge, the electron density is uniform hence
〈ρ(y)〉 = n.
6Notice that the expression of the current can be also determined through the functional derivative
J(x, t) = −c δHδA(x, t)
.
142
The external charge induces a deviation of the electron density with respect to the unperturbedvalue n, which we denote as the induced charge
〈ρ(y)〉 − n = ρind(y, t). (4.121)
Its Fourier transform is, for q 6= 0,
ρind(q, t) = 〈ρ(q)〉,
while for q = 0 it vanishes for charge conservation. The induced charge density at frequency ωis given, within linear response theory, by
ρind(q, ω) = −eχ(q, ω)φ(q, ω) = −4πe2
q2χ(q, ω) ρext(q, ω), (4.122)
where
χ(q, ω) =1
V
∫ ∞−∞
dt eiωt (−iθ(t)〈[ρ(q, t), ρ(−q)]〉) , (4.123)
is the Fourier transform of the density-density linear response function. Commonly it is denotedas the improper density-density linear response function, as opposed to the proper one whichis defined through
ρind(q, ω) = −4πe2
q2χ(q, ω) (ρext(q, ω)− ρind(q, ω)) . (4.124)
The proper function gives the response of the electrons to the internal field, which is generatedboth by the external charge and by the same electrons. 7 Comparing (4.122) with (4.124) wefind that
χ(q, ω) =χ(q, ω)
1− 4πe2
q2χ(q, ω)
. (4.125)
By definition
iq ·D||(q, ω) ≡ iq · ε||(q, ω) E||(q, ω) = 4πeρext(q, ω),
iq ·E||(q, ω) = 4πeρext(q, ω)− 4πeρind(q, ω),
henceε||(q, ω) (ρext(q, ω)− ρind(q, ω)) = ρext(q, ω).
In other words
1
ε||(q, ω)= 1− ρind(q, ω)
ρext(q, ω), (4.126)
7The induced charge refers to electrons, which explains the minus sign in (4.124)
143
ε||(q, ω) = 1 +ρind(q, ω)
ρext(q, ω)− ρind(q, ω). (4.127)
Through Eqs. (4.122) and (4.126) it derives that
1
ε||(q, ω)= 1 +
4πe2
q2χ(q, ω), (4.128)
while by the Eqs. (4.124) and (4.127)
ε||(q, ω) = 1− 4πe2
q2χ(q, ω). (4.129)
These two equations relate the longitudinal dielectric constant to the density-density properand improper response functions. Notice that for the dielectric constant the Kramers-Kronigrelations read
Re 1
ε||(q, ω)= 1− P
∫dω′
π
1
ω − ω′Im 1
ε||(q, ω′). (4.130)
Using once more the definition of the displacement field, one finds that
d
dt∇ ·D = ∂t∇ ·E + 4π∇ · J,
where J is the current flowing inside the system.8 In Fourier space it reads
qωD||(q, ω) = qω E||(q, ω) + 4πiq · J(q, ω).
Since D|| = ε||E||, we find
J||(q, ω) =ω
4πi
(ε||(q, ω)− 1
)E||(q, ω) ≡ σ||(q, ω)E||(q, ω),
where σ|| is the longitudinal conductivity which relates the current to the internal electric field.Therefore
σ||(q, ω) =ω
4πi
(ε||(q, ω)− 1
), (4.131)
ε||(q, ω) = 1 +4πi
ωσ||(q, ω). (4.132)
8 Since we assumed an external probing charge ρext(x, t) which depends on time only explicitly, the totalflowing current includes just the induced one.
144
Gauge invariance and its consequences
So far we have adopted a gauge where the external scalar field is finite while the external longi-tudinal vector potential is zero. Now let us do the opposite, keeping the physical displacementfield fixed. By gauge invariance the response of the system should be the same. Therefore letus consider an external charge density eρext(x, t) leading to a displacement field
D||(q, ω) =e
iqρext(q, ω).
We assume that
D||(q, ω) =iω
cA||(q, ω) = ε||(q, ω)E||(q, ω).
A vector potential provides a perturbation
δH =e
c
∫dx A(x, t) · j(x) +
e2
2mc2
∫dx ρ(x) A(x, t) ·A(x, t), (4.133)
where the electronic current j has been defined by Eq. (4.117) and ρ(x) =∑
σ Ψσ(x)†Ψσ(x) isthe density operator. The longitudinal value of j is given in linear response by
j||(q, ω) =e
cχ||(q, ω)A||(q, ω),
where, ifχlm(x, t) = −iθ(t)〈[jl(x, t), jm(0, 0)]〉
is the current-current response function, then its longitudinal component is defined through
χ||(q, ω) =∑lm
∫dt eiωt
∫dx e−iq·x
qlqmq2
χlm(x, t).
On the other hand the true charge current operator, see Eq. (4.117), is
J(x, t) = −ej(x, t)− e2
mc〈ρ(x)〉A||(x, t).
Since the second term in the right hand side is already linear in the external field, the averageof the density operator has to be evaluated on the unperturbed state, where 〈ρ(x)〉 = n. Hence
J||(q, ω) = −ej||(q, ω)− ne2
mcA||(q, ω)
= −e2
c
(χ||(q, ω) +
n
m
)A||(q, ω)
= − e2
iω
(χ||(q, ω) +
n
m
)ε||(q, ω)E||(q, ω) ≡ σ||(q, ω)E||(q, ω).
145
Therefore we find that
σ||(q, ω) = − e2
iω
(χ||(q, ω) +
n
m
)ε||(q, ω),
which, compared with (4.131) leads to
− e2
iω
(χ||(q, ω) +
n
m
)ε||(q, ω) =
ω
4πi
(ε||(q, ω)− 1
),
namely
χ||(q, ω) +n
m= − ω2
4πe2
(1− 1
ε||(q, ω)
)=
ω2
q2χ(q, ω). (4.134)
This shows that gauge invariance implies that the longitudinal current-current response functioncan be expressed in terms of the density-density response function.
In principle we can also introduce a proper longitudinal current-current response functionthrough
J||(q, ω) = − e2
iω
(χ||(q, ω) +
n
m
)ε||(q, ω)E||(q, ω)
≡ − e2
iω
(χ||(q, ω) +
n
m
)E||(q, ω),
which is related by gauge invariance to the proper density-density response function through
q2(χ||(q, ω) +
n
m
)= ω2 χ(q, ω). (4.135)
We end by noticing that, in the gauge in which the scalar potential is zero, the internallongitudinal vector potential, defined by
A|| int(q, ω) =c
iωE||(q, ω),
is related to the external one,
A||(q, ω) = A|| ext(q, ω) =c
iωD||(q, ω),
through
A|| int(q, ω) =1
ε||(q, ω)A|| ext(q, ω). (4.136)
146
4.6.2 Response to a transverse field
By recalling the definition of the displacement field
d
dtD = ∂t E + 4πJ,
its transverse component is given by
−iω D⊥(q, ω) = −iω ε⊥(q, ω)E⊥(q, ω) = −iω E⊥(q, ω) + 4π J⊥(q, ω)
= (−iω + 4πσ⊥(q, ω)) E⊥(q, ω),
which provides the relationship between the transverse dielectric constant and conductivity:
ε⊥(q, ω) = 1 +4πi
ωσ⊥(q, ω). (4.137)
We already said that the transverse vector potential is actually the internal one, so the re-sponse to it is connected with proper response functions. Following the same reasoning as afterEq. (4.133) but now for the transverse current, we get to the conclusion that
J⊥(q, ω) = −e2
c
(χ⊥(q, ω) +
n
m
)A⊥(q, ω), (4.138)
where χ⊥(q, ω) is obtained by the current-current response function through
χlm(q, ω) =ql qmq2
χ||(q, ω) +
(δlm −
ql qmq2
)χ⊥(q, ω). (4.139)
Notice that, unlike the longitudinal one, the transverse component is proper. Since
J⊥(q, ω) = σ⊥E⊥(q, ω) =iω
cσ⊥A⊥(q, ω), (4.140)
we finally obtain
σ⊥(q, ω) = − e2
iω
(χ⊥(q, ω) +
n
m
). (4.141)
By means of (4.115) and (4.116) and the last Maxwell equation we get the usual wave-equation for the internal vector potential
−∇2 A⊥ = − 1
c2
∂2A⊥∂t2
+4π
cJ⊥ tot,
which in momentum and frequency space reads(c2 q2 − ω2
)A⊥(q, ω) = 4πcJ⊥ tot(q, ω). (4.142)
147
The current appearing in (4.142) is the total one, including the external source of the electro-magnetic field. The latter satisfies a similar equation(
c2 q2 − ω2)
A⊥ ext(q, ω) = 4πcJ⊥ ext(q, ω),
in terms of the external current source. Writing J⊥ tot = J⊥ ext + J⊥, with the latter being theinduced current as given by (4.140), we finally get(
c2 q2 − ω2)
A⊥(q, ω) =(c2 q2 − ω2
)A⊥ ext(q, ω) + 4πiωσ⊥(q, ω) A⊥(q, ω).
In other words, the internal vector potential, A⊥ = A⊥ int, is related to the applied one by theequation
A⊥ int(q, ω) =c2 q2 − ω2
c2 q2 − ω2 − 4πiωσ⊥(q, ω)A⊥ ext(q, ω)
=c2 q2 − ω2
c2 q2 − ω2 ε⊥(q, ω)A⊥ ext(q, ω), (4.143)
to be compared with (4.136) valid for the longitudinal component.Let us introduce back the Zeeman contribution to the transverse current given in Eq. (4.119).
Within linear response theory and assuming a uniform isotropic system
σ(x, t) = gµB
∫dy dt′ χσ(x− y, t− t′) B(y, t′) = gµB
∫dy dt′ χσ(x− y, t− t′)∇y ∧A⊥(y, t′),
where χσ is the spin-density response function. Therefore (4.119) is, at linear order,
δJ(x) = −c (gµB)2∫dy dt′∇x ∧
[χσ(x− y, t− t′)∇y ∧A⊥(y, t′)
]= c (gµB)2
∫dy dt′∇2χσ(x− y, t− t′) A⊥(y, t′), (4.144)
which, after Fourier transform, is
δJ⊥(q, ω) = −c q2 (gµB)2 χσ(q, ω) A⊥(q, ω).
This term modifies (4.143) into
A⊥ int(q, ω) =c2 q2 − ω2
c2 q2 − ω2 ε⊥(q, ω) + 4πc2q2 (gµB)2 χσ(q, ω)A⊥ ext(q, ω). (4.145)
4.6.3 Limiting values of the response functions
The values of the response functions in some particular limits of physical significance can beobtained without making any complicated calculation.
148
Consequences of the f-sum rule
The charge density operator at q = 0 is the total number of electrons N , which is a conservedquantity, independent upon time. Therefore
χ(q = 0, t) = −i 1
Vθ(t) 〈[ρ(0, t), ρ(0)]〉 = −i 1
Vθ(t) 〈[N,N ]〉 = 0.
In other words due to charge conservation it holds that
χ(0, ω) = 0 . (4.146)
The dissipative density-density response function is by definition
χ′′(q, t) =
1
2V〈[ρ(q, t), ρ(−q)]〉 =
∫dω
2πe−iωt χ
′′(q, ω).
By means of the continuity equation
i∂χ′′(q, t)
∂t=
1
2V〈[i∂ρ(q, t)
∂t, ρ(−q)
]〉
=1
2V〈[q · J(q, t), ρ(−q)]〉
=
∫dω
2πe−iωt ω χ
′′(q, ω).
On the other hand, for t = 0, the following expression holds
1
2V〈[q · J(q), ρ(−q)]〉 =
nq2
2m,
which leads to the so-called f -sum rule∫dω
2πω χ
′′(q, ω) =
nq2
2m. (4.147)
By the Kramers-Kronig relations
Re χ(q, ω) = P∫dω′
π
χ′′(q, ω′)
ω − ω′.
We know that the dissipative response function is finite only for frequencies which correspondto the energies of excitations which may be created by the density operators. In other wordsχ′′(q, ω′) is bounded from above. Therefore if we consider a frequency much above this upper
149
bound, and we recall that the dissipative response function is odd in frequency, then we findthat
limω→∞
Re χ(q, ω) =
∫dω′
π
χ′′(q, ω′)
ω
(1 +
ω′
ω
)=
1
ω2
∫dω′
πω′ χ
′′(q, ω′) =
nq2
mω2,
namely
limω→∞
Re χ(q, ω) =nq2
mω2. (4.148)
Alternatively, we can re-write the sum-rule (4.147) as
n
2m=
1
q2
∫dω′
2πω′ χ
′′(q, ω′) = − 1
q2
∫dω′
2πω′ Imχ(q, ω′)
= −∫dω′
2π
1
ω′Im
(χ||(q, ω
′) +n
m
).
In the limit q→ 0, there is no distinction between longitudinal and transverse response, so that,through Eq. (4.139), we obtain
n
2m= − lim
q→0
∫dω′
2π
1
ω′Im
(χ||(q, ω
′) +n
m
)
= − limq→0
∫dω′
2π
1
ω′Im
(χ⊥(q, ω′) +
n
m
)
=
∫dω′
2πIm
(i
e2σ⊥(0, ω′)
).
In other words, the f -sum rule can also be written as∫ ∞0
dωRe σ⊥(0, ω) =ne2
2m, (4.149)
which is the so-called optical f -sum rule.
By Eq. (4.128) we also find that
limω→∞
Re 1
ε||(q, ω)= 1 +
ω2p
ω2, (4.150)
150
− +
x
Figure 4.1: Pictorial representation of a plasmon excitation.
as well as ∫dω′
πω′ Im 1
ε||(q, ω′)= −ω2
p , (4.151)
where
ωp =
√4πne2
m(4.152)
is the so-called electron plasma frequency.
What is the meaning of the plasma frequency? Let us take a piece of metal and let usdisplace uniformly, i.e. at q = 0, the valence electrons with respect to the positive background9
by a distance x, as in Fig. 4.1. As a result a surface charge will appear σ = n e x, so that x issubject to a restoring Coulomb force
md2x
dt2= −4πeσ = −4πe2nx, (4.153)
leading to oscillations, so-called plasmons, with frequency ωp. In other words the only longitu-dinal modes at q = 0 allowed in a metal are plasmons. Taking into account (4.151), which isvalid for all q’s, hence also for q → 0, we conclude that
limq→0Im 1
ε||(q, ω)= −π
2ωp (δ(ω − ωp)− δ(ω + ωp)) , (4.154)
which, by Kramer-Kronig transformation, leads to
Re 1
ε||(0, ω)=
ω2
ω2 − ω2p
,
9In a real solid we have to imagine that the positive background includes the ions and the bound electrons.In other words the derivation implicitly assumes that we explore excitations at frequency smaller than interbandtransitions.
151
or, equivalently,
1
ε||(0, ω)= 1 +
ωp2
(1
ω − ωp + iη− 1
ω + ωp + iη
), (4.155)
with η an infinitesimal positive number. In addition
Re ε||(0, ω) = 1−ω2p
ω2.
Through (4.129) we also find that
limq→0
χ(q, ω) =n
m
q2
ω2, (4.156)
which, through the gauge relation (4.135), leads to
limq→0
χ||(q, ω) = 0 . (4.157)
Finally, using the definition of the longitudinal conductivity (4.131), we also find that
Re σ||(0, ω) = Re i ne2
m
1
ω + iη= π
ne2
mδ(ω) ,
which implies an infinite conductivity of an ideal metal.
Zero frequency response
Now suppose we change by the same amount the charge density of the positive background andof the electrons, for instance by changing the volume V . From the point of view of the electrons,this is analogous to changing the chemical potential, hence∫
dx 〈ρ(x)〉 − n = δN =∂N
∂µδµ.
The compressibility is defined by
K = − 1
V
∂V
∂P,
where P is the pressure. The free-energy F (N,V, T ) can be written as
F (N,V, T ) = V f(n, T ),
152
so that the pressure
P = −∂F∂V
= −f + n∂f
∂n,
hence1
K= n2 ∂
2f
∂n2.
On the other hand
µ =∂F
∂N=∂f
∂n,
∂ν
∂N=
1
V
∂2f
∂n2,
implying that∂N
∂µ= n2K V. (4.158)
From the point of view of the Hamiltonian, the action of changing the density of the back-ground and in the meantime changing the electron chemical potential so to maintain chargeneutrality corresponds to a time-independent perturbation
δH =1
V
∑q
Vext(−q, ω = 0) ρ(q),
where
Vext(q, ω = 0) = −4πe2
q2δρback(q, ω = 0)− V δµ δq,0.
The internal field is instead
Vint(q, ω = 0) = −4πe2
q2(δρback(q, ω = 0)− δρel(q, ω = 0))− V δµ δq,0,
and, by definition,δρel(q, ω = 0) = χ(q, ω = 0)Vint(q, ω = 0).
Since charge neutrality is maintained, in the limit q→ 0 we find
limq→0
Vint(q, ω = 0) = −V δµ,
hencelimq→0
δρel(q, ω = 0) = δN = −V δµ limq→0
χ(q, ω = 0) = n2K V δµ,
leading to the relationship
limq→0
χ(q, 0) = −n2K . (4.159)
153
In terms of the dielectric constant it implies that
limq→0
ε||(q, 0) = 1 +4πn2e2
q2K →∞ . (4.160)
The dielectric constant at zero frequency diverges when the momentum vanishes, provided thecompressibility is finite. What is the meaning of a finite or vanishing compressibility? We haveshown that
K =1
n2V
∂N
∂µ.
It the system is a metal, the number of electrons is a continuos and increasing function ofthe chemical potential, hence K is finite. On the contrary, if the system is an insulator, thechemical potential lies within an energy gap so that the number of electrons stays constantupon small changes of µ, implying K = 0. In other words a finite or vanishing compressibilitydiscriminates bewteen metals and insulators. Equivalently, a metal has a longitudinal dielectricconstant at zero frequency which diverges as q→ 0, while for an insulator this limit is finite. Tobetter appreciate this difference, let us consider the response to an external point-like charge,for instance at position x0
ρext(x, t) = Z δ(x− x0), ρext(q, ω) = Z δ(ω) e−iq·x0 .
According to our definitions, the induced density is
ρind(q, ω) = −4πe2
q2χ(q, ω) ρext(q, ω) = −Z 4πe2
q2
χ(q, ω)
ε||(q, ω)δ(ω) e−iq·x0
= Z δ(ω) e−iq·x0ε||(q, 0)− 1
ε||(q, 0)= ρext(q, ω)
ε||(q, 0)− 1
ε||(q, 0).
Therefore the net charge inside the system is
e (ρext(q, ω)− ρind(q, ω)) = e ρext(q, ω)1
ε||(q, 0).
Since ε||(q, 0) diverges for small q in a metal, then, in the same limit, the total charge inside thesystem vanishes, which implies that in a metal the induced charge is always such as to perfectlyscreen the external one,
Transverse response
We already know thatlimq→0
χ||(q, ω) = 0.
154
Since for q → 0 at fixed frequency there is no difference bewteen transverse and longitudinalfields, this also implies that
limq→0
χ⊥(q, ω) = 0 .
Through Eqs. (4.141) and (4.137) we then find
limq→0
σ⊥(q, ω) = −ω2p
4πiω, (4.161)
limq→0
ε⊥(q, ω) = 1−ω2p
ω2. (4.162)
On the other hand we also know that
limq→0
χ(q, 0) = −n2K,
so that from the gauge relation we should have
limq→0
χ||(q, 0) = − nm
.
The transverse response at ω = 0 is the response to a finite magnetic field in the absence ofelectric one, as E⊥ = iω A⊥/c. Since the transverse conductivity is not expected to be singularin the presence of a magnetic field, we have to conclude from (4.141) that also
limq→0
limω→0
χ⊥(q, ω) = − nm
.
The approach to this limiting value should be regular, so that we expect that
limq→0
limω→0
χ⊥(q, ω) = − nm
(1− q2
Ak2F
), (4.163)
with A a dimensionless constant and kF the Fermi momentum.10. Through Eq. (4.141) we thenfind
limq→0
−4πiω σ⊥(q, ω) = ω2p
q2
Ak2F
,
which inserted in (4.145) gives
A⊥ int(q, 0) =Ac2k2
F
Ac2k2F + ω2
p + 4π c2Ak2F (gµB)2 χσ
A⊥ ext(q, 0), (4.164)
10A = 4 for free electrons
155
being χσ the magnetic susceptibility, which we know is negative. Since an ω = 0 transversevector potential corresponds to a constant magnetic field in the absence of electric one, (4.164)implies that the magnetic field can penetrate inside the metal. The Zeeman term would leadto an increased amplitude with respect to the applied one (Landau paramagnetism), while thefree-carrier term to a decrease (Landau diamagnetism).11
On the contrary, since ε||(q, 0) diverges as q→ 0, from (4.136) one realizes that a longitudinalfield does not propagate freely inside a metal. Indeed, using Eq. (4.160), we can write
ε||(q, 0) = 1 +4πn2e2
q2K f(q),
with f(0) = 1. If f(q) were analytic, then through
A|| int(q, 0) =q2
q2 + 4πn2e2Kf(q)A|| ext(q, 0),
and upon Fourier transform in real space, we would find a longitudinal vector potential decayingexponentially inside the sample, with a decay length
λTF =1
4πn2e2,
which is called the Thomas-Fermi screening length.12 In reality the function f(q) is non-analytic.It has a branch cut starting from q = 2kF due to the Fermi surface. The reason is that it is notpossible to create a particle-hole excitation at zero-frequency if the transfered momentum q >2kF , hence the imaginary part of the density-density response function contains a step-functionwhich turns into a log-singularity in the real part. The final result is that the longitudinal fieldfar away from the external source decays as cos(2kF r)/r
3, so called Friedel oscillatory behavior.
4.6.4 Power dissipated by the electromagnetic field
In the presence of an external longitudinal field, the power dissipated according to the generalformula is
W =(ec
)2 ω
2
∣∣A|| ext(q, ω)∣∣2 χ′′||(q, ω) = −
(ec
)2 ω
2
∣∣A|| ext(q, ω)∣∣2 Imχ||(q, ω)
11For free electrons the diamagnetic term is 1/3 of the paramagnetic one, so the field is actually enhancedwithin the interior of the system.
12Indeed this is the same screening length one would obtain within the Thomas-Fermi approximation. Namely,if we consider the equation for the internal scalar potential in the presence of a time-independent external charge
∇2φ(x) = −4πρext(x)− 4πρind(x),
and we assume a very weak space-dependence so that
ρind(x) ' − e
V
∂N
∂µ(−eφ(x)) = e2n2K φ(x),
as if (−eφ(x)) acts like a chemical potential shift, we do find the Thomas-Fermi screening length.
156
−( eω
)2 ω
2
∣∣E|| ext(q, ω)∣∣2 Imχ||(q, ω),
where χ′′
||(q, ω) is the dissipative current-current response function which is equal to minus theimaginary part of the linear response function. On the other hand
Imχ||(q, ω) = Im(−i ωe2
σ||(q, ω)
ε||(q, ω)
)= − ω
e2Re
σ||(q, ω)
ε||(q, ω)= − ω
e2Re
(ω
4πi
(ε||(q, ω)− 1
) 1
ε||(q, ω)
)=
ω2
4πe2Im 1
ε||(q, ω),
Imχ||(q, ω) = − ωe2Re
σ||(q, ω)
ε||(q, ω)
= − ωe2Re
σ||(q, ω)
1 +4πi
ωσ||(q, ω)
= − ωe2
1
|ε||(q, ω)|2Re σ||(q, ω),
which, through |E|| ext| = |ε||| |E|| int|, implies that
W = − ω
8πIm 1
ε||(q, ω)
∣∣E|| ext(q, ω)∣∣2 =
1
2Re σ||(q, ω)
∣∣E|| int(q, ω)∣∣2 . (4.165)
Therefore the power dissipation in the presence of a longitudinal field is controlled either bythe imaginary part of the inverse dielectric constant or by the real part of the conductivity,depending whether one has access to the applied field or to the full internal one.
In the case of a transverse field
W = −(ec
)2 ω
2|A⊥ int(q, ω)|2 Imχ⊥(q, ω),
where
Imχ⊥(q, ω) = − ωe2Re σ⊥(q, ω) = − ω2
4πe2Imε⊥(q, ω).
Therefore
W =ω2
2c2Re σ⊥(q, ω) |A⊥ int(q, ω)|2 =
1
2Re σ⊥(q, ω) |E⊥ int(q, ω)|2
=ω
8π
ω2
c2Imε⊥(q, ω) |A⊥ int(q, ω)|2 =
ω
8πImε⊥(q, ω) |E⊥ int(q, ω)|2 . (4.166)
157
E rE
i
k i k r
z
x
E t
k t
Figure 4.2: Experimental geometry.
4.6.5 Reflectivity
Let us suppose to shoot a beam of light normally to the surface of the sample, which is in contactwith the vacuum. The experimental geometry is drawn in Fig. 4.2. The normal direction is thez-direction, so that z > 0 is the vacuum and z < 0 the sample, and the electric field is polarizedalong the x-direction. By the geometry of the scattering experiment, the incident, reflected andtransmitted waves have all the wave-vector along z, ki, kr and kt, respectively.
Through Eq. (4.143) it derives that the electromagnetic field propagates inside the sample if
q2 =(ωc
)2ε⊥(q, ω).
Since tha vacuum has ε⊥ = 1, the conservation of frequency implies
k2i = k2
r =k2t
ε⊥(kt, ω),
hence kr = −ki > 0. If the sample is non-magnetic, the electric and magnetic fields, which areboth parallel to the surface, should be continuous through the interface vacuum-sample:
Exi + Exr = Ext , Byi +By
r = Byt .
Notice thatByi =
c
ωkiE
xi ,
and analogously for the reflected and transmitted components. Hence Byi + By
r = Byt implies,
thorugh Exi + Exr = Ext , that
kr (Exr − Exi ) = ktExt = −
√ε⊥(kt, ω)kr (Exi + Exr ) ,
158
which leads to
Exr =1−
√ε⊥(kt, ω)
1 +√ε⊥(kt, ω)
Exi .
Therefore the reflection coefficient is
R =
∣∣∣∣ErEi∣∣∣∣2 =
∣∣∣∣∣1−√ε⊥(kt, ω)
1 +√ε⊥(kt, ω)
∣∣∣∣∣2
. (4.167)
Since the propagating wave-vector of the light is negligible at the typical excitation frequenciesof the sample, we can approximate ε⊥(kt, ω) ' ε⊥(0, ω). In a metal we showed that
ε⊥(0, ω) = 1−ω2p
ω2,
which is real and negative for |ω| < ωp. Through Eq. (4.167) it implies that a metal reflectsperfectly for frequencies below the plasma frequency.
159
4.7 Random Phase Approximation for the electron gas
So far we have introduced the linear response functions of an electron gas and discussed somelimiting values which can be obtained without resorting to any explicit calculation. In this sectionwe will evaluate the response functions within the time-dependent Hartree-Fock approximation.
As before we consider an electron gas in the presence of a positive background which neu-tralizes its charge. The Hamiltonian is
H =∑kσ
εk c†kσckσ +
1
2V
∑kpq
U(q) (ρ(q)− V nδq0) (ρ(−q)− V nδq0) , (4.168)
where εk = ~2|k|2/2m is the kinetic energy, U(q) = 4πe2/|q|2 and ρ(q) the Fourier transforms ofthe Coulomb interaction and of the electron density, respectively, and n the positive backgrounddensity. By definition the Fourier transform of the electron density at q = 0 is equal to V n,thus canceling the positive background charge. Therefore the interaction term is actually
Hint =1
V
∑q 6=0
∑kp
U(q) ρ(q)ρ(−q). (4.169)
Let us start with the Hartree-Fock approximation. If we assume that spin-rotational andspace-translational symmetries are preserved in the actual ground state, then we are forced tosearch for an Hartree-Fock wave-function whose only variational parameters are
〈c†kσckσ〉 = nkσ, (4.170)
with nk↑ = nk↓ = nk. Then, for q 6= 0,
ρ(q)ρ(−q) =∑kp
∑σσ′
c†kσck+qσ c†p+qσ′cpσ′
→ −∑kσ
nk c†k+qσck+qσ + nk+q c
†kσckσ,
so that the Hartree-Fock Hamiltonian is
HHF =∑kσ
εk − 1
V
∑q 6=0
U(q)nk+q
c†kσckσ, (4.171)
which is already diagonal with eigenvalues
εkHF = εk −1
V
∑q 6=0
U(q)nk+q.
160
Let us assume that the Fermi-sea solution, i.e. nk = 1 if |k| < kF and zero otherwise, is stillthe choice which minimizes the total energy, and let us apply the time-dependent Hartree-Fockmachinery to evaluate the response to an external field Vext(q, t) which couples to the density.We readily find that the equation of motion of the average values
∆k,k+q;σ = 〈c†kσck+qσ〉,
is, at linear order,
(i~∂t − εk+qHF + εkHF ) ∆k,k+q;σ =
(nk − nk+q)
Vext(q, t) +1
V
∑pσ′
∆p,p+q;σ′ (U(q)− δσσ′ U(k− p))
.Let us solve this equation neglecting the second term, i.e. the Fock contribution, in the
right hand side, which corresponds to the so-called Random Phase Approximation (RPA). Tobe consistent, the RPA approximation also amounts to neglect the interaction correction to theHF eigenvalues, which only derives from the Fock term, so that εkHF → εk. Within RPA wefind that
∆k,k+q;σ =nk − nk+q
ω − εk+q + εk
Vext(q, t) +U(q)
V
∑pσ′
∆p,p+q;σ′
. (4.172)
The induced charge is defined for as
ρind(q, ω) =∑kσ
〈c†kσck+qσ〉 − V nδq0,
and satisfy the RPA equation
ρind(q, ω) =
(∑kσ
nk − nk+q
ω − εk+q + εk
) Vext(q, t) +U(q)
V
∑pσ′
∆p,p+q;σ′
≡ V χRPA(q, ω)
[Vext(q, t) +
U(q)
Vρind(q, ω)
],
where
χRPA(q, ω) =1
V
∑kσ
nk − nk+q
ω − εk+q + εk + iη(4.173)
corresponds to the proper density-density response function within RPA, the infinitesimal η > 0playing the role of the adiabatic switching rate.
χRPA(q, ω) is also the density-density response function of free electrons, also called theLindhart function, which also clarifies the meaning of the RPA approximation. As we know the
161
proper response function is the response to the external field plus the electric field generated bythe same electrons. In order not to double-count this term, the proper response function hasto be evaluated neglecting the contribution of the electron-electron Coulomb repulsion whichrepresents the coupling of the electrons to the average electric field generated by the sameelectrons, which is roughly as if the eletrons were actually non-interacting. We leave as anexercise the evaluation of the Lindhart function in a three-dimensional. We just mention thatthe RPA approximation preserves the sum-rule (4.147), hence all the results which derives fromthat. Concerning the limiting value (4.159), within RPA the compressibility turns out to bethat one for free electrons.
One can perform the same analysis with an external probe which couples to the transversecurrent. The result would be a transverse current-current response function which, within theRPA approximation, coincides with that of free electrons. Also in this case the exact limitingvalues we have previously extracted are reproduced. In other words the Random Phase is aconsistent Approximation.
Excercises
(3) Calculate the Lindhart function for an εk = ~2k2/2m dispersion in three dimensions.
(4) Calculate thelimq→0
limω→0
χ⊥(q, ω),
of the transverse current-current response function for free electrons. Show that the limitcoincides with (4.163) with A = 4. Prove that the diamagnetic term is actually 1/3 of theparamagnetic one.
162
Chapter 5
Feynman diagram technique
Time-dependent Hartree-Fock is a relatively simple way to access excitation properties thusgoing beyond Hartree-Fock theory. A more systematic scheme, which includes time-dependentHartree-Fock as an approximation, is to perform perturbation theory. However the perturbationexpansion in a many-body problem is not as straightforward as in a single-body case but can besimplified substantially by means of the so-called Feynman diagram technique. In this chapterwe present this technique at finite temperature.
5.1 Preliminaries
First of all we need some preliminary definitions and results.
5.1.1 Imaginary-time ordered products
Let us introduce first the imaginary time evolution of an operator A(x) through
A(x, τ) = eH τ A(x) e−H τ , (5.1)
where H is the fully interacting Hamiltonian and τ ∈ [0, β], where β = 1/T is the inversetemperature (we use ~ = 1 and KB = 1).
Given two operators A(x) and B(y) we define the imaginary time Green’s function through
GAB(x,y; τ − τ ′) = −〈Tτ(A(x, τ)B(y, τ ′)
)〉
= −θ(τ − τ ′) 〈A(x, τ)B(y, τ ′)〉 ∓ θ(τ ′ − τ) 〈B(y, τ ′)A(x, τ)〉. (5.2)
where Tτ denotes the time-ordered product, and the minus sign applies when one or both theoperators are bosonic-like, namely contain bosons or an even product of fermionic operators,while the plus sign when both operators are fermionic-like, i.e. contain an odd product of
163
fermionic operators. Due to time-translation invariance, the Green’s functions only depend onthe time difference τ − τ ′ ∈ [−β, β].
Analogously, given n operators, Ai(xi, τi), i = 1, . . . , n, we can define the multi-operatorGreen’s function
GA1,...,An(x1, . . . ,xn; τ1, . . . , τn) = −〈Tτ (A1(x1, τ1) . . . An(xn, τn))〉, (5.3)
which is the average of the time ordered product, namely operators with earlier times appear onthe right of those at later times, having a minus sign if the number of crossings needed to bringfermionic-like operators in the correct time-ordered sequence is even and the sign plus otherwise.
5.1.2 Matsubara frequencies
Let us consider the two-operator Green’s function GAB(x,y; τ), with τ ∈ [−β, β] being the timedifference. Since the time domain is bounded, we can introduce a discrete Fourier transform,with frequencies ωn = (2π/2β)n = π nT , with n an integer. We define, dropping for simplicitythe spatial dependence,
GAB(iωn) =1
2
∫ β
−βdτ eiωnτ GAB(τ),
and, consequently,
GAB(τ) = T∑n
e−iωnτ GAB(iωn).
From the properties of the trace it derives that
〈B(0)A(τ)〉 =1
ZTr[e−β H B eH τ A eH− τ
]=
1
ZTr[e−β H eβ H eH τ A e−H τ e−β H B
]= 〈A(β + τ)B(0)〉,
which implies that
GAB(iωn) =1
2
∫ β
−βdτ eiωnτ GAB(τ)
= −1
2
∫ β
0dτ eiωnτ 〈A(τ)B(0)〉 ∓ 1
2
∫ 0
−βdτ eiωnτ 〈B(0)A(τ)〉
= −1
2
∫ β
0dτ eiωnτ 〈A(τ)B(0)〉 ∓ 1
2
∫ 0
−βdτ eiωnτ 〈A(β + τ)B(0)〉
= −1
2
∫ β
0dτ eiωnτ 〈A(τ)B(0)〉 ∓ 1
2e−iωnβ
∫ β
0dτ eiωnτ 〈A(τ)B(0)〉
=1
2
(1± e−iωnβ
)∫ β
0dτ eiωnτ GAB(τ).
164
Since exp (−iωnβ) = (−1)n then a non-zero Fourier transform requires for bosonic-like operatorseven integers n’s and for fermionic-like odd n’s. We define the bosonic Matsubara frequencies
Ωm = 2mπ T (5.4)
and the fermionic Matsubara frequencies
ωm = (2m+ 1)π T, (5.5)
so that the bosonic- and fermionic-like Fourier transform of GAB(τ) become, respectively,
GAB(Ωm) =
∫ β
0dτ eiΩmτ GAB(τ), GAB(ωm) =
∫ β
0dτ eiωmτ GAB(τ), (5.6)
hence just require the knowledge of the Green’s functions for positive τ .
Connection between bosonic Green’s functions and linear response functions
Let us assume that both A and B are observable quantities, which implies they are hermitean,hence bosonic-like operators. Then, for positive τ , we have
GAB(τ) = −〈A(τ)B(0)〉 = − 1
Z
∑n
e−βEn 〈n|eH τ A e−H τ B|n〉
= − 1
Z
∑nm
e−βEn e−(Em−En)τ 〈n|A|m〉〈m|B|n〉.
Since ∫ β
0dτ eiΩlτ e−(Em−En)τ =
1
iΩl − (Em − En)
(e−(Em−En)β − 1
),
we finally find that
GAB(iΩl) =1
Z
∑nm
(e−βEn − e−βEm
) 1
iΩl − (Em − En)〈n|A|m〉〈m|B|n〉. (5.7)
Comparing this expression with the Fourier transform in frequency of the linear response func-tion χAB(ω), see Eq. (3.33), we easily realize that the two coincide if we analytically continueGAB(iΩl) on the real axis, iΩl → ω+ iη. In other words the knowledge of GAB(iΩl) means thatwe know the value of the function GAB(z) of the complex variable z on an infinite set of pointson the imaginary axis zl = iΩl. This is sufficent to determine the full GAB(z) in the complexplane, with the supplementary condition that, since GAB(z) = χAB(z), it has to be analytic inthe upper half-plane.
165
Useful formulas
There is a very useful trick to perform summation over Matsubara frequencies. Let us startfrom the fermionic case. Suppose we have to calculate
T∑n
F (iεn),
where we assume that F (z) vanishes faster than 1/|z| for |z| → ∞ and has poles but on theimaginary axis. We notice that the Fermi distribution function, as function of a complex variable,
f(z) =1
eβ z + 1,
has poles at
zn = i (2n+ 1)π
β= iεn,
namely right at the Matsubara frequencies, with residue −T . Let us consider the integral alongthe contour shown in Fig. 5.1
Re(z)
Im(z)
Figure 5.1: Integration countour, see text.
I =
∮dz
2πif(z)F (z).
This countour integral can be calculated by catching all poles of the integrand in the regionenclosed by the countour which includes the imaginary axis, shaded area in Fig. 5.1. Since thisarea is enclosed clockwise the integral gives
I = −∑zn
Res (f(zn)) F (zn) = T∑n
F (iεn),
166
which is just the sum we want to calculate. On the other hand, I can be equally calulated bycatching all poles of the integrand in the area which does not include the imaginary axis, thenon-shaded region, which, being enclosed anti-clockwise gives
I =∑
z∗= poles of F (z)
f(z∗) Res (F (z∗)) .
ThereforeT∑n
F (iεn) =∑
z∗= poles of F (z)
f(z∗) Res (F (z∗)) . (5.8)
Notice that, if F (z) has branch cuts the calculation is slightly more complicated since we haveto deform the contour as to avoid branch cuts. In this case, instead of catching poles we haveto integrate along branch cuts. For instance, suppose that F (z) has a branch cut at z = x+ iω,with x ∈ [−∞,∞], as shown in Fig. 5.2. Let us consider the non-shaded area enclosed inside
Im(z)
Re(z)
Figure 5.2: Integration countour in the presence of a branch cut, see text.
the countour depicted in Fig. 5.2. In this area there are no poles hence the contour integral iszero. On the other hand this contour integral is also equal to the contribution of the poles insidethe shaded area, which includes the imaginary axis, plus the integral along the contour whichencloses the branch cut. Therefore
T∑n
F (iεn) = −∫
dx
2πif(x+ iω)
[F (x+ iω + i0+)− F (x+ iω − i0+)
]. (5.9)
In the case in which the summation is performed over bosonic frequencies, we can use similartricks once we recognize that the poles of the Bose distribution function
b(z) =1
eβ z − 1,
167
coincide with the bosonic Matsubara frequencies,
zn = i 2nπ
β,
with residue T .
5.1.3 Single-particle Green’s functions
Among the averages of time-ordered products an important role in the diagrammatic techniqueis played by the so-called single-particle Green’s functions.
Fermionic case
If Ψσ(x, τ) is the imaginary-time evolution of the Fermi field, then the single-particle Green’sfunctions is defined through
Gσσ′(x,y; τ) = −〈Tτ(
Ψσ(x, τ) Ψσ′(y)†)〉
= −θ(τ)〈Ψσ(x, τ) Ψσ′(y)†〉+ θ(−τ)〈Ψσ′(y)†Ψσ(x, τ) 〉. (5.10)
If we make a spectral representation as before and calculate the Fourier transform using thefermionic Matsubara frequency, we readily find that
Gσσ′(x,y; iωl) =1
Z
∑nm
(e−βEn + e−βEm
) 1
iΩl − (Em − En)〈n|Ψσ(x) |m〉〈m|Ψσ′(y)†|n〉.
Let us take σ = σ′ and x = y, which correspond to the so-called local Green’s function, andintroduce the real and positive spectral function
Aσ(x, ε) =1
Z
∑nm
(e−βEn + e−βEm
) ∣∣∣〈n|Ψσ(x) |m〉∣∣∣2 δ (ε− Em + En) ,
through which, after continuation in the complex plane iωl → z
Gσσ(x,x; z) =
∫dεAσ(x, ε)
1
z − ε.
As function of the complex frequency, G(z) has generally branch cut singularities along the realaxis. Indeed
Gσσ(x,x; z = ω + i0+)−Gσσ(x,x; z = ω − i0+) = −2π iAσ(x, ω).
168
What is the physical meaning of the spectral function? Let us rewrite Aσ(x, ω) in the followingequivalent way:
Aσ(x, ε) =∑nm
e−βEn
Z
∣∣∣〈m|Ψσ(x)†|n〉∣∣∣2 δ (ε− Em + En)
+∑nm
e−βEn
Z
∣∣∣〈m|Ψσ(x) |n〉∣∣∣2 δ (ε+ Em − En) , (5.11)
which shows more explicitly that Aσ(x, ω) is just the local density of states for adding, firstterm in the right hand side, or removing, second term, a particle at position x with spin σ. Thisquantity can be for instance measured in tunneling experiments. Finally we notice that∫
dεAσ(x, ε) =∑nm
e−βEn
Z
( ∣∣∣〈m|Ψσ(x)†|n〉∣∣∣2 +
∣∣∣〈m|Ψσ(x) |n〉∣∣∣2 )
=∑n
e−βEn
Z
(〈n|Ψσ(x) Ψσ(x)†|n〉+ 〈n|Ψσ(x)†Ψσ(x) |n〉
)=
∑n
e−βEn
Z〈n|
Ψσ(x) ,Ψσ(x)†|n〉 = 1,
namely that the integral of the spectral function is normalized to one.
In all this chapter we are going to assume a grand-canonical ensamble for the fermions, whichis equivalent to add to the fermionic Hamiltonian H a chemical potential term, i.e. H → H−µN ,where µ is such that the average number of particles has the desired value N0. At T = 0 µbecomes the Fermi energy and selects as the ground state the lowest energy state with N0
particles. Therefore, at T = 0 the sum over n is reduced just to the ground state, |0, N0〉, withN0 particles, hence
Aσ(x, ε) =∑m
δ(ε+ E0(N0)− Em(N0 + 1)
) ∣∣∣〈m,N0 + 1|Ψσ(x)†|0, N0〉∣∣∣2
+∑m
δ(ε− E0(N0) + Em(N0 − 1)
) ∣∣∣〈m,N0 − 1|Ψσ(x) |0, N0〉∣∣∣2 ,
where we explicitly indicate that the ground state energy refers to N0 particles and the first termin the right hand side involves moving to the subspace with N0 + 1 electrons, and the second tothe subspace with N0 − 1 electrons. The first δ-function implies that
ε = Em(N0 + 1)− E0(N0) = (Em(N0 + 1)− E0(N0 + 1)) + (E0(N0 + 1)− E0(N0)) ≥ 0,
169
because the energy of the state m with N0 + 1 electrons is greater or equal to the ground stateenergy E0(N0 + 1), and, by definition, E0(N0 + 1) − E0(N0) ≥ 0 since the lowest energy statefor any N occurs at N0. Analogously, in the second δ-function
ε = − (E0(N0 − 1)− E0(N0))− (Em(N0 − 1)− E0(N0 − 1)) ≤ 0.
In other words, for positive ε the spectral function is the density of states upon adding anelectron and for negative values that one for removing one particle.
Let us suppose now that the system is translationally invariant. If we then expand the Fermifields in eigenstates of the crystalline momentum, and introduce the space Fourier transform ofthe Green’s function, we find
Gσσ′(k,p; τ) = −〈Tτ(ckσ(τ) c†pσ′
)〉 = δkpGσσ′(k, τ),
whereGσσ′(k, τ) = −〈Tτ
(ckσ(τ) c†kσ′
)〉. (5.12)
In this case and for σ = σ′ the Fourier transform in the complex frequency plane would be
Gσσ(k, z) =
∫dεAσ(k, ε)
1
z − ε,
where
Aσ(k, ε) =1
Z
∑nm
(e−βEn + e−βEm
) ∣∣∣〈n|ckσ|m〉∣∣∣2 δ (ε− Em + En) , (5.13)
is the density of states for adding and removing a particle at momentum k. Since A(k, ε) is realand positive, G(k, z) has generally branch cuts on the real axis, since
Gσσ(k, ε+ i0+)−Gσσ(k, ε− i0+) = −2πiAσ(k, ε). (5.14)
Notice that, as before, the spectral function is normalized to one. In addition one can readilyverify that
Aσ(x, ε) = Aσ(ε) =1
V
∑k
Aσ(k, ε). (5.15)
Let us consider non-interacting electrons described by the Hamiltonian
H0 =∑kσ
εk c†kσckσ,
where, as discussed previously, εk is measured with respect to the chemical potential µ, whichis the Fermi energy at T = 0. In this case it is easy to show that
G(0)σσ′(k, iωn) = δσσ′ G
(0)(k, iωn),
170
where
G(0)(k, iωn) =1
iωn − εk. (5.16)
The spectral function is simplyA(0)(k, ε) = δ(ε− εk), (5.17)
and is finite for ε > 0 if k is outside the Fermi surface, and for ε < 0 otherwise. Notice that asmall frequency ε corresponds to small εk, i.e. to momenta close to the Fermi surface.
Bosonic case
Let us define as Φ(x) the Bose field for spinless bosons. Analogously to the fermionic case wecan define a single-particle Green’s function through
G(x,y; τ) = −〈Tτ(
Φ(x, τ) Φ(y)†)〉
= −θ(τ)〈Φ(x, τ) Φ(y)†〉 − θ(−τ)〈Φ(y)†Φ(x, τ) 〉.
Let us again assume that the model is translationally invariant and introduce the bosonic oper-ators aq and a†q, as well as the conjugate variables
xq =
√1
2
(aq + a†−q
),
pq = −i√
1
2
(aq − a
†−q
).
In most common situations, the object which appears within perturbation theory is, rather thanthe single-particle Green’s function, the x− x time-ordered product, namely
D(q, τ) = −〈Tτ (xq(τ)x−q)〉. (5.18)
Let us consider the bosonic non-interacting Hamiltonian
H0 =∑q
ωq a†q aq,
with ωq = ω−q. Then, for positive τ ,
D(q, τ) = −1
2〈aq(τ) a†q〉 −
1
2〈a†−q(τ) a−q〉
= −1
2e−ωqτ (1 + b(ωq))− 1
2eωqτ b(ωq),
171
where
b(ωq) = 〈a†q aq〉 =(
eωqβ − 1)−1
,
is the Bose distribution function, and
〈aq a†q〉 = 1 + b(ωq) = −(
e−ωqβ − 1)−1
.
Noticing that ∫ β
0dτ eiΩmτ e∓ωqτ =
1
iΩm ∓ ωq
(e∓ωqβ − 1
),
one readily finds that
D(0)(q, iΩm) =1
2
(1
iΩm − ωq− 1
iΩm + ωq
)= − ωq
Ω2m + ω2
q
. (5.19)
5.2 Perturbation expansion in imaginary time
Let us suppose that the full Hamiltonian
H = H0 + V,
where H0 is a single particle Hamiltonian which can be diagonalized exactly while V is a per-turbation which makes the model not solvable, e.g. it is the electron-electron interaction. Wealready showed in Section 4.2 that
e−τ H = e−τ H0 S(τ),
where, see Eq. (4.34),
S(τ) ≡ S(τ, 0) = Tτ
[exp
(−∫ τ
0dτ1 V (τ1)
)], (5.20)
being V (τ) the Heisenberg evolution of V with the non-interacting Hamiltonian. It is easy toshow that, for τ ≥ τ ′,
S(τ − τ ′, 0) = Tτ
[exp
(−∫ τ−τ ′
0dτ1 V (τ1)
)]= Tτ
[exp
(−∫ τ
τ ′dτ1 V (τ1 − τ ′)
)]= e−τ
′H0 Tτ
[exp
(−∫ τ
τ ′dτ1 V (τ1)
)]eτ′H0 ≡ e−τ
′H0 S(τ, τ ′) eτ′H0 ,
namely thate−(τ−τ ′)H = e−τ H0 S(τ, τ ′) eτ
′H0 . (5.21)
172
Suppose we have to calculate the multi-operator Green’s function (5.3), and further assumethat, e.g., τ1 ≥ τ2 ≥ · · · ≥ τn−1 ≥ τn, then
GA1,...,An(τ1, . . . , τn) = −〈A1(τ1) . . . An(τn)〉
=1
ZTr[e−βH eτ1H A1 e−τ1H eτ2H A2 e−τ2H . . . e−τn−1H eτnH An e−τnH
]=
1
ZTr[e−βH0 S(β, τ1) eτ1H0 A1 e−τ1H0 S(τ1, τ2) eτ2H0 A2 e−τ2H0 . . .
. . . S(τn−1, τn) eτnH0 An e−τnH0 S(τn, 0)].
We readily see that all times remain ordered, so that for a generic time-ordering we would get
GA1,...,An(τ1, . . . , τn) = −〈Tτ (A1(τ1) . . . An(τn))〉
=1
ZTr[e−βH0 Tτ (S(β)A1(τ1) . . . An(τn))
],
where the time evolution of the Ai’s operators in the last equation is through the non-interactingHamiltonian. Recalling that
Z = Z0 〈S(β)〉,
we therefore conclude that
GA1,...,An(τ1, . . . , τn) = − 1
〈S(β)〉〈Tτ (S(β)A1(τ1) . . . An(τn))〉, (5.22)
where the thermal averages as well as the imaginary-time evolution are done with the non-interacting Hamiltonian H0. This expression is now suitable for an expansion in V .
5.2.1 Wick’s theorem
Upon expanding S(β) in powers of the perturbation V , the calculation of any Green’s functionreduces to evaluate the average value of a time-ordered product of Fermi or Bose fields with anon interacting Hamiltonian. It is therefore essential to know how to perform this calculation.
Let us suppose to evaluate with a non-interacting Hamiltonian for free fermions the averagevalue
−〈Tτ(
Ψ(x1, τ1) Ψ(x2, τ2) . . . Ψ(xn, τn) Ψ(yn, τ′n)† . . . Ψ(y1, τ
′1)†)〉.
For any time-ordering this amounts to average a product of creation and annihilation operatorsover a non interacting Hamiltonian. We already know that this is the sum of the products ofall possible contractions of an annihilation with a creation operator. Suppose that the operatorΨ(xi, τi) is contracted with Ψ(yj , τ
′j)†. If all other times but τi and τ ′j are the same, there are
two cases: if τi ≥ τ ′j we will have to evaluate the contraction
〈Ψ(xi, τi) Ψ(yj , τ′j)†〉;
173
if τi ≤ τ ′j we need to interchange the two operators, which leads to a minus sign hence to
−〈Ψ(yj , τ′j)†Ψ(xi, τi)〉.
Both cases can be described by the single formula
−G(xi,yj ; τi − τ ′j).
This argument can be extended to all other contractions, justifying the result that
The average value with a non-interacting Hamiltonian of the time-ordered productof Fermi fields is obtained by contracting in all possible ways an annihilation with acreation operator and regarding any contraction as a non-interacting single-particleGreen’s function. The sign of each product of contractions is (−1)n (−1)L, where nis the number of annihilation operators, which has to be equal to the number ofcreation operators, and L is the number of crossings needed to bring each creationoperator to the right of the annihilation operator with which it is contracted.
This is the so-called Wick’s theorem. For instance
−〈Tτ(
Ψ(x1, τ1) Ψ(x2, τ2) Ψ(y2, τ′2)†Ψ(y1, τ
′1)†)〉
= −G(0)(x1,y1; τ1 − τ ′1)G(0)(x2,y2; τ2 − τ ′2) +G(0)(x1,y2; τ1 − τ ′2)G(0)(x2,y1; τ2 − τ ′1).
In the case of non interacting bosons the Wick’s theorem does not hold rigorously. Yet, ifwe work with a translationally invariant model and if there is no Bose condensation, the Wick’stheorem still works, this time without any phase-factor (−1)L, apart from corrections whichvanish in the thermodynamic limit.
5.3 Perturbation theory for the single-particle Green’s functionand Feynman diagrams
Let us consider the Hamiltonian for free electrons
H0 =∑kσ
εk c†kσckσ, (5.23)
in the presence of an interaction
Hint =1
2
∑σσ′
∫dx dy Ψσ(x)†Ψσ′(y)† U(x− y) Ψσ′(y) Ψσ(x) ,
174
which will be our perturbation V . In the S-matrix it appears∫dτ Hint(τ) =
1
2
∑σσ′
∫dx dy
∫dτ Ψσ(x, τ)†Ψσ′(y, τ)† U(x− y) Ψσ′(y, τ) Ψσ(x, τ)
=1
2
∑σσ′
∫dx dy
∫dτ dτ ′Ψσ(x, τ)†Ψσ′(y, τ
′)† U(x− y) δ(τ − τ ′) Ψσ′(y, τ′) Ψσ(x, τ)
≡ 1
2
∑σσ′
∫dx dyΨσ(x)†Ψσ′(y)† U(x− y) Ψσ′(y) Ψσ(x) ,
where we have introduced the multicomponent coordinates x = (x, τ), y = (y, τ ′) and bydefinition
U(x− y) = U(x− y) δ(τ − τ ′).
Therefore
S(β) = Tτ
[exp
(−1
2
∑σσ′
∫dx1 dx2 Ψσ(x1)†Ψσ′(x2)† U(x1 − x2) Ψσ′(x2) Ψσ(x1)
)].
The single-particle Green’s function is, as demonstrated previously,
Gσ(x, y) = − 1
〈S(β)〉〈Tτ
(S(β) Ψσ(x) Ψσ(y)†
)〉, (5.24)
where the averages are done with the non interacting Hamiltonian. Let us calculate it up tofirst order in perturbation theory, namely with
S(β) ' S(0)(β) + S(1)(β) = 1− 1
2
∑σσ′
∫dx1 dx2 Ψσ(x1)†Ψσ′(x2)† U(x1 − x2) Ψσ′(x2) Ψσ(x1) .
We need to calculate up to first order the numerator and the denominator of (5.24). About thenumerator and using Wick’s theorem we find
−〈Tτ(S(β) Ψσ(x) Ψσ(y)†
)〉 = G(0)
σ (x, y)
+1
2
∑αβ
∫dx1 dx2 U(x1 − x2) 〈Tτ
[Ψσ(x) Ψσ(y)† Ψα(x1)†Ψβ(x2)†Ψβ(x2) Ψα(x1)
]〉
= G(0)σ (x, y)
(1 + 〈S(1)(β)〉
)+
1
2
∑αβ
∫dx1 dx2 U(x1 − x2)
[δσαG
(0)σ (x, x1)G(0)
σ (x1, y)G(0)β (x2, x2)
−δσα δαβ G(0)σ (x, x1)G(0)
σ (x1, x2)G(0)σ (x2, y)
]175
G (x,y)(0)
x y
x y=
= G(x,y)
x
y
U(x−y)=
Figure 5.3: Graphical representation of the Green’s functions and the interaction.
+1
2
∑αβ
∫dx1 dx2 U(x1 − x2)
[δσβ G
(0)σ (x, x2)G(0)
σ (x2, y)G(0)α (x1, x1)
−δσβ δβαG(0)σ (x, x2)G(0)
σ (x2, x1)G(0)β (x1, y)
]The last two integrals are actually equal since we can interchange x1 ↔ x2, so that
−〈Tτ(S(β) Ψσ(x) Ψσ(y)†
)〉 = G(0)
σ (x, y)(
1 + 〈S(1)(β)〉)
+∑αβ
∫dx1 dx2 U(x1 − x2)
[δσαG
(0)σ (x, x1)G(0)
σ (x1, y)G(0)β (x2, x2)
−δσα δαβ G(0)σ (x, x1)G(0)
σ (x1, x2)G(0)σ (x2, y)
]The denominator gives simply 1 + 〈S(1)(β)〉 so that, up to first order
Gσ(x, y) = G(0)σ (x, y) +
∫dx1 dx2 U(x1 − x2)
G(0)σ (x, x1)G(0)
σ (x1, y)∑β
G(0)β (x2, x2)
− G(0)σ (x, x1)G(0)
σ (x1, x2)G(0)σ (x2, y)
]. (5.25)
Let us give a graphical representation of this result. We represent the fully interacting Green’sfunction as a line with an arrow, the non-interacting one as a tiner line and the interaction as awavy line with four legs, see Fig. 5.3. An incoming vertex represent a creation operator, whilean outgoing one an annihilation operator. With these notations the Green’s function up tofirst order in the interaction can be represented as in Fig 5.4. The conventions are that anyinternal coordinate is integrated, the spin is conserved along a Green’s function as well as at anyinteraction-vertex, and a wavy line is the interaction U . There are two first order diagrams. Thetadpole one has a plus sign while the other one a minus sign. We notice that both diagrams arefully connected. Actually the disconnected pieces cancel with the denominator 〈S(β)〉. One cango on and calculate the second order corrections to infer the rules for constructing diagrams.Here we just quote the final answer.
176
The n-th order diagrams for the single particle Green’s functions are all fully con-nected and topologically not equivalent diagrams which can be constructed with ninteraction lines and 2n+ 1 non-interacting Green’s functions, with external pointsat x and y. All internal coordinates are integrated. The sign of each diagram is(−1)n (−1)L, where L is the number of internal loops. Spin is conserved at each ver-tex, hence each loop implies a spin sum. Finally, if a Green’s function is connectedby the same interaction line it has to be interpreted as the limit of τ → 0−, since inthe interaction creation operators are on the left of the annihilation ones.
One easily realizes that the following rules reproduce the two first order corrections we havejust derived.
In Fig. 5.5 we draw all topologically inequivalent diagrams at second order in perturbationtheory. Let us follow the above rules to find the expression of the last two, (h) and (i). Theformer has no loops, hence its sign is simply (−1)n = (−1)2 = 1. The spin is the same for alllines hence its expression is
(h) =
∫ 4∏i=1
dxi U(x1−x3)U(x2−x4)G(0)σ (x, x1)G(0)
σ (x1, x2)G(0)σ (x2, x3)G(0)
σ (x3, x4)G(0)σ (x4, y).
Diagram (i) has a loop, hence (−1)n (−1)L = (−1)2 (−1)1 = −1. The internal loop implies aspin summation, hence
(i) = −∑β
∫ 4∏i=1
dxi U(x1−x3)U(x2−x4)G(0)σ (x, x1)G(0)
σ (x1, x2)G(0)σ (x2, y)G
(0)β (x3, x4)G
(0)β (x4, x3).
5.3.1 Diagram technique in momentum and frequency space
Let us assume that, besides time-translational invariance, the system is also space translationalinvariant. Therefore let us derive the perturbation expansion of the Fourier transform of the
x y=
x y
x y
x 1
x2
x y
x x1 2
+
+
Figure 5.4: Graphical representation of the Green’s function up to first order.
177
(a)(b) (c)
(d)(e) (f)
(g)
(h)(i)
x yx y
x x x x x
x
x1 12 3 4
x
2
3 4
Figure 5.5: Graphical representation of the second order corrections to the Green’s function.
Green’s function Gσ(p, iεn). We further need to introduce the Fourier transform of the inter-action line as well as we need to indicate a direction to this line, which is the direction along
iω lq,
iω lq,U( )
iε np, iε n iω l+p+q,
iεmk,iεm iω l+k+q,
Figure 5.6: Graphical representation of the interaction in Fourier space.
which momentum and frequency flow, see Fig. 5.6. Notice that the frequency carried by theinteraction is the difference between two fermionic Matsubara frequencies, hence it is a bosonicone, namely an even multiple of π T . Moreover, since U(x− y) = U(x−y) δ(τ − τ ′), the Fouriertransform is U(q, iωl) = U(q), independent of frequency. Moving to the perturbation expansion,since the spatial and time coordinates of each internal vertex are integrated out, momentum andfrequency are conserved at each vertex, namely the sum of the incoming values is equal to thatof the outgoing ones. Therefore the rules are:
The n-th order diagrams for the single particle Green’s functions with momentum pand frequency iεn are all fully connected and topologically not equivalent diagramswhich can be constructed with n interaction lines and 2n+1 non-interacting Green’sfunctions, with external lines at (p, iεn). At each vertex, the sum of momenta and
178
that of frequencies of the incoming lines must be equal to the those of outgoinglines. All internal momenta and frequency are summed up, sum over momentum,e.g. k, being
1
V
∑k
,
and over frequency, e.g. iεm,
T∑m
.
The sign of each diagram is (−1)n (−1)L, where L is the number of internal loops.Spin is conserved at each vertex, hence each loop implies a spin sum. Finally, if aGreen’s function is connected by the same interaction line it has to be interpretedas the inverse Fourier transform at τ = 0−, i.e.
T∑m
Gσ(k, iεm) e−iεm 0− .
For instance up to first order the diagrams are those in Fig. 5.7 and are explicitly
Gσ(p, iεn) = G(0)σ (p, iεn) +G(0)
σ (p, iεn)2 U(0)T∑m
1
V
∑k
∑β
G(0)β (k, iεm) e−iεm 0−
−G(0)σ (p, iεn)2 T
∑l
1
V
∑q
U(q)G(0)σ (p− q, iεn − iωl) e−i(εn−ωl) 0− ,
where we use the convention that ε denote fermionic Matsubara frequencies and ω bosonic ones.
ε nip,ε nip,
εi m
0, 0
k,
ε nip, ε nip,
ε ni i ω l
i ω lq,
p−q, −
ε niε ni
= +p,p,
+
Figure 5.7: Diagrams for the single-particle Green’s function up to first order in Fourier space.
179
+
+ +
++Σ = =
Figure 5.8: Diagrams for the single-particle self-energy up to second order.
5.3.2 The Dyson equation
Among all possible diagrams for the single particle Green’s function, we can distinguish twoclasses. The first includes all diagrams which, by cutting an internal Green’s function line,transform in two lower-order diagrams of the same Green’s function. These kind of diagramsare called single-particle reducible. For instance diagrams (a), (b), (c) and (d) in Fig 5.5 aresingle-particle reducible. The other class contains all diagrams which are not single-particlereducible, also called single-particle irreducible. For instance both first order diagrams as wellas the second order ones (e) to (i) are irreducible. Let us for convenience introduce a generalizedmomentum p = (p, iεn) and let us define as the single-particle self-energy Σσ(p) the sum of allirreducible diagrams without the external legs. For instance, the self-energy diagrams up tosecond order are drawn in Fig. 5.8, where the self-energy is represented by a rounded box. It
Σ( )p
Σ( )p Σ( )p
Σ( )p
= +
p p p p
+ + .....p p p
= +
p p p
Figure 5.9: Dyson equation for the single-particle Green’s function.
is not difficult to realize that, in terms of the self-energy, the perturbation expansion can be
180
rewritten as in Fig. 5.9, which has the formal solution
Gσ(p, iεn) =G(0)σ (p, iεn)
1−G(0)σ (p, iεn) Σσ(p, iεn)
=1
iεn − εp − Σ(p, iεn), (5.26)
the last expression being valid for H0 in Eq. (5.23), which, being spin-independent like theinteraction, can not produce two different self-energies for spin-up and down electrons. This isthe so-called Dyson equation for the single-particle Green’s function. There are therefore twopossible ways of doing perturbation theory. The simplest is just to do perturbartion theory,say up to order n, directly for the Green’s function. The second is to obtain up to order n theself energy and insert its expression in the Dyson equation (5.26). This provides in principle anapproximate Green’s function which contains all order in perturbation theory, so (5.26) actuallyrepresents a way to sum up perturbation theory. This latter is in reality the most physical wayto proceed. The reason is that the perturbation is going to change the branch cut singularities ofthe Green’s function, which can be traced easier using (5.26) than directly from the perturbativeexpansion of G.1
Physical meaning of the self-energy
The single particle Green’s function has a branch cut on the real axis, see (5.14), which impliesthat the real part is continuous but the imaginary part has a jump. This also means that thereal part of the self-energy is continuous but the imaginary part is not, so that we can define
Σ′(k, ε) = ReΣ(k, ε+ iη) = ReΣ(k, ε− iη),
1A simple way to explain it is the following. Let us take the Green’s function for non-interacting fermions
G(iεn) =1
iεn − E(V ),
where E(V ) are the single-particle eigenvalues which depend on some Hamiltonian parameter V , so that
E(V = 0) = E0 +∑n
En Vn,
and the unperturbed Green’s function at V = 0 is
G(0)(iεn) =1
iεn − E0.
The calculation of the self-energy in perturbation theory gives simply the perturbation expansion of the eigenvalueshence tells us how these eigenvalues changes with V . On the contrary the perturbation expansion of the Green’sfunction
G = G(0) + V(G(0)
)2E1 V1 + V 2
((G(0)
)2E2 +
(G(0)
)3E2
1
)+ . . . ,
is of difficult interpretation since at each order in perturbation theory all lower order corrections to the eigenvalueintervene.
181
Σ′′(k, ε) = ImΣ(k, ε+ iη) = −ImΣ(k, ε− iη) ≤ 0,
where η → 0+ and the last inequality follows from the fact that the spectral function is positiveand given by
A(k, ε) =1
π
η − Σ′′(k, ε)(ε− εk − Σ′(k, ε)
)2+(η − Σ′′(k, ε)
)2 .For non-interacting electrons Σ = 0 and the spectral function tends towards a δ-function δ(ε−εk)for η → 0. For interacting electrons this δ-function broadens on a width which is controlled byΣ′′(k, ε). This broadening reflects the fact that, for instance, a particle, εk > 0, can decay by theinteraction into a particle plus several particle-hole pairs provided that momentum is conserved.The product of this decay will have an energy which is spread around the non-interacting valueεk. Therefore Σ′′(k, ε) can be regarded as the decay rate of a particle with momentum k into anobject with the same momentum but energy ε. This interpretation can be justified by analyzingthe structure of the self-energy diagrams which contribute to the imaginary part. Let us considerfor instance the second-order diagram for the self-energy Σ(k, iε) drawn in Fig. 5.10, Neglecting
p1
ω1
,i
k1 ε 1,i
k1p
1k − + ε,i ε 1i ω
1i− +
Figure 5.10: Second-order diagram contributing to the imaginary part of the self-energy.
the momentum dependence of the interaction, its expression is
δΣ(k, iε) = −2U2 1
V 2
∑k0 k1 p1
T 2∑ε1 ω1
δ(k0 − k− p1 + k1
)1
iε1 − εk1
1
iω1 − εp1
1
iε+ iω1 − iε1 − εk0
,
where the δ-function imposes the momentum conservation and the factor 2 comes from the loopspin summation. Let us first sum over iε1, which gives
T∑ε1
1
iε1 − εk1
1
iε+ iω1 − iε1 − εk0
= f(εk1)1
iε+ iω1 − εk1 − εk0
−f(iε+ iω1 − εk0)1
iε+ iω1 − εk0 − εk1
182
=(f(εk1)− f(−εk0)
) 1
iε+ iω1 − εk0 − εk1
,
where we used the fact the iε + iω1 is a bosonic frequency. Notice that for T = 0 this term isfinite only if k1 and k0 are both particles or holes, namely if εk0εk1 > 0. Now let us sum overiω1,
T∑ω1
1
iω1 − εp1
1
iε+ iω1 − εk0 − εk1
= f(εp1)1
iε+ εp1 − εk0 − εk1
+f(εk0 + εk1 − iε)1
εk0 + εk1 − iε− εp1
=(f(εp1) + b(εk0 + εk1)
) 1
iε+ εp1 − εk0 − εk1
.
Here we used the fact that, since iε is fermionic, then
f(εk0 + εk1 − iε) =(
eβ(εk0+εk1−iε) + 1)
=(− eβ(εk0+εk1 ) + 1
)= −b(εk0 + εk1).
Since at T = 0 b(x) = −θ(−x) and f(x) = θ(−x), the sum over iω1 vanishes if εp1 < 0 andεk0 + εk1 < 0 or if εp1 > 0 and εk0 + εk1 > 0. Since εk0εk1 > 0 for the sum over iε1 to be finite,the only possibilities are: (1) εk0 > 0, εk1 > 0 and εp1 < 0 or (2) εk0 < 0, εk1 < 0 but nowεp1 > 0.
Now we send iε→ ε+ i0+ with ε > 0 and just consider the imaginary part, namely
1
iε+ εp1 − εk0 − εk1
→ 1
ε+ i0+ + εp1 − εk0 − εk1
= −iπδ(ε+ εp1 − εk0 − εk1
).
The δ-function implies thatεk0 + εk1 − εp1 = ε > 0,
which is only compatible with the above possibility (1), i.e. εk0 > 0, εk1 > 0 and εp1 < 0, thusleading to
ImδΣ(k, ε+i0+) = −2π iU2 1
V 2
∑k0 k1 p1
δ(k0−k−p1+k1
)δ(ε+εp1−εk0−εk1
)f(−εk1) f(−εk0) f(εp1).
This expression coincides with the Fermi golden rule for the probability of a particle at mo-mentum k to decay into two particles and one hole with the same total momentum but energyε.
More generally, the diagrams which contribute to the imaginary part of the self-energy canbe represented as in Fig. 5.11, where a particle with momentum k and frequency iε decays by
183
..
..
.
k k
Figure 5.11: Graphical representation of a self-energy diagram which contribute to the imaginarypart.
a matrix element drawn as a box into a particle plus a certain number, say m ≥ 1, of particle-hole pairs,2 which recombine through the other box into the original particle. Let us neglectfor simplicity the momentum and frequency dependence of the boxes, and define as ki(εi) andpi(ωi), i = 1,m, the momenta(frequencies) of the particles and holes, respectively, of the mparticle-hole pairs, and as k0(ε0) that of the additional particle. By momentum and frequencyconservation
k0 +∑i
ki − pi = k, ε0 +∑i
εi − ωi = εn.
As before, if we sum over all the 2m independent internal frequencies and finally send iε →ε+ i0+, with ε > 0, we end up with expression for the imaginary part proportional to
∑k0
∑ki pi
δ(ε−
∑j
(εkj − εpj
)− εk0
)δ(k0 +
∑j
(kj − pj)− k)f(−εk0)
m∏j=1
f(−εkj ) f(εpj ).
Since εki ≥ 0 and εpi ≤ 0, the energy conservation implies that, for small ε, all particles andholes should lye very close to the Fermi surface, on a strip of width at most δk = ε/vF , where vFis the Fermi velocity. Therefore the phase space available for the decay process grows at mostlike ε2m, since there are only 2m free summations over momentum, k0 being fixed by momentumconservation.
The final conclusion is therefore that, within perturbation theory,
limε→0
Σ′′(k, ε) = limε→0ImΣ(k, ε+ iη) = 0, (5.27)
2Two lines which propagate in opposite directions correspond to two Green’s functions one at positive timeand the other at negative time. Positive time means that first we create an electron and later we annihilate it,which denote a so-called particle excitation. On the contrary, negative time means that first we annihilate anelectron and later we create it back, so-called hole excitation.
184
5.4 Other kinds of perturbations
So far we have just constructed and analysed the perturbation expansion in terms of the electron-electron interaction. Let us consider now what changes for different types of perturbations. Inparticular we will just briefly mention two perturbations: a scalar potential and the coupling tobosonic modes.
5.4.1 Scalar potential
Let us suppose to have non-interacting electrons identified by a quantum label a, with non-interacting Green’s functions
G(0)ab (x,y; τ) = δabG
(0)a (x,y; τ),
in the presence of the scalar potential
V =∑ab
∫dx Ψa(x)V (x)ab Ψb(x). (5.28)
The S-matrix is now
S(β) = Tτ
(exp
(−∑ab
∫ β
0dτ
∫dx Ψa(x, τ) V (x)ab Ψb(x, τ)
))
≡ Tτ
(exp
(−∑ab
∫dxΨa(x) V (x)ab Ψb(x)
)),
where, as before, we have introduced the four-dimensional coordinate x = (x, τ). Upon ex-panding S(β) up to first order and keeping only connected diagrams, one can readily obtain theGreen’s function expansion
Gab(x, y) = δabG(0)a (x, y) +
∑cd
∫dz V (z)cd 〈Tτ
(Ψa(x) Ψb(y)†Ψc(z)
†Ψd(z))〉conn
= δabG(0)a (x, y) +
∫dz V (z)abG
(0)a (x, z)G
(0)b (z, y). (5.29)
If we represent graphically the perturbation as in Fig. 5.12(a), then the Green’s function (5.29)up to first order can be drawn as the first two diagrams in the right hand side of Fig. 5.12(b).Higher order terms can be simply obtained by inserting other potential lines, as the second orderterm in Fig. 5.12(b). All diagrams have a positive sign. The Dyson equation can be easily readout:
Gab(x, y) = δabG(0)a (x, y) + +
∑c
∫dz V (z)acG
(0)a (x, z)Gcb(z, y), (5.30)
and is drawn in Fig. 5.12(b).
185
Vab
a b
(a)
a b a b a bc= + + + ... = +
a a a c b
(b)
Figure 5.12: (a) Graphical representation of the scalar potential perturbation. (b) Perturbationexpansion of the Green’s function and Dyson equation.
5.4.2 Coupling to bosonic modes
D (q)(0)
p+q p
k k+q
|g(q)| 2
p+q p
k k+q
g(q)
g(−q)
=
x
x
q
−q
Figure 5.13: On the left: electron-phonon vertices at q and −q. The external dotted vertex linerepresents phonon coordinates. On the right: the effective electron-electron interaction aftercontracting phonons.
Let us imagine now that our system of electrons is coupled to phonons described by the freeHamiltonian
Hphon =∑q
ωq a†qaq.
The coupling term is assumed to be
V =∑q
g(q)xq ρ(−q), (5.31)
186
where g(q)∗ = g(−q) is the coupling constant, the phonon coordinate is
xq =
√1
2
(aq + a†−q
),
andρ(q) =
∑σ
∑k
c†kσck+qσ,
is the density operator. This perturbation is actually more general and describes any kind ofelectron-boson coupling.
= +
+ +
+
+ + . . .
Figure 5.14: First orders in the diagrammatic perturbation expansion of D(q). The exact D(q)is represented by a bold dashed line, while D(0)(q) by a thiner dashed line. The solid lines areelectron Green’s functions.
We can perform perturbation theory in (5.31) using the Wick theorem both for the electronsand for the bosons. The latter amounts to contract an xq with x−q which leads to the freepropagator D(0)(q, iΩn) of Eq. (5.19). Therefore, if we represent the electron-phonon couplingas in Fig. 5.13, in which a dotted vertex line represents the phonon-coordinate, by contractingtwo vertices one recovers an effective electron-electron interaction, see also Fig. 5.13, which ismediated by the phonons and given by
|g(q)|2D(0)(q, iωl) = −|g(q)|2 ωq
ω2l + ω2
q
. (5.32)
187
This effective interaction is retarded, namely depends on the frequency, and attractive. Morespecifically, if we move on the real axis, then
|g(q)|2D(0)(q, iωl → ω)→ −|g(q)|2 ωq
ω2q − ω2
,
which is attractive if |ω| < ωq and repulsive otherwise. The rules for constructing diagrams aretherefore the same as for the electron-electron interaction, with the additional complication thatinteraction is frequency dependent.
However, in this particular case one may also investigate the effects of the electron-phononcoupling to the phonon Green’s function D(q, iωl). The perturbation expansion is shown up tosecond order in Fig. 5.14. Two kinds of diagrams can be identified. The first class includes dia-grams which can be divided into two diagrams of the same expansion by cutting non-interactingD(0)(q)-line, like the first second order diagram shown in Fig. 5.14. These diagrams are re-ducible. The other class includes all other diagrams which are irreducible. If we define aself-energy |g(q)|2 Π(q, iωn) through these irreducible diagrams, shown up to second order inFig. 5.15, we can easily derive the Dyson equation, also shown in the same figure, whose solutionis
D(q, iωn) =D(0)(q, iωn)
1− |g(q)|2D(0)(q, iωn) Π(q, iωn). (5.33)
= +
= + + + + . . .
Figure 5.15: Upper panel: Phonon self-energy, represented by a filled circle, up to second order.Lower panel: Dyson equation for D(q).
188
5.5 Two-particle Green’s functions and correlation functions
Let us consider back the Hamiltonian of interacting electrons, which we assumed to be
H =∑kσ
εk c†kσckσ +
1
2V
∑kpq
∑αβ
U(q) c†pαc†k+qβ ckβcp+qα. (5.34)
The Heisenberg imaginary time evolution of an annihilation operator is
−∂ckσ(τ)
∂τ= [ckσ(τ), H(τ)] = εk ckσ(τ)
+1
V
∑pqα
U(q) c†p+qα(τ)cpα(τ)ck+qσ(τ).
Therefore the Green’s function satisfies the equation of motion
− ∂
∂τGσ(k, τ) = − ∂
∂τ
[−〈Tτ
(ckσ(τ) c†kσ
)〉]
= − ∂
∂τ
[−θ(τ) 〈ckσ(τ) c†kσ〉+ θ(−τ) 〈c†kσ ckσ(τ)〉
]= δ(τ)− 〈Tτ
(∂ckσ(τ)
∂τc†kσ
)〉
= δ(τ) + εkGσ(k, τ)− 1
V
∑pqα
U(q) 〈Tτ(c†p+qα(τ) cpα(τ) ck+qσ(τ) c†kσ
)〉.
This equation can be written as(− ∂
∂τ ′− εk
)Gσ(k, τ ′) = δ(τ ′)− 1
V
∑pqα
U(q) 〈Tτ(c†p+qα(τ ′) cpα(τ ′) ck+qσ(τ ′) c†kσ
)〉. (5.35)
The non-interacting Green’s function satisfies on the contrary(− ∂
∂τ ′− εk
)G(0)σ (k, τ ′) = δ(τ ′).
If we multiply both sides of Eq. (5.35) by G(0)σ (k, τ − τ ′) and integrate over τ ′, we obtain∫
dτ ′G(0)σ (k, τ − τ ′)
(− ∂
∂τ ′− εk
)Gσ(k, τ ′) = G(0)
σ (k, τ)
− 1
V
∑pqα
U(q)
∫dτ ′G(0)
σ (k, τ − τ ′) 〈Tτ(c†p+qα(τ ′) cpα(τ ′) ck+qσ(τ ′) c†kσ
)〉,
189
and upon integrating by part the left hand side, one obtains
Gσ(k, τ) = G(0)σ (k, τ)
− 1
V
∑pqα
U(q)
∫dτ ′G(0)
σ (k, τ − τ ′) 〈Tτ(c†p+qα(τ ′) cpα(τ ′) ck+qσ(τ ′) c†kσ
)〉.(5.36)
Therefore the single-particle Green’s function can be expressed in terms of a two-particle Green’sfunction. Namely, let us define the two-particle Green’s function
Kσ1σ2;σ3σ4(p1τ1,p2τ2; p3τ3,p4τ4) = −〈Tτ(cp1σ1(τ1)cp2σ2(τ2) c†p3σ3(τ3)c†p4σ4(τ4)
)〉, (5.37)
where, by momentum conservation,
p1 + p2 = p3 + p4,
in terms of which
Gσ(k, τ) = G(0)σ (k, τ)
− 1
V
∑pqα
U(q)
∫dτ G(0)
σ (k, τ − τ ′)Kσα;ασ(k + q τ ′,p τ ′; p + q τ ′,k 0).(5.38)
5.5.1 Diagrammatic representation of the two-particle Green’s function
Let us write formally
Kσ1σ2;σ3σ4(p1τ1,p2τ2; p3τ3,p4τ4) = −δσ1σ4δσ2σ3δp1p4δp2p3 Gσ1(p1, τ1 − τ4)Gσ2(p2, τ2 − τ3)
+δσ1σ3δσ2σ4δp1p3δp2p4 Gσ1(p1, τ1 − τ3)Gσ2(p2, τ2 − τ4)
+
∫ 4∏i=1
dτ ′i Gσ1(p1, τ1 − τ ′1)Gσ2(p1, τ2 − τ ′2)Gσ3(p3, τ′3 − τ3)Gσ4(p4, τ
′4 − τ4)
Γσ1σ2;σ3σ4(p1τ′1,p2τ
′2; p3τ
′3,p4τ
′4), (5.39)
which is represented graphically to Fig. 5.16. We notice that, in the absence of interaction, onlythe first two terms survive with the Green’s functions being the non-interacting ones. If westart doing perturbation theory, which has the same rules as before, we will (1) dress the non-interacting Green’s functions, which explains the first two terms; (2) couple them by interactionlines, which justify the last contribution and provides the definition of the interaction vertex Γ,which in lowest order perturbation theory is shown in Fig. 5.17.
In conclusion the equation (5.38) of the single particle Green’s function in terms of the two-particle one can be expressed as function of the single particle Green’s function itself and theinteraction vertex as shown in Fig. 5.18. Notice that the interaction vertex acts as the bareinteraction, so it carries a (-1) sign. Since there is a loop in the third diagram of Fig. 5.18, theoverall sign is plus. This result also provides an expression for the single-particle self-energy, asshown in Fig. 5.19.
190
1 4
2 3− +
1 4
2 3
Γ(1,2;3,4)2
1 4
3
+
K(1,2;3,4) =
Figure 5.16: Graphical representation of the two-particle Green’s function.
p
k+qk
p+q
k+qk
p+q p
+
+ .....+ +
p+q p
k k+q
= −U(q) U(p−k)
p+q p
p+q
h+q h
h+q
k k+q
p
p+k−hk k+q
Figure 5.17: Lowest order expansion of the interaction vertex. p, k, p+ q and k + q label bothmomenta and frequencies.
5.5.2 Correlation functions
We already showed that the average value of the imaginary-time ordered product of two bosonic-like operators can provide, after analytic continuation on the real axis, the linear responsefunctions. Let us then consider two single-particle density operators:
A(q) =∑kαβ
λAk,k+q;αβ c†kαck+qβ, B(q) =
∑kαβ
λBk,k+q;αβ c†kαck+qβ.
191
= +
+ +
Figure 5.18: Dyson equation in terms of the interaction vertex Γ.
+ +Σ =
Figure 5.19: Single-particle self-energy in terms of the interaction vertex Γ.
Their Green’s function, usually called correlation function, is (we use the notation GAB = χABsince we know that the two coincide on the frequency real axis):3
χAB(q, τ) = − 1
V〈Tτ (A(q, τ)B(−q))〉 = − 1
V
∑kαβ
∑p γδ
λAk,k+q;αβ λBp+q,p;γδ
〈Tτ(c†kα(τ)ck+qβ(τ)c†p+qγcpδ
)〉
=1
V
∑kαβ
∑p γδ
λAk,k+q;αβ λBp+q,p;γδ Kβδ;γα(k + q τ,p 0; p + q 0,k τ)
= − 1
V
(∑kα
λAk,k;ααGα(k, 0−)
)(∑p γ
λBp+q;γγ Gγ(p, 0−)
)
+1
V
∑kαγ
λAk,k+q;αγ λBk+q,k;γα Gα(k + q, τ)Gγ(k,−τ)
3 In the equation the Green’s functions at equal times have to be interpreted as the limit of τ → 0−, since inthe definition of A and B creation operators are on the left of the annihilation ones.
192
+1
V
∑kαγ
λAk,k+q;αγ λBk+q,k;γα
∫ 4∏i=1
dτi Gβ(k + q, τ − τ1)Gδ(p, 0− τ2)
Gγ(p + q, τ3 − 0)Gα(k, τ4 − τ) Γβδ ; γα(k + q τ − τ1,p − τ2; p + q τ3,k τ4 − τ)
= − 1
V〈A(q)〉 〈B(−q)〉
+1
V
∑kαγ
λAk,k+q;αγ λBk+q,k;γα Gα(k + q, τ)Gγ(k,−τ)
+1
V
∑kαγ
λAk,k+q;αγ λBk+q,k;γα
∫ 4∏i=1
dτi Gβ(k + q, τ − τ1)Gδ(p, 0− τ2)
Gγ(p + q, τ3 − 0)Gα(k, τ4 − τ) Γβδ ; γα(k + q τ − τ1,p − τ2; p + q τ3,k τ4 − τ).
The disconnected term 〈A(q)〉 〈B(−q)〉 vanishes for q 6= 0. In general it can be moved onthe left hand side, in which case the correlation function is the average of the time-orderedproduct minus the product of the averages. With this definition, the correlation function hasthe graphical representation shown in Fig. 5.20. In the figure, the triangular vertices represent
λ A λ Bλ A λ B
Γ(k
+q
,p;p
+q
,k)k+q p+q
pk
χAB
(q) =
k+q
k
+
Figure 5.20: Graphical representation of the correlation function χAB(q), where q includes bothmomentum q and a bosonic frequency iωl.
the matrix elements λA(B)k,k+q;αγ and the loops imply as usual summation over the spin-indices.
Perturbation theory has the same rules as before with the only exception that the loop phase-factor is now (−1)L−1 due to the minus sign in the definition of χ. If we Fourier transform alsoin imaginary time, the correlation function
χAB(q, iωl),
depends on the transferred momentum, q, and frequency, iωl. Since the operators are bosonic-like, iωl is a bosonic Matsubara frequency. By time-translation invariance, the frequency isconserved at any vertex, as usual, and the interaction vertex becomes consequently
Γβδ ; γα(k + q iεn + iωl,p iεm; p + q iεm + iωl,k iεn), (5.40)
as shown in Fig. 5.21.
193
ε ni ω li+k+q, ε nik,
εmip, εmi ω li+p+q,
β α
γδ
Γ
Figure 5.21: Graphical representation of the interaction vertex of Eq. (5.40).
Non-interacting values
In the absence of interaction only the first term in Fig. 5.20 survives with the Green’s functionlines being the non-interacting ones. Since there is a loop, according to what we said before, thesign is (−1)L−1 = (−1)1−1 = 1 hence
χ(0)AB(q, iωl) =
1
V
∑k
T∑n
∑αγ
λAk,k+q;αγ λBk+q,k;γα G
(0)α (k + q, iεn + iωl)G
(0)γ (k, iεn)
=1
V
∑k
T∑n
∑αγ
λAk,k+q;αγ λBk+q,k;γα
1
iεn + iωl − εk+q
1
iεn − εk. (5.41)
By means of Eq. (5.8) we can easily perform the sum over frequencies in Eq. (5.41), whichgives
T∑n
1
iεn + iωl − εk+q
1
iεn − εk= f(εk+q − iωl)
1
εk+q − iωl − εk+ f(εk)
1
iωl − εk+q + εk
=f(εk)− f(εk+q)
iωl − εk+q + εk,
where we used the fact that, if iωl is bosonic, then f(ε± iωl) = f(ε). The final result is therefore
χ(0)AB(q, iωl) =
1
V
∑k
∑αγ
λAk,k+q;αγ λBk+q,k;γα
f(εk)− f(εk+q)
iωl − εk+q + εk. (5.42)
The linear response function is simply obtained by iωl → ω + iη.
5.6 Coulomb interaction and proper and improper response func-tions
Let us consider the case in which the electron-electron interaction is just the Coulomb repulsion
U(q) =4πe2
q2,
194
which is singular for q → 0. In order to cure this singularity, it is convenient to recast per-turbation theory in a different manner, which naturally brings to identify proper and improperresponse functions. In Fig. 5.22 we draw the diagrammatic expansion of the density-densitycorrelation function χ(q, iωn), which is also the improper response function, up to first order.We can distinguish already two kinds of diagrams: those which can be cut into two by cutting
= + + + + + . . . qχ( )
Figure 5.22: Diagrammatic expansion of the density-density correlation function up to firstorder.
an interaction line, as the last diagram, and those which can not, all the others, which we callirreducible with respect to the interaction. Let us define as χ(q, iωn) the sum of all irreduciblediagrams, in the above sense, whose first order corrections are drawn in Fig. 5.23. We canformally define χ(q, iωn) as
χ(q, iωn) = 2T∑m
1
V
∑k
G(k+q, iεm+ iωn)G(k, iεm) Λ(k+q iεm+ iωn , k iεm; q iωn), (5.43)
by introducing the proper density-vertex function Λ(k, iεm; q, iωn), also shown in the samefigure, so to distinguish between Green’s function corrections and vertex corrections. Noticethat without interaction the density-vertex function is the identity matrix in spin space.
In terms of χ(q, iωn) the perturbation expansion of χ(q, iωn) can be recast as in the lowerpanel of Fig. 5.23, which has the formal solution4
χ(q, iωn) =χ(q, iωn)
1− 4πe2
q2χ(q, iωn)
, (5.44)
proving that χ(q, iωn) is actually the proper response function.
5.6.1 Screened interaction and corresponding Dyson equation
The proper density-density correlation function allows us to define a screened Coulomb interac-tion and a dielectric constant through
W (q, iωn) =4πe2
q2 ε(q, iωn)=
4πe2
q2
1
1− 4πe2
q2χ(q, iωn)
, (5.45)
195
~ qχ( )
==qχ( ) +
+ + +==
k+q
k
= +k+q,k;qΛ( )~
= + . . .
+ . . .
Figure 5.23: Upper panel: The proper χ(q) up to first order. Middle panel: The proper density-vertex function up to first order. Lower panel: Perturbation expansion of the improper χ(q, iωn)in terms of the proper χ(q).
= +
= +−Γ
Figure 5.24: Upper panel: Screened Coulomb interaction W . Lower panel: Interaction vertexin terms of the screened interaction and the proper vertex.
which is drawn as a bold wavy line in Fig. 5.24. The interaction vertex Γ which we previouslyintroduced can be also expressed in terms of the screened Coulomb interaction and the properdensity-vertex, as also shown in Fig. 5.24. Using this definition for the interaction vertex in theexpression of the self-energy, see Fig. 5.19, one finds the alternative definition of the self-energywhich is drawn in Fig. 5.25
4Notice both the interaction and χ(q, iωn), which is a loop, bring a minus sign, hence the sign is plus.
196
Σ = +
Figure 5.25: Self energy in terms of the screened W and the proper density-vertex.
5.7 Irreducible vertices and the Bethe-Salpeter equations
Let us consider again the interaction vertex (5.40). Let us for instance focus on the upper or thelower incoming and outgoing lines which differ by the momentum/frequency transfer q. Thisidentifies the particle-hole channel at momentum/frequency transfer q. By this definition, wecan distinguish two classes of diagrams in the perturbation expansion. The first includes thosediagrams which can be divided in two by cutting two fermionic lines within the same particle-hole channel at momentum/frequency q. These type of diagrams are called reducible in theparticle-hole channel. For instance the third diagram in Fig. 5.17 belongs to this class. Theother class includes all other diagrams which are called irreducible, as the fourth diagram inFig. 5.17. Let us denote the sum of all irreducible diagrams as
Γ0βδ ; γα(k + q iεn + iωl,p iεm; p + q iεm + iωl,k iεn). (5.46)
In terms of Γ0 the perturbation expansion of Γ can be formally written as
Γαβ ; γδ(k + q iεn + iωl,p iεm; p + q iεm + iωl,k iεn)
= Γ0αβ ; γδ(k + q iεn + iωl,p iεm; p + q iεm + iωl,k iεn)
+1
V
∑h
T∑h
∑µν
Γ0αµ ; νδ(k + q iεn + iωl,h iεh; h + q iεh + iωl,k iεn)Gν(h + q, iεh + iωl)
Gµ(h, iεh) Γνβ ;µδ(h + q iεh + iωl,p iεm; p + q iεm + iωl,h iεh), (5.47)
which is graphically shown in Fig. 5.26. This is the so-called Bathe-Salpeter equation whichrelates the fully reducible interaction vertex Γ with its irreducible part Γ0 in the particle-holechannel. Actually, we can proceed in a different way. Namely, instead of selecting the particle-hole channel at momentum transfer q, we can select the two incoming lines, particle-particlechannel, which have total momentum/frequency P = k + p + q. P is also conserved in thescattering process represented by the interaction vertex. Then we can introduce the concept ofreducibility/irreducibility in the particle-particle channel at total momentum/frequency P andidentify the irreducible vertex in that channel. Eventually we end up to another Bethe-Salpeterequation, which is shown in Fig. 5.27. In this figure we have denoted the irreducible vertex in
197
Γ0
Γ0
k+q k+q k+q
k k k
p+q p+q p+q
pp p ph
h+q
= +Γ Γ
Figure 5.26: Bethe-Salpeter equation for the interaction vertex in the particle-hole channel.
pp Γ0
pp
P−k
k k kpp p ph
= +Γ Γ
P−k P−p P−k P−p P−h
Γ
P−p
0
Figure 5.27: Bethe-Salpeter equation for the interaction vertex in the particle-particle channel.
the particle-particle channel as Γ0pp to distinguish it from Γ0.
5.7.1 Bethe-Salpeter equation for the vertex functions
Let us now introduce the vertex function for a density operator A(q), which is a particle-holeoperator, through
〈Tτ(ckα(τ1)c†k+qβ(τ2)A(q, τ)
)〉 ≡
∫ 2∏i=1
dτ ′i Gα(k, τ1 − τ ′1)Gβ(k + q, τ ′2 − τ2)
ΛAαβ(k τ ′1 , k + q τ ′2 ; q , τ). (5.48)
In the absence of interaction, one readily realizes that
ΛAαβ(k τ ′1 , k + q τ ′2 ; q , τ) = δ(τ ′1 − τ) δ(τ ′2 − τ)λAk,k+q;αβ.
In the presence of interaction the formal expression of Λ (in what follows we drop the labelA) in terms of λ and the interaction vertex can be readily inferred from the expression of thecorrelation function, and it is graphically shown in Fig. 5.28.
198
= +
Γ0 Γ
0
Γ0
= + +
= +
Λ λ λΓ
λ λ Γ λ
λ Λ
Figure 5.28: Bethe-Salpeter equation for the particle-hole vertex function Λ.
In Fourier space it reads:
Λαβ(k iεn , k + q iεn + iωl ; q , iωl) = λk,k+q;αβ
+1
V
∑p
T∑m
∑γδ
Γ0αγ;δβ(k iεn , p + q iεm + iωl ; p iεm , k + q iεn + iωl)
Gγ(p + q, iεm + iωl)Gδ(p, iεm) Λδγ(p iεm , p + q iεm + iωl ; q , iωl) (5.49)
= λk,k+q;αβ +1
V
∑p
T∑m
∑γδ
Γαγ;δβ(k iεn , p + q iεm + iωl ; p iεm , k + q iεn + iωl)
Gγ(p + q, iεm + iωl)Gδ(p, iεm)λp,p+q;δγ . (5.50)
Notice that, in the case of a Coulomb repulsion, the above vertex function for the densityoperator, i.e. λAk,k+q;αβ = δαβ, has not to be confused with the proper density-vertex shownin Fig. 5.23. The latter is in fact irreducible with respect to cutting an interaction line, whilethe former is not. One can easily show that Λ satsifies the Bethe-Salpeter equation graphicallyshown in Fig. 5.29.
Γ0k+q,k;qΛ( )~
= = −+
Figure 5.29: Bethe-Salpeter equation for the proper density-vertex function Λ.
199
5.8 The Ward identities
Let us suppose that our model has a set of conserved quantities, and let us select one of them,whose density operator we define as
J0(q) =∑k
∑ab
c†ka λ0k,k+q;ab ck+qb, (5.51)
where to be as general as possible we assume that the roman indices which identify the fermionicoperators include both spin as well as other internal labels, like e.g. the band index. We willfurther assume that the single-particle Green’s function are generally non-diagonal in these in-dices. We can associate to the density operator (5.51) a current density operator J = (J1, J2, J3)given by
J(q) =∑k
∑ab
c†ka λk,k+q;ab ck+qb, (5.52)
such that the continuity equation in imaginary time is satisfied:
∂
∂τJ0(q, τ) + q · J(q, τ) = 0.
If we consider J0 and J as the time and space components of a four dimensional current,(J0, J1, J2, J3), we can introduce for each component a vertex function through
〈Tτ(cka(τ1) c†k+qb(τ2) Ji(q, τ)
)〉.
Let us take the time derivative with respect to τ of the zeroth component. The derivative actseither directly on the charge density operator or on the θ-functions which define the time-orderedproduct. The latter is, dropping for simplicity all indices,
[∂τθ(τ1 − τ2)θ(τ2 − τ)] 〈c (τ1) c†(τ2) J0(τ)〉+ [∂τθ(τ1 − τ)θ(τ − τ2)] 〈c (τ1) J0(τ) c†(τ2)〉+ [∂τθ(τ − τ1)θ(τ1 − τ2)] 〈J0(τ) c (τ1) c†(τ2)〉− [∂τθ(τ2 − τ1)θ(τ1 − τ)] 〈c†(τ2) c (τ1) J0(τ)〉− [∂τθ(τ2 − τ)θ(τ − τ1)] 〈c†(τ2) J0(τ) c (τ1)〉− [∂τθ(τ − τ2)θ(τ2 − τ1)] 〈J0(τ) c†(τ2) c (τ1)〉
= −θ(τ1 − τ2)δ(τ2 − τ) 〈c (τ1) c†(τ2) J0(τ)〉+ [−δ(τ1 − τ)θ(τ − τ2) + θ(τ1 − τ)δ(τ − τ2)] 〈c (τ1) J0(τ) c†(τ2)〉+δ(τ − τ1)θ(τ1 − τ2) 〈J0(τ) c (τ1) c†(τ2)〉+θ(τ2 − τ1)δ(τ1 − τ) 〈c†(τ2) c (τ1) J0(τ)〉− [−δ(τ2 − τ)θ(τ − τ1) + θ(τ2 − τ)δ(τ − τ1)] 〈c†(τ2) J0(τ) c (τ1)〉
200
−δ(τ − τ2)θ(τ2 − τ1) 〈J0(τ) c†(τ2) c (τ1)〉
= δ(τ − τ2) 〈Tτ(c (τ1)
[J0(τ), c†(τ)
] )〉
+δ(τ − τ1) 〈Tτ( [J0(τ), c (τ)
]c†(τ2)
)〉.
It can be readily shown that∑p
∑cd
[c†pc λ
0p,p+q;cd cp+qd , c
†k+qb
]=
∑c
c†kc λ0k,k+q;cb∑
p
∑cd
[c†pc λ
0p,p+q;cd cp+qd , cka
]= −
∑d
λ0k,k+q;ad ck+qd,
so that the above time-derivative becomes∑c
δ(τ − τ2) 〈Tτ(cka(τ1) c†kc(τ)λ0
k,k+q;cb
)〉
−∑d
δ(τ − τ1) 〈Tτ(λ0k,k+q;ad ck+qd(τ) c†k+qb(τ2)
)〉
= −∑c
δ(τ − τ2)Gac(k, τ1 − τ)λ0k,k+q;cb
+∑d
δ(τ − τ1)λ0k,k+q;adGdb(k + q, τ − τ2).
In conclusion we find, making use of the continuity equation,
∂
∂τ〈Tτ
(cka(τ1) c†k+qb(τ2) J0(q, τ)
)〉 = 〈Tτ
(cka(τ1) c†k+qb(τ2)
∂J0(q, τ)
∂τ
)〉
−∑c
δ(τ − τ2)Gac(k, τ1 − τ)λ0k,k+q;cb
+∑d
δ(τ − τ1)λ0k,k+q;adGdb(k + q, τ − τ2)
= −〈Tτ(cka(τ1) c†k+qb(τ2) q · J(q, τ)
)〉
−∑c
δ(τ − τ2)Gac(k, τ1 − τ)λ0k,k+q;cb
+∑d
δ(τ − τ1)λ0k,k+q;adGdb(k + q, τ − τ2).
Upon introducing the vertex functions for the charge and current densities, Λ0 and Λ, re-spectively, and Fourier transforming in frequency we find∑
cd
Gac(k, iεn)Gdb(k + q, iεn + iωl)
201
(− iωl Λ0
cd(k iεn , k + q iεn + iωl ; q iωl) + q ·Λcd(k iεn , k + q iεn + iωl ; q iωl))
= −∑c
Gac(k, iεn)λ0k,k+q;cb +
∑d
λ0k,k+q;adGdb(k + q, iεn + iωl). (5.53)
By means of the Bethe-Salpeter equation for the vertex functions (5.49) we can rewrite thisequation as∑
cd
Gac(k, iεn)Gdb(k + q, iεn + iωl)(− iωl Λ0
cd(k iεn , k + q iεn + iωl ; q iωl) + q ·Λcd(k iεn , k + q iεn + iωl ; q iωl))
=∑cd
Gac(k, iεn)Gdb(k + q, iεn + iωl)(− iωl λ0
k,k+q;c d + q · λk,k+q;c d
)+
1
V
∑p
T∑m
∑cdefgh
Gac(k, iεn)Gdb(k + q, iεn + iωl)
Γ0ce;fd(k iεn,p + q iεm + iωl; p iεm,k + q iεn + iωl)
Gfg(p, iεm)Ghe(p + q, iεm + iωl)(− iωl Λ0
gh(p iεm , p + q iεm + iωl ; q iωl) + q ·Λgh(p iεm , p + q iεm + iωl ; q iωl))
= −∑c
Gac(k, iεn)λ0k,k+q;cb +
∑d
λ0k,k+q;adGdb(k + q, iεn + iωl)
=∑cd
Gac(k, iεn)Gdb(k + q, iεn + iωl)(− iωl λ0
k,k+q;c d + q · λk,k+q;c d
)+
1
V
∑p
T∑m
∑cdef
Gac(k, iεn)Gdb(k + q, iεn + iωl)
Γ0ce;fd(k iεn,p + q iεm + iωl; p iεm,k + q iεn + iωl)(
−∑g
Gfg(p, iεm)λ0p,p+q;ge +
∑h
λ0p,p+q;fhGhe(p + q, iεm + iωl)
).
If we multiply both sides by the matrix product G−1(k, iεn) G−1(k + q, iεn + iωl)5, we get(
− iωl Λ0ab(k iεn , k + q iεn + iωl ; q iωl) + q ·Λab(k iεn , k + q iεn + iωl ; q iωl)
)=(− iωl λ0
k,k+q;a b + q · λk,k+q;a b
)+
1
V
∑p
T∑m
∑ef
Γ0ae;fb(k iεn,p + q iεm + iωl; p iεm,k + q iεn + iωl)
5The Green’s function Gab can be interpreted as the ab component of a matrix G
202
(−∑g
Gfg(p, iεm)λ0p,p+q;ge +
∑h
λ0p,p+q;fhGhe(p + q, iεm + iωl)
)= −
∑d
λ0k,k+q;adG
−1db (k + q, iεn + iωl) +
∑c
G−1ac (k, iεn)λ0
k,k+q;cb.
We notice that the non-interacting Green’s functions satisfy the same equation with Γ0 = 0,namely (
− iωl λ0k,k+q;a b + q · λk,k+q;a b
)= −
∑d
λ0k,k+q;ad
(G(0)
)−1
db(k + q, iεn + iωl)
+∑c
(G(0)
)−1
ac(k, iεn)λ0
k,k+q;cb,
so that we can write
1
V
∑p
T∑m
∑ef
Γ0ae;fb(k iεn,p + q iεm + iωl; p iεm,k + q iεn + iωl)(
−∑g
Gfg(p, iεm)λ0p,p+q;ge +
∑h
λ0p,p+q;fhGhe(p + q, iεm + iωl)
)= −
∑d
λ0k,k+q;ad
(G−1db (k + q, iεn + iωl)−
(G(0)
)−1
db(k + q, iεn + iωl)
)+∑c
(G−1ac (k, iεn)−
(G(0)
)−1
ac(k, iεn)
)λ0k,k+q;cb.
By definition
G−1 −(G(0)
)−1= −Σ,
where Σ is the self-energy matrix, so that
∑c
λ0k,k+q;ac Σcb(k + q, iεn + iωl)− Σac(k, iεn)λ0
k,k+q;cb
=1
V
∑p
T∑m
∑cd
Γ0ac;db(k iεn,p + q iεm + iωl; p iεm,k + q iεn + iωl)∑
e
(−Gde(p, iεm)λ0
p,p+q;ec + λ0p,p+q;deGec(p + q, iεm + iωl)
). (5.54)
203
This is the so-called Ward identity which is a consequence of conservation laws. There are othertwo equivalent ways to rewrite this identity. Indeed, through (5.53), we can also write∑
c
λ0k,k+q;ac Σcb(k + q, iεn + iωl)− Σac(k, iεn)λ0
k,k+q;cb
=1
V
∑p
T∑m
∑cd
Γ0ac;db(k iεn,p + q iεm + iωl; p iεm,k + q iεn + iωl)
Gde(p, iεm)Gfc(p + q, iεm + iωl) (5.55)(− iωl Λ0
ef (p iεm,p + q iεm + iωl; q iωl) + q ·Λef (p iεm,p + q iεm + iωl; q iωl)).
Futhermore, using the fully reducible vertex Γ instead of Γ0, see Eq. (5.50), we also find∑c
λ0k,k+q;ac Σcb(k + q, iεn + iωl)− Σac(k, iεn)λ0
k,k+q;cb
=1
V
∑p
T∑m
∑cd
Γac;db(k iεn,p + q iεm + iωl; p iεm,k + q iεn + iωl)
Gde(p, iεm)Gfc(p + q, iεm + iωl)(− iωl λ0
p,p+q;ef + q · λp,p+q;ef
). (5.56)
5.9 Consistent approximation schemes
Since it is usually impossible to sum up all orders in perturbation theory, one is forced tomake some approximation on the self-energy. In doing that, one would like not to spoil anyconservation law. In this section we show what is the proper way to define an approximationscheme which is consistent with the conservation laws.
In general we can recast the perturbation expansion for the self-energy in terms of the fullyinteracting Green’s functions instead of the non-interacting ones. This kind of expansion is calledskeleton expansion, the diagrams up to second order being drawn in Fig. 5.30. This allows us
+[G] = + + ....+Σ
Figure 5.30: Skeleton expansion for the self-energy up to second order in the interaction.
to assume formally that the self-energy is a functional of the Green’s function Σ[G]. Let us
204
calculate the functional derivative of Σ with respect to G. Let us imagine to do it graphically,as shown in Fig. 5.31. Namely, we take out of the rounded box, which represents the whole
ba
c d
δG
a b
c d
=p p+q
k+q k
Figure 5.31: Graphical representation of δΣ/δG.
skeleton expansion of Σ, a Green’s function G and we vary it, δG. Therefore δΣ/δG is a fourleg vertex. What can it be? One easily realizes that, since the functional derivative is done withrespect to the fully interacting Green’s function, this four leg vertex is just the irreducible Γ0 inthe particle-hole channel. Therefore
δΣab(p, p+ q)
δGdc(k, k + q)= Γ0
ac;db(p, k + q; k, p+ q). (5.57)
We further notice that
δGdc(k, k + q) = −∑ef
Gde(k)[δ(G
(0)ef (k, k + q)
)−1− δΣef (k, k + q)
]Gfc(k + q),
which inserted into (5.57) and solving for δΣ leads to
δΣab(p, p+ q) =
∫dk Γ0
ac;db(p, k + q; k, p+ q) δGdc(k, k + q) (5.58)
= −∑ef
∫dk Γac;db(p, k + q; k, p+ q)Gde(k) δG
(0)ef (k, k + q)−1Gfc(k + q),
where we used the Bethe-Salpeter equation for Γ versus Γ0.
We notice that (5.57) is perfectly consistent with the Ward identity (5.54). Since the Wardidentities are consequence of conservation laws, we get to the following conclusion:
If the self-energy is approximated with a functional Σappx[G], this approximation isconsistent with the conservation laws if the irreducible vertex is also approximatedby
Γ0appx =
δΣappx[G]
δG.
205
Finally we notice that, given an approximate Σappx[G], then the actual Green’s function isobtained by solving the self-consistency equation:
G−1 =(G(0)
)−1− Σappx[G]. (5.59)
5.9.1 Example: the Hartree-Fock approximation
Let us consider the interacting Hamiltonian
H =∑ab
tab c†acb +
1
2
∑abcd
Uacdb c†ac†c cdcb.
Let us approximate the skeleton expansion of the self-energy with the two first-order diagrams,as shown in Fig. 5.32. This amounts to the following expression of the self-energy
HFΣ [ ] =G a b + a b
c
d c
d
Figure 5.32: Skeleton expansion in the Hartree-Fock approximation.
Σab(iεn) = T∑m
∑cd
Uacdbe−iεm0− Gdc(iεm)− T
∑m
∑cd
Uacbd e−iεm0− Gdc(iεm). (5.60)
We notice thatT∑m
e−iεm0− Gdc(iεm) = Gdc(τ = 0−) = 〈c†c cd〉 ≡ ∆cd.
The Green’s function satisfies the self-consistency equation (5.59), which in this case reads(G−1
)ab
= iεn − tab − Σab = iεn − tab −∑cd
∆cd (Uacdb − Uacbd) ,
which is diagonalized by diagonalizing
tab +∑cd
∆cd (Uacdb − Uacbd) = (HHF )ab ,
which is nothing but the Hartree-Fock Hamiltonian. Namely, the self-consistency requirement(5.59) is just the Hartree-Fock self-consistency equation.
206
In order to have a consistent scheme, we need to approximate Γ0 as
Γ0acdb =
δΣab
δGdc= Uacdb − Uacbd.
One can readily show that the correlation functions calculated with the above Γ0 coincide withthe time-dependent Hartree-Fock, which is therefore a consistent approximation.
207
5.10 Some additional properties and useful results
In this final section we derive some properties that may be useful in several contexts.
5.10.1 The occupation number and the Luttinger-Ward functional
By the definition of the single-particle Green’s function in momentum space, one can derive theaverage values
∆k ab = 〈c†kb cka〉,
where the labels a and b include spin and additional internal indices, through
∆k = G(k, τ = 0−) = T∑n
G(k, iεn) e−iεn0−
= T∑n
1
iεn − εk − Σ(k, iεn)e−iεn0− ,
where ∆k, εk and Σ(k, iεn) are matrices in the a-space. We define ε0 = πT such that, forT → 0, ε0 → 0, which allows to formally introduce the derivative with respect to the Matsubarafrequencies. One finds that
G(k, iεn) =1
iεn − εk − Σ(k, iεn)
=ln G−1(k, iεn + iε0)− ln G−1(k, iεn)
iε0+ G(k, iεn)
Σ(k, iεn + iε0)− Σ(k, iεn)
iε0
=∂ ln G−1(k, iεn)
∂iεn+ G(k, iεn)
∂Σ(k, iεn)
∂iεn,
hence
∆k = T∑n
(∂ ln G−1σ (k, iεn)
∂iεn+ G(k, iεn)
∂Σ(k, iεn)
∂iεn
)e−iεn0− . (5.61)
Let us concentrate on the second term in (5.61). By the definition of the derivatives it followsthat
T∑n
G(k, iεn)∂Σ(k, iεn)
∂iεn= −T
∑n
∂G(k, iεn)
∂iεnΣ(k, iεn).
The interacting Hamiltonian allows for several conserved quantities. Let us write a generic oneof them like
M =∑k
∑ab
c†kaMab(k) ckb.
208
It turns out that the average value of M is given by
〈M〉 =∑k
mk,
where
mk = Tr(M(k) ∆k
)= T
∑n
[Tr
(M(k)
∂ ln G−1(k, iεn)
∂iεn
)
−Tr
(M(k)
∂G(k, iεn)
∂iεnΣ(k, iεn)
)e−iεn0−
].
We draw in Fig. 5.33 the first orders in the skeleton expansion of the functional of the fully-interacting Green’s function X[G], introduced originally by Luttinger and Ward. It is obtainedfrom the skeleton expansion of the self-energy by connecting the external vertices with a fully-interacting Green’s function and dividing each term by 1/2n, where 2n is the number of Green’sfunctions, n being the number of interaction lines. 6 By its definition, one can readily verify
1/2 [G] =X + 1/2 + 1/4 + 1/4 + ....
Figure 5.33: Graphical representation of the functional X[G].
that X[G] satisfies
δX[G(k)] = T∑n
Tr(
Σ(k, iεn) δG(k, iεn)), (5.62)
namely the self-energy is the functional derivative of X. One can readily show that, for any
6The self-energy itself is a functional of G, which guarantees that X is a well defined functional.
209
conserved quantity, the following result holds:7
δX[G(k)] = T∑n
Tr
[Σ(k, iεn)M(k)
∂G(k, iεn)
∂iε0
]δiε0 = 0. (5.63)
7To prove Eq. (5.63), let us assume to rotate the basis into the one in which the hermitean matrix M(k) isdiagonal, with eigenvalues mα(k). The fermion operators in the new basis are accordingly ckα and c†kα, hence theconserved operator becomes
M =∑kα
mα(k) c†kα ckα.
The fully interacting Green’s function must be diagonal in α for M to commute with the Hamiltonian
0 =[H,M
]=[H0,M
]+[Hint,M
].
Note that, because the first commutator is a one-body operator while the second a two-body one, it follows that[H0,M
]= 0,
[Hint,M
]= 0.
Now suppose that, within the perturbative calculation of the Luttinger-Ward functional X, which just amountsto calculate the average value of products of Hint(τ) in the interaction representation at different times, eachinteraction vertex
Hint(τ) = eH0τ Hint e−H0τ ,
is transformed intoH′int(τ) = eiε0Mτ eH0τ Hint e−H0τ e−iε0Mτ ,
where ε0 = π T . Since Hint and H0 commute separately with M, it follows that H′int(τ) = Hint(τ); indeed
H′int(τ) = eiε0Mτ eH0τ Hint e−H0τ e−iε0Mτ
= eH0τ eiε0Mτ Hint e−iε0Mτ e−H0τ
= eH0τ Hint e−H0τ = Hint(τ).
Therefore the Luttinger-Ward functional X is invariant to this transformation. On the other hand, insteadof transforming the interaction as a whole, I can also transform each fermionic operator inside the interactionseparately, that implies, in the Heisenberg representation,
ckα(τ) = eHτ ckα e−Hτ → eiε0Mτ ckα(τ) e−iε0Mτ
= e−iε0mα(k)τ ckα(τ),
and accordingly that the Green’s function changes into
Gα(k, τ − τ ′)→ e−iε0mα(k)(τ−τ ′) Gα(k, τ − τ ′),
namelyGα(k, iεn)→ Gα (k, iεn + iε0mα(k)) .
Since the final result must not change, we get to the conclusion that
δX[G] = X [Gα (k, iεn + iε0mα(k))]−X [Gα(k, iεn)]
= T∑n
∑α
Σα(k, iεn)mα(k)∂Gα(k, iεn)
∂iε0δiε0 = 0.
After back-rotation in the original basis, we recover Eq. (5.63).
210
As a result we obtain that
mk = T∑n
Tr
(M(k)
∂ ln G−1(k, iεn)
∂iεn
)e−iεn0− . (5.64)
We already showed that the Green’s function, hence also its logarithm, has generally branchcuts on the real axis. By means of (5.9) we then find8
mk = T∑n
Tr
(M(k)
∂ ln G−1(k, iεn)
∂iεn
)e−iεn0−
= −∫dε
πf(ε)
∂
∂εArg Tr
(M(k) ln G−1(k, ε+ i0+)
)(5.65)
= −∫dε
π
∂f(ε)
∂εIm Tr
(M(k) ln G(k, ε+ i0+)
). (5.66)
We emphasize that this result holds only for conserved quantities.
5.10.2 The thermodynamic potential
A simple way to determine the thermodynamic potential is by means of the Hellmann-Feynmantheorem, according to which, if the Hamiltonian depends on some parameter λ, i.e. H = H(λ),then the derivative of the thermodynamic potential Ω(λ) with respect to λ is
∂Ω(λ)
∂λ= 〈 ∂H(λ)
∂λ〉λ, (5.67)
where 〈. . . 〉λ means quantum and thermal average with the Hamiltonian at finite λ. Let usconsider for simplicity the case of interacting electrons, the cases of bosons or electrons plusbosons being a straightforward generalization. The Hamiltonian is the free electron Hamiltonian
H0 =∑kσ
εk c†kσckσ,
plus the electron-electron interaction
Hint =1
2V
∑kpq
∑αβ
U(q) c†k+qα c†pβ cp+qβ ckα.
Let us consider now a λ-dependent Hamiltonian
H(λ) = H0 + λHint,8The exponential factor in (5.64) guarantees that the contour integral is well behaved at Re z → −∞, while
when Re z → +∞ is the Fermi distribution function which does the job.
211
then, through the Hellmann-Feynman theorem, it holds that
∂Ω(λ)
∂λ= 〈 ∂Hint
∂λ〉λ
=1
λ
1
2V
∑kpq
∑αβ
λU(q) 〈 c†k+qα c†pβ cp+qβ ckα 〉λ.
One might in principle calculate directly this average value by using the fluctuation-dissipationtheorem that allows to relate the average value of a four fermion operator with the sum overfrequency of the imaginary part of appropriate correlation functions, that can be accessed pertur-batively. Alternatively, by means of the equation of motion of the Green’s function, Eq. (5.35),we can write
1
2V
∑kpq
∑αβ
λU(q) 〈 c†k+qα c†pβ cp+qβ ckα 〉λ =
1
2T∑n
∑kα
[(iεn − εk) Gα(k, iεn)− 1
]eiεn 0+
= T∑n
∑k
(G(0)α (k, iεn)
)−1 [Gα(k, iεn)−G(0)
α (k, iεn)]
eiεn 0+ ,
having made use of the fact that
limτ→0
δ(τ) = limτ→0
T∑n
eiεnτ = T∑n
.
Therefore we finally obtain that
∂Ω(λ)
∂λ=
T
2λ
∑n
∑kα
(G(0)α (k, iεn)
)−1 [Gα(k, iεn)−G(0)
α (k, iεn)]
eiεn 0+
=T
2λ
∑n
∑kα
Σα(k, iεn)Gα(k, iεn) eiεn 0+ , (5.68)
where the interacting Green’s function and self-energy are calculated at finite λ, and we usedthe Dyson equation. Since for λ = 0, the thermodynamic potential is that one of free electrons,Ω0, that is known, we can evaluate the interacting potential Ω = Ω(λ = 1) by
Ω = Ω0 +
∫ 1
0dλ
T
λ
∑n
∑k
(G(0)(k, iεn)
)−1 [G(k, iεn)−G(0)(k, iεn)
]eiεn 0+ .
Now, let us define a functional X[Σ] through
X [Σ] = −T∑n
∑kσ
[ln(
1−G(0)σ (k, iεn) Σσ(k, iεn)
)+ Σσ(k, iεn)Gσ(k, iεn)
]eiεn 0+ +X
[G[Σ]
],
(5.69)
212
where X is the Luttinger-Ward function, see Fig. 5.33, assuming now that G is functional ofΣ (we have explicitly indicated the spin label, eventhough each term is spin-independent). Bymeans of (5.62) we find that
δX
δΣσ=
G(0)σ
1−G(0)σ Σ
−G− ΣδG
δΣσ+δX
δΣσ
=G(0)σ
1−G(0)σ Σ
−G− ΣδG
δΣσ+
δX
δGσ
δGσδΣσ
=G(0)σ
1−G(0)σ Σ
−G− ΣδG
δΣσ+ Σ
δGσδΣσ
=1(
G(0)σ
)−1− Σ
−G.
The last expression is nothing but the Dyson equation. This means that X is stationary
δX
δΣσ= 0,
provided G satisfies the Dyson equation. Now, once again, we multiply the interaction by aparameter λ and assume we have calculated all quantities at finite λ, so that the self-energybecomes function of λ, i.e. Σ(λ). We note that
∂X
∂λ=δX
δΣ
∂Σ
∂λ+
(∂X
∂λ
)Σ
=
(∂X
∂λ
)Σ
,
where we have assumed that Σ is the stationary point, and the last term implies that thederivative is at fixed Σ, namely one does not need to derive anymore Σ(λ) with respect to λ.The only other place in which λ appears in X is in the interaction lines that are now λU(q).Since for the n-th order diagram X(n) there are n interaction lines, the derivative is(
∂X(n)
∂λ
)Σ
=n
λX(n)(λ).
Therefore the perturbation expansion of the derivative of X is just the same as of X itselfwithout the 1/n pre-factor of the n-th order term but with a 1/λ pre-factor. Hence, it is easyto realize, see Fig. 5.33, that
∂X
∂λ=
(∂X
∂λ
)Σ
=1
2λT∑n
∑kσ
Σσ(k, iεn)Gσ(k, iεn). (5.70)
213
Comparing with (5.68) we find that
∂Ω(λ)
∂λ=∂X
∂λ. (5.71)
Since, for λ = 0, X(0) = 0, we immediately find that, at λ = 1,
Ω = Ω0 + X [Σ]
= Ω0 − T∑n
∑kσ
[ln(
1−G(0)σ (k, iεn) Σσ(k, iεn)
)+ Σσ(k, iεn)Gσ(k, iεn)
]eiεn 0+ +X [G]
= T∑n
∑kσ
[lnGσ(k, iεn) − Σσ(k, iεn)Gσ(k, iεn)
]eiεn 0+ +X [G] , (5.72)
where we use the expression
Ω0 = T∑n
∑kσ
lnG(0)σ (k, iεn) eiεn 0+ .
5.10.3 The Luttinger theorem
As usually let us assume that the Hamiltonian as well as the ground state are spin-rotationallyinvariant, which also implies that the Green’s function is spin-independent. In this case the totalnumber of electrons per unit volume at zero temperature is
N
V= =
1
V
∑k
∑σ
nkσ = − 2
V
∑k
∫dε
πf(ε)
∂
∂εArg lnG−1(k, ε+ i0+)
= − 2
V
∑k
∫ 0
−∞
dε
π
∂
∂εArg lnG−1(k, ε+ i0+)
= − 2
πV
∑k
(Arg lnG−1(k, 0 + i0+)−Arg lnG−1(k,−∞+ i0+)
)= − 2
πV
∑k
(φ(k, 0)− φ(k,−∞)
), (5.73)
where
φ(k, ε) = tan−1
(ImG(k, ε+ i0+)
ReG(k, ε+ i0+)
)= tan−1
(−ImG−1(k, ε+ i0+)
ReG−1(k, ε+ i0+)
), (5.74)
is the phase of the Green’s function approaching the real axis from above. From Eq. (5.14) itderives that
ImG(k, ε+ i0+) = −ImG(k, ε− i0+) = |G(k, ε)| sinφ(ε) = −π A(k, ε) ≤ 0,
214
which implies that φ is defined within [−π, 0]. For infinite frequency a constant interaction cannot be effective, hence
lim|ε|→∞
G(k, ε+ i0+) = lim|ε|→∞
G(0)(k, ε+ i0+)→ 1
ε+ i0+,
which implies that
φ(k,−∞) = tan−1 0−
−∞= −π.
We will show later that, in the most common situations, ImG−1(k, ε+ i0+ → 0+ i0+)→ 0+
so that G−1(k, 0) is purely real hence
φ(k, 0) = tan−1 0−
G−1(k, 0)
is equal to π if G−1(k, 0) < 0 and 0 otherwise. Using these results to evaluate (5.73) we finallyobtain
N
V=
2
V
∑k
θ(G−1(k, 0)
). (5.75)
This expression, also known as Luttinger’s theorem, states that the number of electrons is equalto the volume in the Brillouin zone which includes all momenta such that the Green’s functionat zero real-frequency is positive.
Remarks
Let us discuss in what cases the above result applies.The simplest example is when the single-particle local spectral function A(ε) has a gap or a
pseudo-gap, i.e. vanishes as a power-law, at the chemical potential, namely at ε = 0. Since
A(ε) =1
V
∑k
A(k, ε),
and both A(ε) and A(k, ε) are by definition real and positive, this implies that for any k A(k, ε→0)→ 0, hence also ImG(k, ε+ i0+ → 0 + i0+) = 0. Then, through (5.74), we find that
φ(k, 0) = tan−1 0−
ReG(k, 0).
Since the sign of the real part of G is the same as that of G−1, the Luttinger sum rule (5.75) isrecovered. This would be for instance the case of a superconductor.
215
On the contrary, in the case of a metal we do expect a finite spectral function at the chemicalpotential A(0) 6= 0, so the above demonstration does not work. We notice that
ReG−1(k, ε+ i0+) = ε− εk −ReΣ(k, ε+ i0+),
ImG−1(k, ε+ i0+) = 0+ − ImΣ(k, ε+ i0+),
so that the question reduces to know how it behaves the imaginary part of the self-energy uponapproaching the real axis. We have already shown, see Eq. (5.27), that within perturbationtheory
limε→0ImΣ(k, ε+ i0+) = 0−,
which justifies the Luttinger sum rule also in this case.
216
Chapter 6
Landau-Fermi liquid theory: amicroscopic justification
The Landau-Fermi liquid theory, that we have introduced at the beginning as a phenomeno-logical description of the low energy excitations of an interacting electron gas, can be justifiedmicroscopically using the Feynman diagram technique that we have just discussed.
6.1 Preliminaries
We are going to discuss the Landau-Fermi liquid theory in a model of N degenerate species offermions, a = 1, . . . , N . At the moment we shall consider only short-range interactions; later wewill extend the analysis to the Coulomb case. The non-interacting Green’s function is definedby
G(0)a (k, iεn) = G(0) (k, iεn) =
1
iεn − ε0k,
and the interacting one by
Ga (k, iεn) = G (k, iεn) =1
iεn − ε0k − Σ (k, iεn),
and both are a-independent. All energies are measured with respect to the chemical potential.We know that, provided perturbation theory is valid, the self-energy analytically continued
on the real axis, Σ(k, ε), has an imaginary part that vanishes at least like ε2 for ε → 0. Thisimplies that, for very small ε,
G (k, ε) ' 1
ε− ε0k −<eΣ (k, ε),
217
which we are going to assume has a simple pole on the real axis at ε = εk such that
εk − ε0k −<eΣ (k, εk) = 0,
with residue
Zk =
∣∣∣∣1− ∂<eΣ (k, ε)
∂ε ε=εk
∣∣∣∣−1
.
This pole represents a particle-like coherent excitation, that has to be identified as the quasipar-ticle. The residue, which is smaller than one as the derivative of the self-energy is negative, isthe weight of a quasi-particle excitation into a real particle one. By consistency, εk must be verysmall, so that the imaginary part of the self-energy at ε = εk is indeed negligible with respectto εk itself. Within this assumption, for complex frequency z very close to the origin,
G (k, z) ' Zk
z − εk+Ginc (k, z) , (6.1)
where Ginc (k, z) is an incoherent component that has no singularity for small |z|. Within thisassumption, the surface identified by εk = 0 corresponds by the Luttinger theorem to the Fermisurface.
Let us now consider the Bethe-Salpeter equation (5.47), which can be written formally as
Γ = Γ0 + Γ0 R Γ, (6.2)
where means the sums over internal indices, including the sum over momenta and Matsubarafrequencies, and
R (k, iεn; q, iωl) = G (k + q, iεn + iωl) G (k, iεn) . (6.3)
We note that, for free electrons and for small q and ωl
T∑n
G(0) (k + q, iεn + iωl) G(0) (k, iεn) F (iεn) = T
∑n
1
iεn + iωl − ε0k+q
1
iεn − ε0kF (iεn)
=
∮dz
2πif(z)
1
z + iωl − ε0k+q
1
z − ε0kF (z)
'f(ε0k)− f
(ε0k+q
)iωl − ε0k+q + ε0k
F(ε0k)
+
∮dz
2πif(z)
(1
z − ε0k
)2 (F (z)
)sing
' −∂f(ε0k)
∂ε0k
ε0k+q − ε0kiωl − ε0k+q + ε0k
F(ε0k)
+
∮dz
2πif(z)
(1
z − ε0k
)2 (F (z)
)sing
' δ(ε0k) ε0k+q + ε0k
iωl − ε0k+q + ε0kF (0) +
∮dz
2πif(z)
(1
z − ε0k
)2 (F (z)
)sing
,
218
where(F (z)
)sing
means that one has to catch in the contour integral only the singularities of
F . In the above equation, the term
δ(ε0k) v0
k · qiωl − v0
k · q
has a singular behavior for small q and ωl. If q = 0 and ωl finite, it vanishes, while if ωl = 0 atfinite q, it is −δ
(ε0k). In other words, the function
R(0) (k, iεn; q, iωl) = G(0) (k + q, iεn + iωl) G(0) (k, iεn)
regarded as a distribution in iεn, has a singular part at small q and ωl given by
−∂f(ε0k)
∂ε0kδ (iεn)
v0k · q
iωl − v0k · q
.
The formal meaning of δ (iεn) for discrete Matsubara frequencies is
δ (iεn) =1
Tδn0.
Seemingly, the product of two interacting Green’s functions of the form (6.1), regarded as adistribution, can be represented at small q and ωl as
R (k, iεn; q, iωl) = G (k + q, iεn + iωl) G (k, iεn)
' −∂f (εk)
∂εkδ (iεn) Z2
k
εk+q − εkiωl − εk+q + εk
+Rinc (k, iεn; q, iωl)
≡ ∆ (k, iεn; q, iωl) +Rinc (k, iεn; q, iωl) , (6.4)
where Rinc (k, iεn; q, iωl) is non-singular for q → 0 and ωl → 0, and
∆ (k, iεn; q, iωl) = −∂f (εk)
∂εkδ (iεn) Z2
k
εk+q − εkiωl − εk+q + εk
. (6.5)
We note that, if we send first q → 0 and then ωl → 0, so-called ω-limit, then
limωl→0
limq→0
R = Rω = Rinc, (6.6)
while, in the opposite case, so called q-limit,
limq→0
limωl→0
R = Rq =∂f (εk)
∂εkδ (iεn) Z2
k +Rinc = ∆q +Rω. (6.7)
219
Going back to the Bethe-Salpeter equation (6.2), we note that the irreducible vertex Γ0 is bydefinition non-singular at small q and ωl, in other words its ω-limit coincides with the q-limit.Therefore, in both limits,
Γq = Γ0 + Γ0 Rq Γq, (6.8)
Γω = Γ0 + Γ0 Rω Γω, (6.9)
with the same Γ0. Solving for Γ0 we find that
Γ = Γq + Γq (R−Rq) Γ ≡ Γq + Γq ∆ Γ, (6.10)
where
∆ (k, iεn; q, iωl) = ∆ (k, iεn; q, iωl)−∂f (εk)
∂εkδ (iεn) Z2
k
= −∂f (εk)
∂εkδ (iεn) Z2
k
iωliωl − εk+q + εk
, (6.11)
or, alternatively,Γ = Γω + Γω (R−Rω) Γ = Γω + Γω ∆ Γ. (6.12)
In this way we have been able to absorb the unknown Rinc and Γ0 into two scattering vertices,Γq and Γω.
6.1.1 Vertex and Ward identities
The fermionic operators cka, labelled by generic quantum numbers a = 1, N , have, as we as-sumed, interacting Green’s functions diagonal in these numbers and independent of them. Letus now add a perturbation of the form
δH = h(−q) ·∑k
∑ab
mab(k,q) c†kack+qb ≡ h · M,
in the limit of q → 0. In this limit it is convenient for what follows to rotate the basis cka intothe one ckα that diagonalizes the matrix m(k) with elements mab(k), i.e. mab(k)→ mα(k) δαβ.However, even though we are going to use this diagonal basis to derive explicitly several identities,we will always provide for each of these identities also a matrix representation, which is obviouslyindependent of the basis set.
In the diagonal basis the non interacting Green’s function is also diagonal,(G(0)α (k, iεn)
)−1= iεn − εk − hmα(k),
220
while the interacting one might not be so
(Gα(k, iεn))−1αβ = iεn − εk − hmα(k) δαβ − Σαβ(k, iεn). (6.13)
If the operator M refers to a conserved quantity of the fully interacting Hamiltonian, then theinteracting self-energy is diagonal, hence also the Green’s function.
In the presence of the external field, the average value of the operator M, namely
〈M〉 =∑α
mα(k)nkα =1
V
∑k
T∑n
∑α
mα(k)Gαα(k, iεn),
becomes h dependent, which allows to define the thermodynamic susceptibility
κ =∂〈M〉∂h |h=0
=1
V
∑k
T∑n
∑α
mα(k)∂Gαα(k, iεn)
∂h |h=0. (6.14)
We note that, considering Gαβ as elements of a matrix G, as well as mα of a diagonal matrixm, and so on, the following relation holds:
∂G(k, iεn)
∂h= G(k, iεn)
(m(k) +
∂Σ(k, iεn)
∂h
)G(k, iεn). (6.15)
This equation is actually independent of the basis set, namely in holds also in the originalrepresentation where m(k) has elements mab(k). On the other hand, through Eq. (5.58) weknow that
∂Σαβ(k, iεn)
∂h= T
∑m
∑γ
1
V
∑p
Γαγ;δβ(k iεn,p iεm; p iεm,k iεn)Gδν(p, iεm)mν(p)Gνγ(p, iεm),
where the product of the two Green’s functions must be interpreted as
limq→0
Gδν(p + q, iεm)Gνγ(p, iεm),
namely as the q-limit. In the limit h→ 0, that we will consider hereafter, all Green’s functionsbecome diagonal and independent of the index α, i.e. Gαβ = δαβ G, hence
mα(k) δαβ +∂Σαβ(k, iεn)
∂h |h=0= mα(k) δαβ (6.16)
+T∑m
∑γ
1
V
∑p
Γαγ;γβ(k iεn,p iεm; p iεm,k iεn)Rq(p iεm)mγ(p),
where we recall thatRq(p iεm) = lim
q→0G(p + q, iεm)G(p, iεm).
221
We can also introduce the triangular vertex Λαβ(k + q iεn + iωl,k iεn; q iωl) correspondingto the perturbation, with non-interacting value at small q
Λ(0)αβ(k + q iεn + iωl,k iεn; q iωl) = mα(k) δαβ.
As we know Λ satisfies the Bethe-salpeter equation
Λ = Λ(0) + ΓR Λ(0).
Similarly to the above discussion of the interaction vertices, we can define the q and ω limitsof the triangular vertex
Λq = Λ(0) + Γq Rq Λ(0),
Λω = Λ(0) + Γω Rω Λ(0).
Solving for Λ(0) one finds that the vertex is related to its q or ω limits by
Λ = Λω + Γω ∆ Λ = Λω + Γ∆ Λω, (6.17)
Λ = Λq + Γq ∆ Λ = Λq + Γ ∆ Λq, (6.18)
where ∆ and ∆ have been defined in Eqs. (6.5) and (6.11), respectively.Through Eq. (6.16) and by the definition of the triangular vertex, we obtain the following
identity:
mα(k) δαβ +∂Σαβ(k, iεn)
∂h= Λqαβ(k, iεn) , (6.19)
or, in a representation independent of the basis choosen,
m(k) +∂Σ(k, iεn)
∂h= Λq(k, iεn) . (6.20)
Let us assume now that the operator M is a conserved quantity, which implies that, even ifh 6= 0, the Green’s functions remain diagonal in α. In this case the Ward identity (5.56) holds,which reads in this case and for q = 0
mα(k)Σα(k, iεn + iωl)− Σα(k, iεn)
iωl= − 1
V
∑p
T∑m
∑β
Γαβ;βα(k iεn,p iεm + iωl; p iεm,k iεn + iωl)Gβ(p, iεm)Gβ(p, iεm + iωl)mβ(p).
In the limit of ωl → ω0 1 we can write
mα(k)∂Σα(k, iεn)
∂iεn= − 1
V
∑p
T∑m
∑β
Γαβ;βα(k iεn,p iεm + iω0; p iεm,k iεn + iω0)
222
Gβ(p, iεm)Gβ(p, iεm + iω0)mβ(p)
= −Λωαα(k, iεn) +mα(k). (6.21)
In other words, another identity holds, namely:
mα(k)−mα(k)∂Σα(k, iεn)
∂iεn= Λωα(k, iεn) , (6.22)
or, in matrix form,
m(k)− m(k)∂Σ(k, iεn)
∂iεn= Λω(k, iεn) . (6.23)
In the limit of vanishing h and for small εn, this identity becomes
Λ(0)(k)Z−1k = Λω(k, iεn) . (6.24)
We emphasize that this result holds only for a conserved operator.
6.2 Correlation functions
Two single particle operators
A(q) =∑k
∑ab
c†ka Λ(0)Aab(k,q) ck+qb, B(q) =
∑k
∑ab
c†ka Λ(0)B ab(k,q) ck+qb,
can be associated to a correlation function χAB(q, iωl) that is defined by
χAB = Tr(
Λ(0)A RΛ
(0)B
)+ Tr
(Λ
(0)A RΓRΛ
(0)B
)= Tr
(Λ
(0)A RΛB
), (6.25)
where all quantities are calculated at zero external fields.1 Once more we can define q and ωlimits by (we drop for simplicity the subscripts A and B, keeping in mind that the triangularvertex on the left refers to A, and that one on the right to B):
χω = Tr(
Λ(0)Rω Λω),
1The actual meaning of the trace over the internal indices α’s is
Tr(
Λ(0)A RΛB
)=
∑Λ
(0)Aαβ Gβγ ΛB γδ Gδα
=∑
Λ(0)Aαβ G
2 ΛB βα
since at zero field the Green’s functions are diagonal and independent of α.
223
χq = Tr(
Λ(0)Rq Λq).
Through Eqs. (6.17) and (6.18), we can rewrite (6.25) in the following way:
χ = Tr[Λ(0)R (Λω + Γ ∆ Λω)
]= Tr
(Λ(0)RΛω
)+ Tr
(Λ(0)RΓ ∆ Λω
)= Tr
(Λ(0)Rω Λω
)+ Tr
[Λ(0) (R−Rω) Λω
]+ Tr
(Λ(0)RΓ ∆ Λω
)= χω + Tr
(Λ(0) ∆ Λω
)+ Tr
(Λ(0)RΓ ∆ Λω
)= χω + Tr
(Λ ∆ Λω
)= χω + Tr
[(Λω + Λω ∆ Γ) ∆ Λω
]= χω + Tr
(Λω ∆ Λω
)+ Tr
(Λω ∆ Γ ∆ Λω
). (6.26)
Similarly, it also holds that
χ = χq + Tr(
Λq ∆ Λq)
+ Tr(
Λq ∆ Γ ∆ Λq). (6.27)
We note that both ∆(k, iεn; q, iωl) and ∆(k, iεn; q, iωl) contain a delta function in frequency,δ(εn) and a derivative of the Fermi function with respect to its argument. This implies firstthat all sums over the internal Matsubara frequencies drop out, all frequencies being fixed tothe lowest one ε0 = π T , and that all momenta are very close to the Fermi surface. We define,for small q and ωl
λωk = Zk Λω (k, 0; q = 0, iωl → 0) , (6.28)
λqk = Zk Λq (k, 0; q→ 0, iωl = 0) , (6.29)
Akp (q, iωl) = Zk Zp Γ (k + q 0 + iωl,p 0; p + q 0 + iωl,k 0) , , (6.30)
fkp = Zk Zp Γω(k 0,p 0; p 0,k 0), (6.31)
as well as
δk (q, iωl) = −∂f (εk)
∂εk
εk+q − εkiωl − εk+q + εk
, (6.32)
δk (q, iωl) = −∂f (εk)
∂εk
iωliωl − εk+q + εk
. (6.33)
With these definitions,A = f + f δ A, (6.34)
and Eqs. (6.26) and (6.27) can be written as
χ(q, iωl) = χω + Tr(λω δ λω
)+ Tr
(λω δ A δ λω
)224
= χq + Tr(λq δ λq
)+ Tr
(λq δ A δ λq
), (6.35)
where the trace does not include anymore the sum over Matsubara frequencies. The explicitexpressions read
χ(q, iωl) = χω − 1
V
∑k
∑αβ
λωkαβ λωkβα
∂f (εk)
∂εk
εk+q − εkiωl − εk+q + εk
+1
V 2
∑kp
∑αβγδ
∂f (εk)
∂εk
∂f (εp)
∂εpλωkαβ Akβ,p γ;p δ,kα(q, iωl)λ
ωp δγ
εk+q − εkiωl − εk+q + εk
εp+q − εpiωl − εp+q + εp
(6.36)
= χq − 1
V
∑k
∑αβ
λqkαβ λqkβα
∂f (εk)
∂εk
iωliωl − εk+q + εk
+1
V 2
∑kp
∑αβγδ
∂f (εk)
∂εk
∂f (εp)
∂εpλqkαβ Akβ,p γ;p δ,kβ(q, iωl)λ
qp δγ
iωliωl − εk+q + εk
iωliωl − εp+q + εp
. (6.37)
We just note that, through Eqs. (6.20), (6.13) and (6.15), we can relate χq
χq =1
V
∑k
T∑n
Tr
[Λ
(0)A (k,0)G(k, iεn)
(Λ
(0)B (k,0) +
∂Σ(k, iεn)
∂hB |h=0
)G(k, iεn)
]
=1
V
∑k
T∑n
Tr
[Λ
(0)A (k,0)
∂G(k, iεn)
∂hB |h=0
]= κAB,
to the thermodynamic susceptibility, κAB, of the operator A to an external field hB that couplesto B. So far we have been able to express any correlation function in a form similar to thatof free particles (quasi-particles), with R replaced by the “free-particle” quantities δ and δ, allinteraction effects being absorbed into the unknown three- and four-leg vertices. However, whenboth A and B refer to conserved quantities, a lot of simplications arise.
6.2.1 Conserved quantities
Suppose that A is conserved (the same result would hold if B, or both were conserved). Then
χωAB = Tr(
ΛωARω Λ
(0)B
).
225
Through Eq. (6.23), we find that2
χωAB =1
V
∑k
T∑n
Tr
[Λ
(0)A (k,q = 0)
(1− ∂Σ(k, iεn)
∂iεn
)G(k, iεn + iωl)G(k, iεn) Λ
(0)B (k,q = 0)
]
=1
V
∑k
Tr[Λ
(0)A (k,q = 0) Λ
(0)B (k,q = 0)
]T∑n
∂G(k, iεn)
∂iεn= 0, (6.38)
namely χωAB vanishes. Therefore, by means of Eq. (6.24), we finally obtain, when both A andB are conserved, and for small q and ωl,
χAB(q, iωl) = − 1
V
∑k
∑ab
Λ(0)Aab(k,0) Λ
(0)B ba(k,0)
∂f (εk)
∂εk
εk+q − εkiωl − εk+q + εk
+1
V 2
∑kp
∑αβ
Λ(0)Aab(k,0)Ak b,p c;p d,k a Λ
(0)B dc(k,0)
∂f (εk)
∂εk
∂f (εp)
∂εp
εk+q − εkiωl − εk+q + εk
εp+q − εpiωl − εp+q + εp
. (6.39)
We immediately recognize that the above expression coincides with that one obtained withinthe Landau-Fermi-liquid theory, in the general case of quantum numbers a = 1, . . . , N . There-fore, when considering conserved quantities, all interaction effects can be absorbed into thequasiparticle energy εk and into the scattering vertices A.
6.3 Coulomb interaction
So far we have only taken into account short-range electron-electron interaction. What does itchange when the interaction is instead long-ranged?
Let us assume therefore that the electrons interact mutually by a long-range Coulomb re-pulsion
Hint =1
V
∑kpq
∑ab
U(q) c†k+qa c†pb c†p+qb c
†ka,
with
U(q) =4πe2
q2 .
Furthermore we shall assume that there is a background positive charge that compensate exactlythe electron one. This implies that the self-energy, see Fig. 5.19, does not include the Fock term,which otherwise would give an infinite contribution since U(q → 0) → ∞. The problem with
2Recall that at zero field the Green’s functions Gab = δabG.
226
a long-range interaction is that, besides the singular behavior for small q and ωl due to R,additional singularities arise at q → 0 by the interaction. In order to disentagle both sources ofsingularities, it is convenient to adopt the same approach as in section 5.6.
Therefore, in the perturbative expansion of the four-leg vertex Γ(k + q,p; p + q,k), weidentify a subset of proper diagrams that do not include any interaction line with momentum q.This subset defines the proper vertex Γ, which by definition does not include any singularity forq→ 0 arising from the singular behavior of U(q). The proper vertex allows to define a properdensity-density correlation function3
χ = Tr (R) + Tr(R ΓR
),
as well as a proper three-leg vertex
Λ = Λ(0) + ΓR Λ(0).
By exploiting the analytical properties of R, we can follow precisely what we did before, thistime for proper vertices. Notice that, since the self-energy does not include the Fock-term, allvertex and Ward identities still hold in terms of proper vertices. In particular, the expression(6.39) remains valid for the proper correlation function with
Akp = Zk Zp Γω(k,p; p,k).
Once we have been able to express proper correlation functions in terms of few parameters, theimproper ones can be derived straightforwardly just like in section 5.6. The results obviouslycoincide with the Landau-Silin theory.
3Note that in the charge-density channel the non-interacting vertex Λ(0)ab = δab.
227
Chapter 7
Kondo effect and the physics of theAnderson impurity model
The behavior of magnetic impurities (Fe, Mn, Cr) diluted into non-magnetic metals (e.g. Cu, Ag,Au, Al) is likely the simplest manifestation of a breakdown of the conventional band-structuretheory due to strong electron-electron correlations.
We know that the behavior of a good metal with a large Fermi temperature, TF ∼ 104 K,is dominated at low temperature T TF by the Pauli principle. For instance its magneticsusceptibility is roughly constant, χ ∼ 1/TF , and one should in principle heat the sampleto very high temperatures T TF to release the spin entropy and recover a Curie-Weissbehavior χ ∼ 1/T . Moreover the resistivity is an increasing function of temperature, since thechannels which may dissipate current, the coupling to the lattice as well as to multi particle-holeexcitations, become available only upon heating
At odds with this expectation, if one introduces a very diluted (few part per million) con-centration of magnetic impurities the above behavior changes drastically. We just mention threedistinct features.
(1) The magnetic susceptibility shows a Curie-Weiss behavior well below TF , proportional tothe impurity concentration ni and roughly with the same g-factor of the isolated magneticimpurity, apart from corrections due to the crystal field. Around a very low temperature,called Kondo-temperature TK , the Curie-Weiss behavior turns into a logarithmic behaviorand finally the susceptibility saturates at low temperature to a value χ(0) ∼ ni/TK , withχ(T )− χ(0) ∼ −T 2.
(2) The resistivity R(T ) displays a minimum around TK , followed at T < TK by a logarithmicincrease. At very low temperatures R(T ) approaches a constant value R(0) ∝ ni withR(T ) − R(0) ∼ −T 2. The value of the residual resistivity R(0) suggests very strongscattering potential, near the so-called unitary limit.
228
(3) The entropy which is released above the Kondo temperature, which can be obtained bymeasuring the specific heat, includes only spin and eventually orbital degrees of freedomof the isolated impurity but not the charge degrees of freedom. This indicates that themagnetic impurities behave as local moments above TK .
Explaining this behavior amounts actually to understand three different problems.The first concerns the region TK T TF and can be formulated as follows: how is it
possible to sustain a local moment inside a metal ? We will see that to answer this question oneis obliged to abandon the conventional band-structure theory.
The second problem regards what happens around the Kondo temperature, namely to un-derstand the logarithmic crossover.
Finally, the last question concerns the low temperature behavior and the way the localmoments get screened and why the resistivity is a decreasing function of temperature.
Last two questions turn out to represent a complicated many-body problem which has re-quired the full machinary of renormalization group to be solved for the first time by Wilson.Nowadays one has at his disposal other sophisticated methods like the Bethe-Ansatz, whichprovides an exact solution, the abelian bosonization and the Conformal Field Theory.
7.1 Brief introduction to scattering theory
We start by showing why the existence of local moments in a metal for T TF is so puzzling.Let us consider an impurity imbedded in a normal metal. We only consider the valence band,
described by the Hamiltonian
H0 =∑k,σ
εk c†kσckσ. (7.1)
The scattering potential provided by the impurity has the general form
V =∑σ
∑k,p
Vkp c†kσcpσ, (7.2)
where Vkp are the matrix elements of the impurtity potential onto the valence band Block waves.
The single-particle Green’s function in complex frequency for the full Hamiltonian H =H0 + V can be formally written as the matrix
G(z) =1
z − H. (7.3)
Analogously the unperturbed one is
G0(z) =1
z − H0
. (7.4)
229
The above operators have the following interpretation. If we consider for instance the inverse ofthe Green’s function G, namely
G(z)−1 = z − H,
in the basis set of the valence band Block waves, this is the matrix(G(z)−1
)kσ,pσ′
= δσσ′ [δkp z − δkp εk − Vkp] , (7.5)
hence the Green’s function is the inverse of the above matrix. Suppose that we have diagonalizedthe full Hamiltonian and got the eigenvalues εa’s. In the diagonal basis also the Green’s functionis diagonal with matrix elements
G(z)ab = δab1
z − εa.
Therefore we readily find that
− 1
πlimδ→+0
ImTr(G(z = ω + iδ)
)= − 1
πlimδ→+0
Im∑aσ
1
ω − εa + iδ
=∑aσ
δ(ω − εa) ≡ ρ(ω),
where ω is a real frequency and ρ(ω) is the so-called density of states (DOS). Since the trace isinvariant under unitary transformations, hence also under the transformation which diagonalizesthe Hamiltonian, it is generally true that
− 1
πlimδ→+0
ImTr[G(z = ω + iδ)
]= ρ(ω). (7.6)
Analogously
− 1
πlimδ→+0
ImTr[G0(z = ω + iδ)
]= ρ0(ω) =
∑kσ
δ(εk − ω), . (7.7)
gives the DOS of the host metal.Let us formally write the full Green’s function as
G =1
z − H0 − V=
1
G−10 − V
=1
1− G0 VG0 = G0 + G0 V
[1− G0 V
]−1G0
≡ G0 + G0 T G0, (7.8)
which provides the definition of the so-called T -matrix, namely
T (z) = V[1− G0 V
]−1. (7.9)
230
From now on we will assume z = ω + iδ with δ a positive infinitesimal number, so we willnot explicitly indicate the limδ→+0. We notice that
∂
∂zln G(z) = −G(z),
so that
ρ(ω) =1
π
∂
∂z
ImTr ln G(z)
. (7.10)
We are interested in the variation of the DOS induced by the impurity, which, by makinguse of (7.8) and (7.9), is given by
∆ρ(ω) = ρ(ω)− ρ0(ω)
=1
π
∂
∂z
ImTr ln
[G(z) G0(z)−1
]=
1
π
∂
∂z
ImTr ln
[1
1− G0(z) V
]
=1
π
∂
∂z
ImTr ln V −1 T (z)
. (7.11)
We define the matrix of the scattering phase shifts
δ(z) = Im ln V −1 T (z) = Arg(V −1 T (z)
), (7.12)
through which
∆ρ(ω) =1
π
∂
∂ωTrδ(ω). (7.13)
We notice that for |z| → ∞, G0(z)→ 1/z → 0, hence T (z)→ V and δ(z)→ 0.The variation of the total number of electrons, ∆Nels, induced by the impurity at fixed
chemical potential µ is therefore
∆Nels =
∫ µ
−∞dω∆ρ(ω) =
1
πTrδ(µ), (7.14)
which is the so-called Friedel sum rule.Let us go back to the definition of the T -matrix (7.9). One readily finds that its inverse is
given by
T (z)−1 =[V −1 − G0(z)
].
Since the Hamiltonian is hermitean, then[T (z)−1
]†=[V −1 − G0(z∗)
],
231
so that [T (z)−1
]†− T (z)−1 =
[G0(z)− G0(z∗)
].
Therefore
T (z)†[T (z)−1
]†− T (z)−1
T (z) = T (z)− T (z)†
= T (z)†[G0(z)− G0(z∗)
]T (z). (7.15)
SinceG0(z)− G0(z∗) = −2πiδ
(ω − H0
),
the Eq. (7.15) implies the following identity
Tkp(ω)− T †kp(ω) = −2πi∑q
T †kq(ω) δ(ω − εq) Tqp(ω), (7.16)
which is the so-called optical theorem. It also shows that the imaginary part of the T -matrix isfinite only within the conduction band.
Let us now analyse the on-shell T -matrix
Tkp(ε),
where εk = εp = ε. We can rewrite the on-shell optical theorem as follows
−2πi δ (εk − εp)[Tkp(ε)− T †kp(ε)
]= (−2πi)2 δ (εk − εp)
∑q
T †kq(ε) δ(εk − εq) Tqp(ε).
The above equation implies that, if we introduce the so-called on-shell S-matrix through
Skp(ε) = δkp − 2πi δ (εk − εp) Tkp(ε), (7.17)
where, as before, εk = εp = ε, then it follows that the S-matrix is unitary, i.e
S S† = I . (7.18)
Skp(ε) is the transition probability that an electron in state k scatters elastically into state p.Since it is unitary, it follows that only elastic scattering survives.
Let us consider the simpler case of a spherical Fermi surface, i.e. εk = εk depending onlyon the modulous of the wavevector. In addition we assume that the matrix elements Vkp only
232
depend on the angle between the two wavevectors, θkp, hence can be expanded in Lagrangepolynomials
Vkp =∑l
Vl (2l + 1)Pl (cos θkp) . (7.19)
Then also the T -matrix has a similar expansion
Tkp(z) =∑l
tl(z) (2l + 1)Pl (cos θkp) . (7.20)
Since ∫dΩq
4πPl (cos θkq) Pl′ (cos θqp) =
1
2l + 1δll′ Pl (cos θkp) ,
the optical theorem transforms into
tl (ω)− t∗l (ω) = 2iImtl (ω) = −2πi ρ(ω) |tl (ω)|2 , (7.21)
where ρ(ω) is the bare conduction electron density of states per spin. Since
tl(ω) = |tl(ω)| eiδl(ω),
it derives that
tl(ω) = − 1
πρ(ω)sin δl(ω) eiδl(ω). (7.22)
If the concentration of impurity is very low, then the scattering rate suffered by the electrons isgiven by the Fermi-golden rule with the T -matrix, namely
1
τk= 2π ni
∑q
T †kq(ω) δ(ω − εq) Tqk(ω) (1− cos θkq) ,
where ni is the impurity concentration and the last term is a geometric factor which guaranteesthat forward scattering, θ = 0, does not contribute to the current dissipation. Since
(2l + 1) z Pl(z) = (l + 1)Pl+1(z) + l Pl−1(z),
we obtain
1
ni τk(ω)= 2π ρ(ω)
∑ll′
t∗l (ω) tl′(ω)
∫dΩq
4π(2l + 1)Pl(cos θkq)[
(2l′ + 1)Pl′(cos θkq) + (l′ + 1)Pl′+1(cos θkq) + l′ Pl′−1(cos θkq)]
= 2π ρ(ω)∑l
(2l + 1) |tl(ω)|2 + l t∗l (ω) tl−1(ω) + (l + 1) t∗l (ω) tl+1(ω), (7.23)
which as expected gives a scattering rate independent upon momentum. The zero-temperatureresistivity turns out to be proportional to
R(0) ∝ 1
τ(0). (7.24)
233
7.1.1 General analysis of the phase-shifts
Let us keep assuming a spherically symmetric case. In addition we assume that the scatteringcomponents Vl are non-zero only for a well defined l = L. For instance, in the case of magneticimpurities with partially filled d-shell, L = 2. Then the only non-zero phase shift is
δL(ω) = tan−1 ImtL(ω)
Re tL(ω).
An important role is played by the frequencies at which the real part vanishes.The first possibility is that this occurs outside the conduction band, either below or above.
In this case we know that the imaginary part is zero. This implies that the phase-shift jumpsby π at these energies so that the variation of the DOS is δ-like. In this case one speaks aboutbound states which appear outside the conduction band. Clearly this possibility can not explaina Curie-Weiss behavior for T TF . Indeed at very low temperature the bound state is eitherdoubly occupied, if below the conduction band, or empty, if above, and one needs a temperaturelarger than the conduction bandwidth, hence larger that TF , to release its spin entropy.
The other possibility is that the real part vanishes at a frequency ω∗ within the conductionband where the imaginary part is non zero. Around ω∗ and assuming a real part linearlyvanishing and an imaginary part roughly constant we get
δL(ω) ' tan−1 Γ
ω − ω∗,
leading to a Lorentian DOS variation due to the impurity
∆ρ(ω) ∼ 1
π
Γ
(ω − ω∗)2 + Γ2, (7.25)
which is called a resonance. Clearly, in other for this resonance to contribute at T TF , ω∗should be very close to the chemical potential, in particular |ω∗ − µ| ≤ TK , and, in addition,Γ ' TK .
Let us therefore assume Γ = TK and for further simplicity that ω∗ = 0. The contribution ofthe resonant state to the magnetic susceptibility is then given by
∆χ(T ) = −µB g (2L+ 1)
∫dε
∂f(ε)
∂ε
1
π
TKε2 + T 2
K
, (7.26)
where
f(ε) =(
1 + eβε)−1
,
is the Fermi distribution. One readily finds that for T TK ,
∆χ ' µB g2L+ 1
T,
234
while for T TK
∆χ ' µB g2L+ 1
πTK,
similar to the observed behavior. In addition the resistivity would be given through (7.23) and(7.24) by (ρ(ω) ∼ ρ(0) ≡ ρ0)
R(0) ∝ 1
τ(0)= ni
2(2L+ 1)
πρ0sin2 δL(0)
= ni2(2L+ 1)
πρ0, (7.27)
again compatible with the almost unitary limit, δ(0) = π/2, observed experimentally. Thereforethe existence of a narrow resonance near the chemical potential with width exactly given by TKwould seem the natural explanation to what experiments find out.
However this is evidently very strange, since the Kondo behavior is observed in many differenthost metals as well as for different magnetic impurities, hence it would be really surprising thatin all these cases a resonance appear always pinned near the chemical potential.
Moreover this simple single-particle scenario fails to explain the entropy released above TK .Indeed, for T TK the resonance is effectively like an isolated level which can be empty,singly occupied with a spin up or down, or doubly occupied. Therefore its entropy should byS = 2(2L+1) ln 2. On the other hand the experiments tells us that only spin and orbital degreesof freedom are released above TK , which amounts in the above simplified model to an entropyS = 2(2L+ 1) ln 2. In other words a resonance at the chemical potential at temperatures largerthan its broadening is not at all the same as a local moment, since the former does have valencefluctuations which are absent in the latter.
Therefore, although the resonance scenario is suggestive, it is not at all the solution to thepuzzle.
7.2 The Anderson Impurity Model
Something which is obviously missing in the above analysis are local electron-electron correla-tions around the magnetic ions. A narrow resonance induced by an impurity is quite localizedaround the same impurity and is essentially an atomic level broadened by the hybridization withthe conduction electrons of the host metal. Therefore, like an atomic level, such a resonance islikely to accomodate only a fixed number of electrons, say N , paying a finite amount of energy inadding or removing electrons with respect to the reference valency N . As usual it is convenientto define a so-called Hubbard repulsion U through
U = E(N + 1) + E(N − 1)− 2E(N), (7.28)
235
where E(M) is the total energy when the resonance is forced to have M electrons, N being theequilibrium value. In principle one might include this additional ingredient by adding to theHamiltonian a term
U
2(n−N)2 , (7.29)
with n the occupation number operator of the resonance. If it were possible to define suchan operator, then (7.29) would actually solve one of the puzzles. Indeed, if U TK , fortemperatures TK T U valence fluctuations with respect to the equilibrium value N wouldbe suppressed and the only degrees of freedom contributing to the entropy would be those relatedto the degeneracy of the N -electron configurations at the resonance. Namely, if the resonance is2(2L+1) degenerate, 2 from spin and (2L+1) from the orbital, the degeneracy of the N -electronstate is the binomial
C2(2L+1)N =
(2(2L+ 1)
N
)=
(4L+ 2)!
N ! (4L+ 2−N)!,
with entropy S = lnC2(2L+1)N . In other words the resonance would actually behave like a lo-
cal moment. Unfortunately the resonance occupation number operator is an ill defined object.To overcome this difficulty, Anderson had the idea to represent the resonance as an additionalelectronic-level inside the conduction band, even if this would be, rigorously speaking, an over-complete basis. This leads to the so-called Anderson Impurity Model (AIM) defined in thesimplest case of a single-orbital impurity L = 0 as
H =∑kσ
εk c†kσckσ +
∑kσ
Vk c†kσdσ +H.c.
+U
2(nd − 1)2 + εd nd. (7.30)
This model describes a band of conduction electrons with energy dispersion εk, hybridized witha single level, dσ, with energy εd, both εk and εd being measured with respect to the chemicalpotential, and nd =
∑σ d†σdσ. This level suffers from an interaction U which tends to lock no
more no less than a single electron.
Let us start from the non-interacting model, U = 0, to show that it indeed describes aresonance. Before switching the hybridization Vk the conduction electron Green’s function incomplex frequency z is
G(0)k (z) =
1
z − εk= (7.31)
and is represented as a dashed tiny line and the impurity one is
G(0)(z) =1
z − εd= (7.32)
236
VkV k*
VkV k*
= + + ..... =
k
+
k
Vk V *p= +
k
pk k k p
Vk*
Vkk
=k
=
kk
Figure 7.1: Dyson equations for the Green’s functions. Wavy (red) lines are the Green’s functionsin the presence of hybridization, represented as a circle, while tiny (black) lines are unperturbedGreen’s functions.
and represented as a solid tiny line. The corresponding Green’s functions in the presence of Vk,
Gkp(z) =
G(z) = ,
are represented as wavy lines. Notice that the conduction electron Green’s function is nomorediagonal in momentum. In addition one has to introduce the mixed Green’s functions
Gkd(τ) = −〈T(ckσ(τ) d†σ
)〉 =
Gdk(τ) = −〈T(dσ(τ) c†kσ(τ)
)〉 =
Through the Dyson equation shown diagramatically in Fig. 7.1 we find that
G(z) =1
z − εd −∆(z), (7.33)
where the so-called hybridization function is defined through
∆(z) =∑k
|Vk|2
z − εk. (7.34)
237
The conduction electron Green’s function is instead
Gkp(z) = δkpG(0)k (z) +G
(0)k (z)Tkp(z)G
(0)p (z), (7.35)
where the T -matrix is
Tkp = Vk1
z − εd −∆(z)V ∗p . (7.36)
In addition
Gkd(z) =Vk
z − εk1
z − εd −∆(z), (7.37)
Gdk(z) =V ∗k
z − εk1
z − εd −∆(z). (7.38)
We notice that the hybridization function has a branch cut on the real axis since
∆(z → ε± i0+) = ∓ iπ∑k
|Vk|2 δ (ε− εk) ≡ ∓ iΓ(ε). (7.39)
In terms of Γ(ε) > 0 we can write
∆(z) =
∫dε
π
Γ(ε)
z − ε.
Since we are interested in temperatures/frequencies much smaller than the conduction electronFermi energy, we will assume for simplicity that Γ(ε) = Γ for |ε| < W and Γ(ε) = 0 otherwise,with W of order TF . Then, if |z| W one readily finds that
∆(z) = −iΓ sgn Imz. (7.40)
In this approximation the impurity DOS is
ρd(ε) = − 1
πImG(ε+ i0+) =
1
π
Γ
(ε− εd)2 + Γ2, (7.41)
which is just a resonant state trapping a number of electrons of given spin given by
Nd =
∫ +∞
−∞dε f(ε) ρd(ε). (7.42)
238
7.2.1 Variation of the electron number
Quite generally the Green’s function, e.g. of the dσ-electron, in Matsubara frequency is givenby
G(iωn) =1
Z
∑NM
〈N |dσ|M〉〈M |d†σ|N〉e−βEM + e−βEN
iωn − (EM − EN ). (7.43)
We want to calculate T∑
n G(iωn). We notice that the Fermi function in the complex plane
f(z) =(
eβz + 1)−1
,
has poles on the imaginary axis just at the Matsubara frequencies iωn with residue−T . Thereforeif we consider a contour which encircles anticlockwise the imaginary axis then∮
dz
2πif(z)G(z) = T
∑n
G(iωn).
On the other hand the contour integral is also equivalent to the clockwise contour which runsavoiding the imaginary axis, which amounts to sum over the poles of the denominator in (7.43).As a result one gets
T∑n
G(iωn) =1
Z
∑NM
〈N |dσ|M〉〈M |d†σ|N〉 f(EM − EN )(
e−βEM + e−βEN)
=1
Z
∑NM
〈N |dσ|M〉〈M |d†σ|N〉 e−βEM
=1
Z
∑M
e−βEM 〈M |d†σ dσ|M〉 = 〈d†σ dσ〉.
Therefore the sum over Matsubara frequencies gives the occupation number, hence the variationof the electron number per spin due to the impurity is simply given by
∆Nels = T∑n
[G(iωn) +
∑k
Gkk(iωn)−G(0)kk(iωn)
]
= T∑n
1
iωn − εd −∆(iωn)
[1 +
∑k
|Vk|2(
1
iωn − εk
)2]
= T∑n
∂
∂iωnln (iωn − εd −∆(iωn)) , (7.44)
Let us consider the contour drawn in Fig. 7.2. The function S(z) = ln(z − εd −∆(z)) has abranch cut on the real axis and it is analytic everywhere else. Therefore∮
dz
2πif(z)
∂
∂zS(z) = 0,
239
Re z
Im z
Figure 7.2: Integration contour.
because in the region which excludes both axes the function is analytic. On the other hand
0 =
∮dz
2πif(z)
∂S(z)
∂z= T
∑n
∂
∂iωnln (iωn − εd −∆(iωn))
+
∫ +∞
−∞
dε
2πif(ε)
∂
∂ε
[S(ε+ i0+)− S(ε+ i0+)
]Therefore we finally get
∆Nels = −∫ +∞
−∞
dε
2πif(ε)
∂
∂ε
[S(ε+ i0+)− S(ε− i0+)
]. (7.45)
Within the previous approximation
S(ε+ i0+)− S(ε− i0+) = ln (ε− εd + iΓ)− ln (ε− εd − iΓ) = 2i tan−1 Γ
ε− εd,
hence, through Eq. (7.42),
∆Nels =
∫ +∞
−∞dε f(ε)
1
π
Γ
(ε− εd)2 + Γ2= Nd. (7.46)
In other words the change in the number of electrons only comes from those electrons occu-pying the impurity level. If for instance εd↑ is different from εd↓, at zero temperature the numberof electrons at the impurity with spin σ is
Ndσ =1
2− 1
πtan−1 εdσ
Γ. (7.47)
7.2.2 Energy variation
For later convenience, we calculate here the impurity contribution to the total energy. Supposethat we change Vk → λVk. The total energy becomes a function of λ, E(λ), which, by theHelmann-Feynmann theorem, satisfies
∂E(λ)
∂λ= 〈Ψλ|
∂H∂λ|Ψλ〉
240
=∑kσ
〈Ψλ|Vk c†kσdσ +H.c. |Ψλ〉 = 2T∑n
∑k
VkGdk(iωn;λ) + V ∗k Gkd(iωn;λ).
Here |Ψλ〉 is the ground state wavefunction at a fixed λ, and G(iωn;λ) are the correspondingGreen’s functions. Specifically
Gkd(z;λ) = λVk
z − εk1
z − εd − λ2 ∆(z), (7.48)
Gdk(z;λ) = λV ∗k
z − εk1
z − εd − λ2 ∆(z), (7.49)
with ∆(z) defined in Eq. (7.34). Therefore
∂E(λ)
∂λ= 4T
∑n
∑k
λ|Vk|2
iωn − εk1
iωn − εd − λ2 ∆(iωn)
= 4T∑n
λ∆(iωn)1
iωn − εd − λ2 ∆(iωn)= −2T
∑n
∂
∂λln(iωn − εd − λ2 ∆(iωn)
),
so that the energy variation due to the impurity is
E(λ)− E(0) = −2T∑n
∂
∂λln
(iωn − εd − λ2 ∆(iωn)
iωn − εd
). (7.50)
In addition the average value of the hybridization at fixed λ is
Ehyb(λ) =∑kσ
〈Ψλ|λVk c†kσdσ +H.c. |Ψλ〉 = λ∂E(λ)
∂λ, (7.51)
so that the change in the bath energy is given by
Ebath(λ)− Ebath(0) = E(λ)− E(0)− Ehyb(λ) = E(λ)− E(0)− λ ∂E(λ)
∂λ. (7.52)
To calculate (7.50) we use the same integration contour as in Fig. 7.2. By noticing that
ln(ε− εd + i0+)− ln(ε− εd − i0+) = 2i tan−1 0+
ε− εd= 2π i θ(εd − ε),
we find
E(λ)− E(0) = 2
∫dε
πf(ε)
[tan−1 λ2Γ
ε− εd− π θ(εd − ε)
]. (7.53)
Let us evaluate this integral in the simple case εd = 0 at zero temperature, where
E(λ)− E(0) = 2
∫ 0
−∞
dε
π
[tan−1 λ
2Γ
ε− π
]= − 2
πλ2 Γ ln
We
λ2 Γ. (7.54)
241
Σσ
[G]
− σ
U N− σd
= =
Figure 7.3: Self-energy within the Hartree-Fock approximation.
Therefore the full energy gain is
∆E = − 2
πΓ ln
We
Γ, (7.55)
which derives from an hybridization gain
Ehyb = − 4
πΓ ln
W
Γ, (7.56)
and a bath energy cost
∆Ebath =2
πΓ ln
W
eΓ. (7.57)
7.2.3 Mean-field analysis of the interaction
Let us consider the case εd = 0 and switch on the interaction
U
2(nd − 1)2 = U nd↑ nd↓ −
U
2nd.
The second term gives an εd = −U/2. Let us treat the first term within the Hartree-Fockapproximation. The self-energy for the spin σ impurity electrons is given by, see Fig. 7.3,
Σσ = U Nd−σ, (7.58)
whereNd−σ = 〈nd−σ〉,
is the average value, which has to be evaluated self-consistently. Therefore the impurity Green’sfunction is
Gσ(z) =1
z − εd −∆(z)− Σσ(z)=
1
z + U2 −∆(z)− U Nd−σ
, (7.59)
as if εdσ = −U/2 + U Nd−σ. By writing Nd σ = 1/2 + mσ we find through (7.47) the self-consistency condition
m =1
πtan−1 U m
Γ. (7.60)
There are two possibilities. If U ≤ πΓ the only solution is m = 0. It describes a paramagneticimpurity and corresponds to the previous analysis at U = 0. If U > πΓ, there are two stable
242
−U/2 U/20
DO
S
Figure 7.4: DOS of the AIM within the Hartree-Fock approximation.
solutions at finite m = ±m0, which describe a partially polarized impurity. Since these twosolutions are degenerate, the entropy S = ln 2 just reflects the spin degrees of freedom. In otherwords, the magnetic state m 6= 0 describes indeed a local moment. If we assume a density matrixwhich is the average of the two degenerate ground states, then the impurity Green’s functionreads:
G(z) =1
2
[1
z + U m+ iΓsign Imz+
1
z − U m+ iΓsign Imz
], (7.61)
and the DOS is composed by two lorentians, one below and one above the chemical potential,which are called the Mott-Hubbard bands. In particular, if U Γ, m → 1/2 and the twolorentians are separated by an energy roughly of order U , see Fig. 7.4.
The mean-field solution for large U of the Anderson impurity model seems to account forthe behavior above the Kondo temperature and explains how local moments appear. The keyingredient is a large Coulomb repulsion around the magnetic impurity in comparison with theresonance broadening Γ.
Yet we have to understand what happens as the temperature is lowered.
243
7.3 From the Anderson to the Kondo model
The Hartree-Fock solution of the Anderson impurity model does explain in the large U/Γ-limitthe existence of local free moments in a metal well below its Fermi temperature. Nevertheless thedegeneracy of the mean-field solution is an artifact of mean-field theory, since the two solutions,in the above example, are not orthogonal and can be coupled through the conduction electrons.In order to understand the role of this coupling, it is more convenient to introduce the conductionwave-function at the impurity site through
c0σ =
√1
V
∑k
V ∗k ckσ, (7.62)
whereV 2 =
∑k
|Vk|2,
so that the impurity Hamiltonian plus the hybridization becomes
V∑σ
(c†0σ dσ +H.c.
)+U
2(nd − 1)2 . (7.63)
In the large U limit the impurity traps a single electron, either with spin up or down, andwe can treat the hybridization as a small perturbation lifting the degeneracy between the twospin direction. By second order perturbation theory through intermediate states in which theimpurity is empty or doubly occupied, with energy difference equal to U/2 for large U , one getan effective operator lifting the impurity degeneracy given by, apart from constant terms,
−2V 2
U
∑σσ′
c†0σdσ d†σ′c0σ′ + d†σ′c0σ′ c
†0σdσ
= JK S0 · Sd
where
JK =8V 2
U,
and
S0 =1
2
∑αβ
c†0α σαβ c0β
=1
2V 2
∑kpαβ
Vk V∗p c†kα σαβ cpβ, (7.64)
Sd =1
2
∑αβ
d†α σαβ dβ. (7.65)
244
In other words in the large U limit the AIM reduces to the so-called Kondo model describingconduction electrons antiferromagnetically coupled to a local moment, a spin-1/2 in our single-orbital AIM example. The Kondo Hamiltonian is therefore
HK =∑kσ
εk c†kσckσ + JK S0 · Sd. (7.66)
First of all we notice that this model has built-in a local moment, hence by construction correctlydescribes the local moment regime for TK T TF . This local moment provide a scatteringpotential for the conduction electrons which the major difference with respect to a conventionalscalar potential that the impurity has internal degrees of freedom. We will show now theconsequences of this difference.
7.3.1 The emergence of logarithmic singularities and the Kondo temperature
Let us analyse the role of the Kondo exchange
JK S0 · Sd, (7.67)
in perturbation theory. Since (7.67) conserves independently the number of conduction andd-electrons, we use a trick due to Abrikosov and treat the impurity spin in terms of d-electronswith an unperturbed Green’s function
G(0)(z) =1
z,
namely as a resonant state right at the chemical potential with a vanishing broadening, whichwe already know contains in the “unperturbed state” just one electron, number which is notgoing to be changed by (7.67). Since a single electron has a well defined spin, it acts indeed likea local spin-1/2 moment. Moreover the Green’s function for the conduction electron c0 is
G(0)0 (z) =
1
V 2
∑k
|Vk|2
z − εk=
1
V 2∆(z).
For convenience let us rewrite (7.67) in a spin-asymmetric form as∑i=x,y,z
Ji Si0 S
id, (7.68)
and calculate the first order corrections to the exchange as given by the diagrams in Fig. 7.5. Forsimplicity we assume that all external lines are at zero frequencies. Each diagram is multipliedby (-1) because of first order perturbation theory. The explicit expression of the diagram (a) is
(a) = −∑
i,j=x,y,z
Ji Jj∑γc
Siαγ Siac S
jγβ S
jcbT
∑n
G(0)0 (iεn)G(0)(−iεn)
245
J i J j
J i J j
αγ
β
αγ
β
ac
b
ac b
(b)
(a)
ε
ε
−ε
ε
Figure 7.5: First order corrections to the Kondo exchange. Solid and dashed lines indicateimpurity and conduction electron Green’s functions. Also indicated are the spin labels. Allexternal lines are assumed to be at zero frequency ε = 0. The internal frequency, which is goingto be summed over, is also indicated.
= −∑
i,j=x,y,z
JiJjV 2
∑γc
Siαγ Siac S
jγβ S
jcbT
∑n
∆(iεn)
−iεn
= −I(T )∑
i,j=x,y,z
JiJjV 2
∑γc
Siαγ Sjγβ S
iac S
jcb,
where the function of temperature I(T ) is defined through
I(T ) = −T∑n
∆(iεn)
iεn.
The diagram (b) is analogously
(b) = −∑
i,j=x,y,z
Ji Jj∑γc
Siαγ Sicb S
jγβ S
jacT
∑n
G(0)0 (iεn)G(0)(iεn)
= I(T )∑
i,j=x,y,z
JiJjV 2
∑γc
Siαγ Sjγβ S
jac S
icb,
hence the sum is
(a) + (b) = I(T )∑
i,j=x,y,z
JiJjV 2
∑γ
Siαγ Sjγβ
∑c
(Sjac S
icb − Siac S
jcb
)= I(T )
∑i,j,k=x,y,z
JiJjV 2
∑γ
Siαγ Sjγβ i εjik S
kab,
246
where we used the commutation relations between the spin operators[Si, Sj
]= i εijk S
k,
where εijk is the Levi-Civita antisymmetric tensor. Since we sum over i and j, we can also write
(a) + (b) =1
2I(T )
∑i,j,k=x,y,z
JiJjV 2
∑γ
Siαγ Sjγβ i εjik S
kab + (i↔ j)
=1
2I(T )
∑i,j,k=x,y,z
JiJjV 2
i εjik Skab
∑γ
(Siαγ S
jγβ − S
jαγ S
iγβ
)
=∑
k=x,y,z
Skab Skαβ
1
2I(T )
∑i,j
JiJjV 2|εijk|2
,which provides the first order correction to the exchange constants according to
Jk + δJk = Jk +1
2I(T )
∑i,j
JiJjV 2|εijk|2. (7.69)
In order to evaluate I(T ) we use the same integration contour as in Fig. 7.2. We recall that thehybridization function has a branch cut approximately given by
∆(ε+ i0+)−∆(ε− i0+) = −2 iΓ θ(W − |ε|),
where θ(W − |ε|) is the Heavyside function and W a cut-off of the order of the conductionbandwidth. We then find that
I(T ) = −∫ W
−W
dε
πf(ε)
Γ
ε' Γ
πlnW
T,
so that e.g. in the isotropic case Jx = Jy = Jz = J we finally get
J + δJ = J +Γ
π V 2J2 ln
W
T.
In other words the perturbation theory in J generates logarithmic singularities which becomevisible roughly around a temperature, which is therefore to be identified with the Kondo tem-perature, when the correction becomes of the same order as J , namely
J =Γ
π V 2J2 ln
W
TK,
leading to
TK = W exp
(−πV
2
ΓJ
)= W exp
(−πU
8Γ
)W ∼ TF , (7.70)
since U Γ. In agreement with experiments we do find that the Kondo temperature is muchsmaller than the host-metal Fermi temperature. Since perturbation theory becomes meaninglessbelow TK , the next obvious question is how to proceed further.
247
7.3.2 Anderson’s scaling theory
What we have calculated before is the first order correction to the Kondo exchange for conductionand impurity electrons at zero frequency, which we found to be
Jz[W ] + δJz[W ] = Jz[W ] +Γ
π V 2J⊥[W ]2 ln
W
T, (7.71)
J⊥[W ] + δJ⊥[W ] = J⊥[W ] +Γ
π V 2J⊥[W ] Jz[W ] ln
W
T. (7.72)
In (7.71) and (7.72) we have assumed an easy-axis asymmetry Jz 6= Jx = Jy ≡ J⊥, and wehave explicitly indicated the dependence upon the bandwidth cut-off W . Clearly the result doesnot change for finite external frequencies provided they are much smaller than the temperature,otherwise they would cut-off the log-singularity instead of T . Now suppose we have anothermodel with a smaller conduction electron bandwidth cut-off W (s) = W/s with s > 1, differentJ ’s but equal Γ/V 2. In this case, for electrons close to the chemical potential, specifically muchcloser than the temperature, we would get the first order correction
Jz [W (s)] + δJz [W (s)] = Jz [W (s)] +Γ
π V 2J⊥ [W (s)]2 ln
W
sT,
J⊥ [W (s)] + δJ⊥ [W (s)] = J⊥ [W (s)] +Γ
π V 2J⊥ [W (s)] Jz [W (s)] ln
W
sT.
On the other hand, it makes not really a big difference for the low energy behavior whetherthese electrons which lie so close to the chemical potential derive from a band with width Wor W (s) provided they suffer the same scattering off the impurity. Therefore we can ask thefollowing question:
what J [W (s)] should be in order for the effective exchange up to first order to be the same asthat one with bandwidth W ?
The answer is quite simple, since we can re-write e.g. (7.71) as
Jz[W ] + δJz[W ] = Jz[W ] +Γ
π V 2J⊥[W ]2 ln(s) +
Γ
π V 2J⊥[W ]2 ln
W
sT
'(Jz[W ] +
Γ
π V 2J⊥[W ]2 ln(s)
)+
Γ
π V 2
(Jz[W ] +
Γ
π V 2J⊥[W ]2 ln(s)
)2
lnW
sT+O
(J3),
the equality being valid up to the order at which we have stopped the expansion. Thereforethe two models with W and W/s have the same spin exchange up to first order in perturbationtheory provided the bare exchange constants satisfy
Jz [W (s)] = Jz[W ] +Γ
π V 2J⊥[W ]2 ln(s),
248
J⊥ [W (s)] = J⊥[W ] +Γ
π V 2J⊥[W ] Jz[W ] ln(s),
which can be cast in a differential form
djz(s)
d ln s= j⊥(s)2, (7.73)
dj⊥(s)
d ln s= j⊥(s) jz(s), (7.74)
having defined
Ji [W (s)]Γ
π V 2≡ ji(s).
These equations describe how the bare exchange constants have to be modified in order for themodel with a reduced bandwidth W/s to have the low-energy scattering amplitudes equal tothose of the original model. The idea behind is that, if we are able to follow the evolution ofji(s) until W/s ∼ T , at this point we can rely on perturbation theory since ln(W/sT ) ' 0. Thisis more or less how Anderson formulated his poor man’s scaling theory for the Kondo problemin 1970 which is in essence the first implementation of a Renormalization Group transformation.
Let us study the scaling equations. We readily find that, ln s = x,
d2jzdx2
= 2j⊥dj⊥dx
= 2 jzdjzdx
=dj2z
dx.
By integrating from x = ln s = ln 1 = 0, the initial condition W (s) = W , to same x > 0 we get
djz(x)
dx− djz(0)
dx=djz(x)
dx− j⊥(0)2 = jz(x)2 − jz(0)2,
namely
djz(x)
dx= jz(x)2 − jz(0)2 + j⊥(0)2,
d ln j⊥(x)
dx= jz(x).
In Fig. 7.6 it is shown the scaling flow of the exchange constants jz(x) and j⊥(x) as x → ∞.Since the sign of J⊥ can be always changed by rotating of an angle π the impurity spin aroundthe z-axis, we have just drawn the flow with J⊥ > 0, Jz > 0 and Jz < 0 implying antiferro- andferro-magnetic Kondo exchange, respectively. We notice that for ferromagnetic exchange Jz < 0with |Jz| ≥ J⊥, the couplings flow to a line of fixed points, where perturbation theory becomevalid. In particular for isotropic couplings Jz = J⊥ = J < 0, the value at x = lnW/T 1 is
j(T ) =j
1− j lnW/T' − 1
lnW/T→ 0. (7.75)
249
J z
J
Figure 7.6: Scaling flow of the Kondo exchange constants as the cut-off is sent to zero.
This suggests that the impurity asymptotically decouples from the conduction electrons. Forinstance the impurity magnetic susceptibility, which can be calculated perturbatively in thiscase, behaves at low temperatures as
χ(T ) ∼ 1
T lnW/T,
almost the Curie-like behavior of a free spin.
For antiferromagnetic Jz > 0, as well as for Jz < 0 but |Jz| < J⊥, the exchange constantsflow to strong-coupling jz ' j⊥ → +∞. This implies that the electrons very close to thechemical potential feel a very strong antiferromagnetic exchange with the impurity spin. It istherefore reasonable to assume that one has to diagonalize the Kondo spin-exchange first andonly then treats what is left within perturbation theory. It is convenient to rewrite the KondoHamiltonian (7.66) in a form more suitable for perturbation theory in 1/JK . We start from the
definition of c0σ given in Eq. (7.62), and introduce the single-particle wave-function (we dropfor the meanwhile the spin index)
|φ0〉 = c†0 |0〉.
We then define a new single-particle wave-function through∑k
εk c†kck c
†0 |0〉
= Hbath |φ0〉 ≡ ε0 |φ0〉+ t0 |φ1〉,
where |φ1〉 is a normalized single-particle wave-function orthogonal to |φ0〉 satisfying
t0 |φ1〉 = (Hbath − ε0) |φ0〉 ≡ Hbath |φ0〉 − |φ0〉 〈φ0|Hbath |φ0〉
Next we introduce another state orthogonal to the previous ones through
Hbath |φ1〉 = ε1 |φ1〉+ t0 |φ0〉+ t1 |φ2〉,
wheret1 |φ2〉 = Hbath |φ1〉 − |φ1〉 〈φ1|Hbath |φ1〉 − |φ0〉 〈φ0|Hbath |φ1〉.
250
We notice that applying Hbath to |φ2〉 we do not get any component on |φ0〉. In other words
Hbath |φ2〉 = ε2 |φ2〉+ t1 |φ1〉+ t2 |φ3〉,
which provides a definition of |φ3〉, again orthogonal to the other states. By iterating thisprocedure, and after defining
|φn〉 = c†n |0〉,
one can rewrite the Kondo Hamiltonian as
HAIM =∑σ,n≥0
εn c†nσcnσ + tn
(c†nσcn+1σ +H.c.
)+JK S · S0, (7.76)
which describes a one-dimensional chain with nearest neighbor hopping where the first site, site“0”, is spin-coupled to the impurity spin S. For simplicity let us assume that the model isparticle-hole invariant, which means that the Hamiltonian is invariant under the transformation
cn↑ → (−1)n c†n↓, cn↓ → −(−1)n c†n↑.
One readily finds that the hopping term as well as the Kondo exchange are invariant under thistransformation. On the contrary the site-energy term is not, hence we have to assume εn = 0for all n’s.
In the spirit of the scaling approach, we assume that JK is much larger than any tn’s hencewe diagonalize the two-site problem which includes the impurity spin and site “0”, and treatsthe hopping t0 within perturbation theory. By diagonalizing the Kondo exchange we find thatthe ground state is obtained by coupling the impurity spin-1/2 with a single electron at site“0” into a singlet state. All other states are above at least of energy JK . The hopping termt0 couples the singlet state with high energy states, hence, since JK t0, we have to concludethat t0 is an irrelevant perturbation which as first approximation can be simply dropped out.In other words the impurity spin binds one electron in a singlet state making the site “0” nomore available to other electrons. This singlet state is inert hence the net result is that theasymptotic Hamiltonian flows to Eq. (7.76) with t0 = 0. The electron binding implies from theFriedel’s sum rule that
251
Chapter 8
Introduction to abelian Bosonization
In one dimension, many-body models can be handled with by a very powerful tool known asbosonization. In what follows we shall give a brief introduction to its abelian version. 1
8.1 Interacting spinless fermions
Let us consider a one dimensional model of spinless fermions described by a non interactingHamiltonian of the form
H0 =∑k
εkc†kck (8.1)
with a band dispersion of the form in figure 8.1. If the chain has L sites, then
k =2πn
La, n ∈
[−L
2,L
2
]for periodic boundary conditions
k =2π
La(n+ 1), n ∈
[−L
2,L
2
]for antiperiodic boundary conditions.
The scope is to study the model Eq. (8.1) in the presence of a generic interaction
Hint =1
2L
∑q
V (q)ρ(q)ρ(−q) (8.2)
where ρ(q) =∑
k c†kck+q is the Fourier transform of the density.
We will adopt an approach which is substantially similar to what is usually done to studyquantum fluctuations in models which develop, at the mean field level, an order parameter.
1I am grateful to Angelo Russomanno who wrote in LaTex my lecture notes on this topic
252
Figure 8.1:
Let us start by the non interacting Hamiltonian (8.1). If N fermions are present, the groundstate is obtained by occupying the momentum eigenstates up to a Fermi momentum such that
kF∑k=−kF
= N ' L
πkF if N and L are large.
This ground state (Fermi sea, see figure 8.2) has a kind of order parameter, namely the averageof the occupation number in momentum space⟨
c†kck
⟩≡ 〈nk〉 = θ(kF − |k|) .
We assume that the average value of the band energy
1
N〈H0〉 ' εF
is much bigger than the interaction energy between two electrons at their typical distance, i.e.V (q) at q ∼ kF ; then the only degrees of freedom affected by the interaction are those aroundeach Fermi point ±kF . We define as right moving fermions those with positive momentum(cRk) and left moving fermions those with k < 0, (cLk). Analogously, the Right and Leftmoving densities are defined through
ρR(q) =∑k
c†RkcRk+q ρL(q) =∑k
c†LkcLk+q .
253
Figure 8.2:
Since the interaction is weak compared to εF , the Fourier components of the densities which areaffected by the interaction are those with
|q| kF .
In this limit, they are also well defined the Right and Left mover densities in real space ρR(x) andρL(x), they describe the distribution functions R and L of electrons with momentum k ∼ ±kFwhich vary in space over distances ∆x ∼ |1/q| such that, from ∆x∆k ∼ 1
∆k ∼ 1
∆x∼ |q| kF .
The perturbed ground state is identified by the “order parameters”
〈nRk〉 =⟨c†RkcRk
⟩k > 0
〈nLk〉 =⟨c†LkcLk
⟩k < 0 .
Let us study the role of interaction in the same spirit as one studies the role of quantumfluctuations on models which develop an order parameter at the mean field level.
The idea is to derive the equations of motion by making the approximation that, whenevera commutator of two operators has a component proportional to the order parameter, we sub-stitute it with its average value on the ground state. Let us show how this procedure works inthe well known spin wave theory.
254
8.1.1 Spin wave theory
On each site R there is a spin with components Sx(R), Sy(R), Sz(R). The classical ground statehas an order parameter
〈Sz(R)〉 = M(R) ,
then, within spin wave approximation[Sx(R), Sy(R
′)]
= iδR,R′Sz(R) ' iδR,R′M(R)[Sx,y(R), Sz(R
′)]
= ±iδR,R′Sy,x(R) . (8.3)
Therefore, the third component of the spin is equivalent to
Sz(R) ≡ − 1
2M(R)
(Sx(R)2 + Sy(R)2
)+
1
2+M(R) .
Let us apply this idea to our case.We start by calculating[
ρR(q), ρR(q′)]
=∑k, p
[c†RkcRk+q, c
†RpcRp+q′
]=
∑k, p
(δp, k+q c
†RkcRp+q′ − δk, p+q′ c
†RpcRk+q
).
By a change of variables, it is equal to∑k, p
(δp, k+qc
†RkcRk+q+q′ − δp, k+q′c
†RkcRk+q+q′
).
If the sums run for all k’s, the above term would be zero. However, since all the fermions whichare involved should be right movers, with positive momentum, hence the first δ-function impliesk + q > 0 while the second k + q′ > 0, then one finds
[ρR(q), ρR(q′)
]=
max(0,−q′,−q−q′)∑k=max(0,−q,−q−q′)
+
min(π/a,π/a−q,π/a−q−q′)∑k=min(π/a,π/a−q′,π/a−q−q′)
c†RkcRk+q+q′ .
Since all q’s are small, for q 6= −q′, this operator involves excitations very far from kF , eitherclose to k ∼ 0 or k ∼ π/a. Since we expect the ground state occupation number not to bemodified for k ∼ 0 and k ∼ π/a, we can safely assume the commutator to be zero. On thecontrary, for q = −q′, the commutator becomes related to the order parameter, namely
[ρR(q), ρR(−q)] =
max(0,q)∑max(0,−q)
+
min(π/a,π/a−q)∑min(π/a,π/a+q)
nRk .
255
In the spirit of what we previously said, we approximate the operator with its average value onthe Fermi sea; namely
[ρR(q), ρR(−q)] ≈ 〈[ρR(q), ρR(−q)]〉 =
max(0,q)∑max(0,−q)
θ(kF − k) =qL
2π, (8.4)
therefore [ρR(q), ρR(q′)
]= δq,−q′
qL
2π. (8.5)
Analogously for the left movers one finds[ρL(q), ρL(q′)
]= −δq,−q′
qL
2π. (8.6)
In real space
[ρR(x), ρR(y)] =1
L2
∑q, q′
eiqx eiq′y[ρR(q), ρR(q′)
]e−α|q|
=1
2πL
∑q
q eiq(x−y)−α|q| = − i
2π∂x∑q
eiq(x−y)−α|q|
= − i
2π∂xδα(x− y) ,
where α is a cut off (recall that |q| kF was a starting condition) as well as
[ρL(x), ρL(y)] =i
2π∂xδα(x− y) . (8.7)
8.1.2 Construction of the effective Hamiltonian
We start with the commutator of the density in momentum space with the Hamiltonian
[ρR(q), H0] =∑k, p>0
εp
[c†RkcRk+q, c
†RpcRp
]=
∑k, p>0
εp
(δp, k+q c
†RkcRp − δk, p c
†RpcRk+q
)=
∑k
(εk+q − εk) c†RkcRk+q ' vF q ρR(q) , (8.8)
where we have defined the Fermi velocity as vF ≡ ∂kεk|k=kFand we exploited that only the
electrons near the Fermi momentum kF are involved and that |q| kF . Since ∂kεk|k=−kF = −vF
[ρL(q), H0] = −vF q ρL(q) . (8.9)
256
Thanks to the above commutation relations we can write the effective Hamiltonian
H0 =π
LvF∑q
(ρR(q)ρR(−q) + ρL(q)ρL(−q))
= πvF
∫dx (ρR(x)ρR(x) + ρL(x)ρL(x)) . (8.10)
Moreover, the general continuity equation for Right and Left movers implies
i~ρR(q) = [ρR(q), H0] = qJR(q)
i~ρL(q) = [ρL(q), H0] = qJL(q) (8.11)
which, compared with Eqs. (8.8) and (8.9), gives the important relations for the current oper-ators
JR(q) = vFρR(q) (8.12)
JL(q) = −vFρL(q) . (8.13)
The commutation relations Eqs. (8.5) and (8.6) provide a bosonic representation of the longwavelength density fluctuations. We define, for q > 0,
bRq = −i√
2π
qLρR(q) b†Rq = i
√2π
qLρR(−q) (8.14)
as well as
bLq = i
√2π
qLρL(−q) b†Lq = −i
√2π
qLρL(q) , (8.15)
which satisfy, for a, b = R, L [ba q, b
†b q′
]= δa, b δq, q′[
ba q, bb q′]
= 0 .
Notice that, for q = 0, ρR(q = 0) and ρL(q = 0) correspond to the densities of right R and leftL moving fermions, NR/L and NL/L respectively, where NR,L are number operators.
Therfore, we can write
ρR(x) =i
L
∑q>0
√q L
2π
(eiqxbRq − e−iqxb†Rq
)+NR
L(8.16)
ρL(x) = − iL
∑q>0
√q L
2π
(eiqxbLq − e−iqxb†Lq
)+NL
L. (8.17)
257
Figure 8.3:
We can also introduce fields which are conjugate of ρR,L; indeed the fields
φR(x) =1
L
∑q>0
√2π L
q
(eiqxbRq + e−iqxb†Rq
)+ kF,Rx− θR +
π
2NL
(8.18)
φL(x) =1
L
∑q>0
√2π L
q
(e−iqxbLq + eiqxb†Lq
)+ kF,Lx+ θL +
π
2NR
(8.19)
where kF R = 2πNR/L, kF L = 2πNL/L are the right and left Fermi momenta, corresponding tothe deformation shown in figure 8.3. They satisfy
[φR(x), ρR(y)] = −iδ(x− y) ,1
2π∂xφR(x) = ρR(x) , (8.20)
[φL(x), ρL(y)] = iδ(x− y) ,1
2π∂xφL(x) = ρL(x) . (8.21)
Moreover [θR, NR] = [θL, NL] = i, therefore the last term in Eqs. (8.18) and (8.19) implies that
[φR(x), φL(y)] = −iπ (8.22)
which will be useful later. We emphasize that the θ operators which are conjugate of NR,L areneeded because the sum over q does not include q = 0.
Since φR and φL are conjugate of the densities, one can easily construct operators whichcreate or destroy particles. Indeed
e−iφR(x) creates a Right fermion in x since
258
eiφR(x)ρR(y)e−iφR(x) = ρR(y) + i [φR(x), ρR(y)] = ρR(y) + δ(x− y)
and analogously eiφL(x). In turns, these operators can be used to construct Fermi fields. It iscustomary to define shifted φ’s. Given a reference state with
〈NR〉 = 〈NL〉 =N
2k0F R = k0
F L = πN
L≡ kF ,
we define new φR and φL like in Eqs. (8.18) and (8.19), by substituting the number operatorsby
NR,L → ∆NR,L = NR,L −N
2.
Accordingly, we have also to perform the substitution
kF, r → ∆kF, r = kF, r − kF = kF, r − πN
L
implying that ρR,L = 12π∂xφR,L + N
2L . Then the operators
ΨR(x) ≡ 1√2πα
eikF x+iφR(x) (8.23)
ΨL(x) ≡ 1√2πα
e−ikF x−iφL(x) (8.24)
behave like Fermi fields which annihilate a right or left fermion at position x. To show it, werecall that, given two operators A and B both commuting with [A,B], then the BCH formulaholds
eAeB = eBeAe−[B,A]
eA+B = eAeBe−12
[A,B] = eBeAe−12
[B,A] . (8.25)
Therefore (using Eq. (8.22))
ΨR(x)ΨL(y) =1
2παeikF (x−y)eiφR(x)e−iφL(y)
=1
2παeikF (x−y)e−iφL(y)eiφR(x)e[φL,φR] = −ΨL(y)ΨR(x) .
Moreover
ΨR(x)Ψ†R(y) =1
2παeikF (x−y)eiφR(x)e−iφR(y)
=1
2παeikF (x−y)ei(φR(x)−φR(y))e
12
[φR(x),φR(y)] . (8.26)
259
The above commutator is not impossible to evaluate
[φR(x), φR(y)] =2π
L
∑q>0
1
q
(eiq(x−y) − e−iq(x−y)
)+
2π
Li(x− y)
=2π
L
∑q
1
qeiq(x−y)e−α|q| = 2i atan
(x− yα
)' iπ sign(x− y) .
(8.27)
Analogously
[φL(x), φL(y)] = −2i atan
(x− yα
)' −iπ sign(x− y) . (8.28)
Hence
ΨR(x)Ψ†R(y) =1
2παeikF (x−y)ei(φR(x)−φR(y))ei
π2
(x−y) while (8.29)
Ψ†R(y)ΨR(x) =1
2παeikF (x−y)ei(φR(x)−φR(y))e−i
π2
(x−y) . (8.30)
Hence, for |x− y| αΨR(x),ΨR(y) = 0
and the same holds for ΨL
ΨL(x),ΨL(y) = 0 .
What does it happen when we send x → y? We cannot simply take φR(x) − φR(y) → 0 inEqs. (8.29) and (8.30), since the exponential operator is singular. We first have to normal orderit and then take the limit x→ y
φR(x)− φR(y) ' 1
L
∑q>0
√2πL
q
(e−iqx − e−iqy
)b†Rq
+1
L
∑q>0
√2πL
q
(eiqx − eiqy
)bRq = φ
(+)R + φ
(−)R (8.31)
With this decomposition, using relations (8.25), we can write
ei(φ(+)R +φ(−)) = eiφ
(+)R eiφ
(−)R e
12
[φ(+)R , φ
(−)R ]
where
[φ(+)R , φ
(−)R ] = −π
L
∑q>0
1
q
(e−iqx − e−iqy
) (eiqx − eiqy
)e−αq =
260
= −πL
∑q>0
1
qe−αq [2− 2 cos (q(x− y))] =
1
2log
(α2
α2 + (x− y)2
).
(8.32)
The same result is valid for φL. By adding the phase factor
1
2[φR/L(x), φR/L(y)] = ±i atan
(x− yα
)evaluated in Eqs. (8.27) and (8.28), we have in ΨR/L(x)Ψ†R/L(y)
1
2log
(α2
α2 + (x− y)2
)± i atan
(x− yα
)= log
(α
α∓ i(x− y)
).
Therefore, introducing the normal ordering symbol as
: e±i(φR/L(x)−φR/L(y)) : ≡ eiφ(+)
eiφ(−)
,
we can definitely write the anticommutator as
ΨR/L(x),Ψ†R/L(y)
' 1
2παe±ikF (x−y) : e±i(φR/L(x)−φR/L(y)) :
[α
α∓ i(x− y)+
α
α± i(x− y)
](8.33)
= e±ikF (x−y) : e±i(φR/L(x)−φR/L(y)) :1
π
α
α2 + (x− y)2
α→0−→ δ(x− y) .
(8.34)
(We have here the approximation δα(x − y) = 1π
αα2+(x−y)2
to the δ.) So we can see that the
anticommutator gives a δ-function regularized by α. This was predictable since Ψ†R/L(x) doesnot really create a fermion in x, but a wave packet around x of width α.
The previous calculation also shows that a particular care has to be taken when we considerthe product of two bosonized operators at the same point. Given two operators A(x) and B(x),the correct way to do the product is
A(x)B(x) = limx→y
[A(x)B(y)]| |x−y|α1
.
Example : How to recover Ψ†R(x)ΨR(x) = ρR(x).
261
In the spirit of what we have stated few lines above, let us calculate Ψ†R(x)ΨR(y) and thentake the limit. As we can see in Eq. (8.33),
Ψ†R(x)ΨR(y) ' 1
2παe−ikF (x−y) : e−i(φR(x)−φR(y)) :
α
α− i(x− y)
=1
2π
1
α− i(x− y): 1− ikF (x− y)− i∂xφR(x)(x− y) : .
(8.35)
Hence
Ψ†R(x)ΨR(x) = limy→x
[Ψ†R(x)ΨR(y)
]∣∣∣ |x−y|α1
=1
2π∂xφR(x) +
kF2π≡ ρR(x) .
The equations of motion Eqs. (8.11) for ρR and ρL imply that
bRq(t) = bRqe−ivF qt bLq(t) = bLqe
−ivF qt,
hence, thanks to Eqs. (8.16) and (8.17), we have
φR(x, t) = φR(x− vF t)φL(x, t) = φL(x+ vF t) .
Therefore, analogously to Eq. (8.33), we find the following correlation functions⟨ΨR/L(x, t)Ψ†R/L(0, 0)
⟩= e±ikF x
1
2π
[1
α∓ i(x∓ vF t)
]⟨
Ψ†R/L(0, 0)ΨR/L(x, t)⟩
= e±ikF x1
2π
[1
α± i(x∓ vF t)
],
so that the Green function is easily written
GL/R(x, t) = −i⟨←−>(
ΨR/L(x, t)Ψ†R/L(0, 0))⟩
=1
2πe±ikF x
1
±x− vF t+ iα sign(t).
We can even write the density-density correlation function (by using directly formulae Eqs. (8.16)and (8.17))⟨
ρR/L(x, t)ρR/L(0, 0)⟩
=1
4π2
⟨∂xφR/L(x, t) ∂xφR/L(0)
⟩=
1
4π2
1
[α∓ i(x∓ vF t)]2.
For future utility, let us calculate
GβR βL =⟨
eiβRφR(x,t)+iβLφL(x,t)e−iβRφR(0)−iβLφL(0)⟩.
Since [φR(x), φL(y)− φL(z)] = 0 then
GβR βL =⟨
eiβRφR(x,t)e−iβRφR(0)⟩⟨
eiβLφL(x,t)e−iβLφL(0)⟩
=
[α
α− i(x− vF t)
]β2R[
α
α+ i(x+ vF t)
]β2L
(8.36)
262
8.2 Interactions
Let us now switch on interaction
Hint =1
2L
∑q
V (q)ρ(q)ρ(−q) .
Since we are assuming a weak coupling approach; of all the scattering amplitudes generated byV (q), we keep only those which act among Fermions close to the Fermi momenta (see figure 8.4).For generic filling N/L, there are two such scattering processes. In the former we have
V (q) ∼ V (q = 0) (direct interaction)
and in the latter
V (q) ∼ V (q = 2kF ) (exchange interaction) .
Shortly, we have
g2 = V (0)− V (2kF )
andHint =
g2
L
∑q
ρR(q)ρL(−q) .
Therefore, the low energy Hamiltonian is
H = H0 +Hint =πvFL
∑q
[ρR(q)ρR(−q) + ρL(q)ρL(−q)] +g2
L
∑q
ρR(q)ρL(−q) , (8.37)
263
Figure 8.4:
which, written in terms of the bRq and bLq (dropping an infinite though constant therm) hasthe form
H = vF∑q>0
q[b†RqbRq + b†LqbLq
]+g2
2π
∑q>0
[bRqbLq + b†Lqb
†Rq
].
This is a bilinear form which can be easily diagonalized. However, we will adopt a differentapproach, which is more useful for the following. Let us introduce two new fields
Φ(x) ≡ 1√4π
(φR(x) + φL(x))
Θ(x) ≡ 1√4π
(φL(x)− φR(x)) (8.38)
as well asΠ(x) ≡ ∂xΘ(x) =
√4π (ρL(x)− ρR(x)) .
Using the commutators Eqs. (8.22), (8.27), (8.28), we can see that
[Φ(x), Θ(y)] =1
4π
− [φR(x), φR(y)] + [φL(x), φL(y)]
+ [φR(x), φL(y)]− [φL(x), φR(y)]
=1
4π
−4i atan
(x− yα
)− 2πi
' 1
4π−2πi sign(x− y)− 2πi = −iθ(x− y) , (8.39)
where θ is the Heaviside step function. Therefore we have the important relation
[Φ(x), Π(y)] = iδ(x− y) (8.40)
264
stating that that Π and Φ are conjugate variables. From Eq. (8.38) we have
φL(x) =√π (Φ(x) + Θ(x))
φR(x) =√π (Φ(x)−Θ(x)) . (8.41)
Since ρR/L(x) = 12π∂xφR/L(x) + kF
2π we have
H =vF4π
∫dx [∂xφR(x)∂xφR(x) + ∂xφL(x)∂xφL(x)] +
g2
4π2
∫dx ∂xφR(x)∂xφL(x)
=vF2
[∫dx
(1 +
g2
2πvF
)∂xΦ(x) ∂xΦ(x) +
∫dx
(1− g2
2πvF
)Π(x)Π(x)
]
=1
2vF
√1−
(g2
2πvF
)2 ∫dx
[1
K∂xΦ(x) ∂xΦ(x) +KΠ(x)Π(x)
](8.42)
where we have defined
K ≡
√1− g2
2πvF
1 + g22πvF
.
Notice that the density and the current are given by
ρ(x) = ρL(x) + ρR(x) =1√π∂xΦ(x)
J(x) = vF (ρR(x)− ρL(x)) = − 1√π∂xΘ(x) . (8.43)
The canonical transformation
Φ(x) =√KΦ(x)
Π(x) =1√K
Π(x) (8.44)
makes Eq. (8.42) diagonal
H =1
2v
∫dx[∂xΦ(x) ∂xΦ(x) + Π(x)Π(x)
]where v ≡ vF
√1−
(g2
2πvF
)2
. (8.45)
This Hamiltonian describes the normal modes of the low energy particle-hole excitations (inLandau theory the zero sound, which exhausts in one dimension the particle-hole spectrum).The big advantage of Bosonization is that we can also calculate single particle correlations
φR/L(x) =√π
(√KΦ(x)∓ 1√
KΘ(x)
)265
=1
2
√K(φR(x) + φL(x)
)∓ 1
2√K
(φR(x)− φL(x)
)=
1
2
(√K ± 1√
K
)φR(x) +
1
2
(√K ∓ 1√
K
)φL(x) . (8.46)
Using Eqs. (8.36) and (8.45), we can see that
⟨ΨR/L(x, t)Ψ†R/L(0)
⟩=
1
2πα
[α
α− i(x− vt)
] 14
(√K± 1√
K
)2 [α
α+ i(x+ vt)
] 14
(√K∓ 1√
K
)2
=1
2πα
α
α∓ i(x∓ vt)α∓ i(x∓ vt)
α
[α
α− i(x− vt)
] 14(K+ 1
K±2) [ α
α+ i(x+ vt)
] 14(K+ 1
K∓2)
=1
2πα
α
α∓ i(x∓ vt)
[α2
(α− i(x− vt))(α+ i(x+ vt))
] 14K
(K−1)2
. (8.47)
Therefore the single particle Green’s function acquires an anomalous exponent 14K (K − 1)2.
8.2.1 Umklapp processes
Let us now consider the half filling case N/L = 1/2. In this very special case (assuming thelattice spacing a = 1) kF = π/2, and an additional scattering amplitude is allowed close to theFermi points, namely
∼ V (2kF ) = V (π) (Umklapp scattering) .
In real space it corresponds to an interaction term V (x − y) = V (π)(−1)x−yf(|x − y|), wheref(r) is some short range function such that
∫drf(r) = 1 . Hence the Umklapp term gives a
contribution to the Hamiltonian
Umklapp = V (π)
∫dxdy (−1)x−yf(|x− y|)
[Ψ†R(x)Ψ†R(y)ΨL(y)ΨL(x) + H.c.
].
Applying Eqs. (8.23) and (8.25) we can recast this formula as2
Umklapp =V (π)
(2πα)2
∫dxdy
[e−
12
[φR(x), φR(y)]
2To be consistent with eqs. (8.23) the argument of the cosine should be√
16πΦ(x) + 2πx; nevertheless theadditional term is not important because x is an integer number of lattice spacings a = 1.
266
× e−12
[φL(y), φL(x)]e−i(φR(x)+φR(y)+φL(x)+φL(y)) + H. c.]
' V (π)2
(2πα)2
∫dx cos(
√16πΦ(x)) , (8.48)
with g3 = V (π). After applying the canonical transformation Eq. (8.44) we have
HUmklapp =2g3
(2πα)2
∫dx cos
√16πKΦ(x)
and the Hamiltonian reads
H =1
2v
∫dx
[∂xΦ(x) ∂xΦ(x) + Π(x)Π(x) +
1
v
4g3
(2πα)2cos√
16πKΦ(x)
](8.49)
which is a sine-Gordon model. The dimension of the Umklapp is 4K, to be compared with thedimension of ∂xΦ(x) ∂xΦ(x) + Π(x)Π(x) which is 2. Therefore the Umklapp gets marginal atK = 1/2 and relevant for K < 1/2. When it is relevant, the Umklapp locks the field Φ to avalue which minimizes g3 cos
√16πΦ(x), and a gap opens in the spectrum.
How to calculate dimensions
Let us normal order cos√
16πKΦ(x). We get
cos√
16πKΦ(x) = : cos√
16πKΦ(x) : e8πK[Φ+(x), Φ−(x)]
= : cos√
16πKΦ(x) : exp
−4K2π
L
∑q>0
1
qe−αq
.Performing the integration over the second Brillouin zone gives an approximation of the argu-ment of the exponential
' −4K
∫ 1/α
2π/L
dq
q= −4K log
(L
2πα
)and therefore
1
(2πα)2cos√
16πKΦ(x) = (2πα)4K−2
(1
L
)4K
: cos√
16πKΦ(x) : .
The relevance/irrelevance is controlled by the behaviour under the limit α→ 0.
267
8.3 Bosonization of the Heisenberg model
Let us consider a 1d Heisenberg model
H = J
L∑i=1
(Szi S
yi + Syi S
yi+1 + ∆Szi S
zi+1
). (8.50)
We can map this model onto a spinless fermion model via the so called Jordan-Wigner trans-formation. We write
Szi =1
2− c†ici =
1
2− ni . (8.51)
Here ni = 0, 1 is the spinless Fermion number at site i, therefore Szi = 12 corresponds to an
empty site and Szi = −12 to an occupied site. Let us show that
S+i = eiπ
∑i−1j=1 njci , S−i = c†ie
−iπ∑i−1j=1 nj (8.52)
is a good representation for spin operators. Let us define
ξi ≡ eiπ∑i−1j=1 nj =
i−1∑j=1
(1− 2nj) (8.53)
the string operator. Since ξ2i = 1, we easily check that[S+i , S
−i
]=
[ci, c
†i
]= 1− 2ni = 2Szi[
S+i , S
zi
]= −ξi [ci, ni] = −S†i .
Moreover, since for i > jξicj = −cjξi
while for i ≤ jξicj = cjξi ,
then [S+i , S
+(−)j
]= ciξiξjc
(†)j − c
(†)j ξjξici = 0
for i 6= j, as it should be. Let us rewrite (8.50) using (8.51), (8.52) and (8.53). We have
Sxi Sxi+1 + Syi S
yi+1 =
1
2
(S+i S−i+1 + S−i S
+i+1
)=
1
2
(ciξiξ
†i+1c
†i+1 + c†iξ
†i ξi+1ci+1
)=
1
2
(cie−iπnic†i+1 + c†ie
iπnici+1
)268
=1
2
(c†ici+1 + c†i+1ci
)(8.54)
and
Szi+1 =
(1
2− ni
)(1
2− ni+1
)=
1
4+ nini+1 −
1
2ni −
1
2ni+1 . (8.55)
Hence
H =J
2
∑i
(c†ici+1 + H. c.
)+ J∆
∑i
[1
4+ nini+1 −
1
2ni −
1
2ni+1
]. (8.56)
In case of an open chain, i.e. site 1 not coupled to L, the last two terms are
−J∆
2
∑i
(ni + ni+1) =J∆
2(n1 + nL)− J∆
∑i
ni .
Since∑
i ni = N , the total number of fermions, is a conserved quantity, this term introduces ascattering potential at edge sites 1 and L. It is therefore better to deal with a closed chain, byadding the bond
J
2
(S+
1 S−L + S−1 S
+L
)+ J∆Sz1S
zL . (8.57)
We notice that
S+1 S−L = c1e−iπ
∑L−1j=1 njc†L = c1c
†Le−iπ
∑L−1j=1 nj
= c1c†Le−iπ
∑Lj=1 njeiπnL = (−1)Nc1c
†LeiπnL = (−1)Nc1c
†L . (8.58)
Hence, we can recast Eq. (8.57) as
J
2(−1)N
(c1c†L + c†Lc1
)+ J∆
(1
4+ n1nL −
1
2n1 −
1
2nL
). (8.59)
We further apply the transformationci → (−1)ici (8.60)
such that the Hamiltonian Eq. (8.50) (using also Eq. (8.59)) reads (apart from constant terms)
H = −J2
∑i
(c†ici+1 + H. c.
)+ J∆
∑i
nini+1
+J
2(−1)N+L
(c1c†L + c†Lc1
)+ J∆n1nL . (8.61)
The factor (−1)N+L makes the ground state non degenerate, as it should be for bosonic models.For simplicity, let us take N + L odd. In such case, Eq. (8.61) is equivalent to the spinlessfermion Hamiltonian
H = −t∑i
(c†ici+1 + H. c.
)+ V
∑i
nini+1
269
=∑k
εkc†kck +
V
L
∑q
cos (qa) ρqρ−q , (8.62)
with t = J/2, V = J∆ and εk = −2t cos(ka), a being the lattice spacing. The HamiltonianEq. (8.62) can be analysed with bosonization. In this specific case
kF =π
a
N
L=
π
2a− π
LaSztotal ,
εF = 2ta sin kFa ,
g2 = 2V (1− cos 2kFa) .
At half filling, g3 = −2V cos 2kFa = +2V is the Umklapp. As we showed, the Umklapp isirrelevant/marginal/relevant if
K > / < / =1
2;
what does K = 1/2 correspond to? Let us consider the bosonized expression of the spin oper-ators. As before, the spin operators will be related to phase field operators which are assumedto be slowly varying, namely varying in distances ∆x 1
kF∼ a. In this approximation a
continuum limitx ≡ j · a = continuous variable
is well defined, and
S+j ' S
+x ' (−1)x/aeiπ
∫ x−a dyρ(y) [ΨR(x) + ΨL(x)] , (8.63)
where (−1)x/a comes from Eq. (8.60), and
ρ(y) = ρR(y) + ρL(y) =1
2π∂yφR(y) +
1
2π∂yφL(y) +
N
L.
More explicitly
S+x ' (−1)x/a√
2παei(φR(x−a)+φL(x−a))/2eikF (x−a)
[eikF x+iφR(x) + e−ikF x−iφL(x)
]' (−1)x/ae−ikF a√
2πα
[e2ikF xei(3φR(x)+φL(x))/2 + ei(φR(x)−φL(x))/2
]=
(−1)x/ae−ikF a√2πα
[e2ikF xei
√π(2Φ(x)−Θ(x)) + e−i
√πΘ(x)
]=
(−1)x/ae−ikF a√2πα
[e2ikF xei
√π(2√KΦ(x)−Θ(x)/
√K) + e−i
√πΘ(x)/
√K]
; (8.64)
270
therefore (applying Eqs. (8.33) and (8.36)) we obtain (for |x− y| α)
⟨S+x S−y
⟩∼ (−1)(x−y)/aα1/(2K)−1
(1
|x− y|
)1/(2K)
. (8.65)
On the other hand (using Eqs. (8.25), (8.22) and (8.44))
Szi ∼ Szx ' Ψ†(x)Ψ(x) = Ψ†R(x)ΨR(x) + Ψ†L(x)ΨL(x) + Ψ†R(x)ΨL(x) + Ψ†L(x)ΨR(x)
=1√π∂xΦ(x) +
N
Lπ+
1
2πα
[e−2ikF xe−iφR(x)e−iφL(x) + H.c.
]=
1√π∂xΦ(x) +
N
Lπ+
i
2πα
[e−2ikF xe−i
√4πKΦ(x) + H.c.
]. (8.66)
Sztot = 0, which includes the ground state for isotropic Hamiltonians, then N = L/2, i.e.kF = π/2, hence the leading contribution to Szx is
Szx ∼i
2πα(−1)x/a
[e−i√
4πKΦ(x) + H.c.].
Therefore, the correlation is
⟨SzxS
zy
⟩∼ (−1)x/a
2π2α2(K−1)
(1
|x− y|
)2K
. (8.67)
If ∆ = 1, spin isotropy is a symmetry of Eq. (8.50); hence the ground state should have Sztot = 0and Eq. (8.67) should equal Eq. (8.65) up to a coefficient; this implies
2K =1
2K=⇒ K =
1
2;
therefore for ∆ < 1, K > 1/2 (K = 1 for ∆ = 0) and Umklapp processes are irrelevant. Thespin excitations are massless and the correlation functions decay as power laws. If ∆ > 1 thenK < 1/2 and Umklapp is relevant. A gap opens if
√16πΦ is locked to values π mod (2π) (see
Eq. (8.49) for g3 > 0). In this case⟨SzxS
zy
⟩∼ (−1)(x−y)/a · constant
which corresponds to a Neel state.What does it happen if K = 1/2, namely at the isotropic point? In this case Umklapp is
marginal and V = 2t (see Eq. (8.62)). This implies a strong coupling limit which is outside thelimit of applicability of bosonization model (Eq. (8.62)). We must resort to better controlledmodels which represent Eq. (8.50).
271
8.4 The Hubbard model
This model is described by the Hamiltonian
H = −t∑i, σ
c†i σci σ + U∑i
ni ↑ni ↓ . (8.68)
The interaction term represents an on-site Coulomb repulsion which prevents two electrons, withopposite spin, to be on the same site. If the number of electrons N is equal to the number ofsites L and U t, the Hubbard model is insulating, each site being occupied by an electronwhich does not move, since it would cost energy U t. Therefore, at energy U , charge degreesof freedom are frozen and only the spin degrees of freedom matter. Indeed, it can be mappedonto the Heisenberg antiferromagnet, namely spin isotropic, with J = 4t2/U .
However, at half filling, the model Eq. (8.68) is insulating, not only for U t, but for anyvalue of U > 0. This happens because of the nesting property of the Fermi surface together withthe filling being commensurate. Therefore, charge degrees of freedom are frozen for any U andsince there is no phase transition as a function of U , we expect the spin degrees of freedom tostill be described by an Heisenberg model. Therefore, we have the possibility to access the spinisotropic point of the Heisenberg model in the weak coupling, U t, regime of the Hubbardmodel.
Let us bosonize Eq. (8.68). We have to introduce four fields φR ↑, φL ↑, φR ↓ and φL ↓.Moreover, to preserve the proper anticommutation relations among the Fermi fields we mustimpose that
[φp(x), φq(y)] = ±iπ for p 6= q p, q = R ↑, L ↑, R ↓, L ↓ .
We write (σ =↑, ↓)
φRσ(x) =1
L
∑q>0
√2π L
q
(eiqxbRσ q + e−iqxb†Rσ q
)+ kF,Rσx− θRσ +ARσ
(8.69)
φLσ(x) =1
L
∑q>0
√2π L
q
(e−iqxbLσ q + eiqxb†Lσ q
)+ kF,Lσx+ θLσ +ALσ
(8.70)
where
AR ↑ ≡π
2(NL ↑ +NR ↓ +NL ↓)
AL ↑ ≡π
2(NR ↑ −NR ↓ −NL ↓)
AR ↓ ≡π
2(−NR ↑ −NL ↑ +NL ↓)
272
AL ↓ ≡π
2(NR ↑ +NL ↓ +NR ↓) .
This choice implies that (remembering the relation[θL/R, σ, NL/R, σ
]= −iπ)
[φR ↑, φL ↑] = [φR ↓, φL ↓] = −iπ[φR ↑, φR ↓] = [φL ↑, φL ↓] = iπ
[φR ↑, φL ↓] = [φL ↑, φR ↓] = −iπ .
The advantage of this choice is that, if we introduce
Φσ ≡1√4π
(φRσ + φLσ) ; Θσ ≡1√4π
(φLσ − φRσ)
we have[Φ↑, Φ↓] = [Φ↑, Θ↓] = [Θ↑, Φ↓] = 0 while [Θ↑, Θ↓] = i .
We introduce the charge and spin combinations
Φc/s =1√2
(Φ↑ ± Φ↓) , Θc/s =1√2
(Θ↑ ±Θ↓) (8.71)
as well as Πc/s = ∂xΘc/s which satisfy, thanks to our choice,
[Φc, Φs] = [Πc, Πs] = [Φc, Πs] = [Πc, Φs] = 0[Φc/s(x), Πc/s(y)
]= iδ(x− y) .
In terms of these fields, the non interacting part of the Hamiltonian can be written as
H0 =vF2
∑p=c,s
∫dx (∂xΦp(x)∂xΦp(x) + Πp(x)Πp(x)) , (8.72)
with vF = 2ta sin kFa. Let us bosonize the interaction
Hint = U∑i
ni ↑ni ↓ =U
L
∑q
ρq ↑ρ−q ↓ .
The term with q ∼ 0 is
Ua
∫dx
(1
2π
)2
(∂xφR ↑(x) + ∂xφL ↑(x))(∂xφR ↓(x) + ∂xφL ↓(x))
=Ua
2π
∫dx (∂xΦc(x)∂xΦc(x)− ∂xΦs(x)∂xΦs(x)) . (8.73)
273
There are two processes at q ∼ 2kF . The first is the so called backscattering
.By bosonization this term reads
2Ua
(2πα)2
∫dx cos(
√8πΦs(x)) . (8.74)
The other process is the Umklapp
which reads
− 2Ua
(2πα)2
∫dx cos
(√8πΦc(x) + 4kFx
). (8.75)
At half filling, kF = π/2a, and 4kFx = 2πx/a can be dropped out of Eq. (8.75) since x/a =integer.
The full bosonized Hamiltonian at half filling is therefore
H =vF2
∑p=c,s
∫dx (∂xΦp(x)∂xΦp(x) + Πp(x)Πp(x))
+gs2π
∫dx∂xΦc(x)∂xΦc(x) +
gs2π
∫dx∂xΦs(x)∂xΦs(x)
+2g1
(2πα)2
∫dx cos(
√8πΦs(x))− 2g3
(2πα)2
∫dx cos(
√8πΦc(x)) ,
(8.76)
where gc ≡ Ua, gs = −Ua, g1 = g3 = Ua. Both charge and spin fields are described by asine-Gordon Hamiltonian with marginal cos(
√8πΦ). Since spin isotropy holds, the sine-Gordon
model in the spin sector has to be equivalent to the Heisenberg model with ∆ = 1, which
274
corresponds in the spinless fermion model to a strong coupling point while in this model to aweak coupling one. The bilinear terms can be diagonalised by a canonical transformation likeEq. (8.44) where
Kc =1√
1 + gcπvF
' 1− gc2πvF
= 1− Ua
2πvF< 1
Ks =1√
1 + gsπvF
' 1− gs2πvF
= 1 +Ua
2πvF> 1 ,
and the velocity of charge/spin collective excitations is
vc/s = vF
√1± Ua
πvF.
Since all interaction terms in Eq. (8.76) are marginal, one can do Renormalization group analysis,which leads to the following equations
gs =1
πvsg2
1, g1 =1
πvsg1gs
gc =1
πvcg2
3, g3 =1
πvcg3gc
whose flux is represented in figure 8.5. From the starting values, one finds that the Umklappis marginally relevant coupling; therefore a charge gap opens and the field Φc is locked to 0mod (2π/
√8π). As expected, the model is insulating for any U > 0.
On the contrary, the backscattering is marginally irrelevant. The fixed point corresponds tog∗s = g∗1 = 0, hence K∗s = 1. Namely, the spin fixed point Hamiltonian reduces simply to
H∗s =vs2
∫dx(∂xΦs(x)∂xΦs(x) + Πs(x)Πs(x))
which describes gapless spin excitations. This analysis gives us the opportunity to derive theexpression of the spin operators directly at the isotropic point.
S+(x) = Ψ†↑(x)Ψ↓(x) = Ψ†R ↑(x)ΨR ↓(x) + Ψ†L ↑(x)ΨL ↓(x) + Ψ†R ↑(x)ΨL ↓(x) + Ψ†L ↑(x)ΨR ↓(x) .
Ψ†R ↑(x)ΨR ↓(x) =1
2παe−iφR ↑eiφR ↓ =
i
2παe−i√
2π(Φs−Θs)
(8.77)
Ψ†R ↑(x)ΨR ↓(x) =i
2παei√
2π(Φs+Θs)
275
Figure 8.5:
(8.78)
Ψ†R ↑(x)ΨL ↓(x) =i
2πα(−1)xe−i
√2πΦcei
√2πΘs .
Since the charge field is locked to Φc = n π√2π
, then
Ψ†R ↑(x)ΨL ↓(x) =i
2πα(−1)xCei
√2πΘs , C ≡
⟨e−i√
2πΦc⟩.
An analogous result holds for Ψ†L ↑ΨR ↓, so that
S†(x) =i
2παe−i√
2π(Φs−Θs) +i
2παei√
2π(Φs+Θs) +i
πα(−1)xCei
√2πΘs .
Finally
Sz(x) =1
2(Ψ†↑Ψ↑ −Ψ†↓Ψ↓)
=1
2(Ψ†R ↑ΨR ↑ + Ψ†L ↑ΨL ↑ −Ψ†R ↓ΨR ↓ −Ψ†L ↓ΨL ↓)
+1
2(Ψ†R ↑ΨL ↑ −Ψ†R ↓ΨL ↓ + H.c.)
276
=1√2π∂xφs +
(−1)x
παcos(√
2πΦc) sin(√
2πΦs)
' 1√2π∂xφs +
(−1)xC
παsin(√
2πΦs) (8.79)
277
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