Gabriel Kotliar Center for Materials Theory Rutgers University

THE STATE UNIVERSITY OF NEW JERSEY

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The Mott transition in transition metal oxides and in organic materials:a dynamical mean field theory (DMFT) perspective. Part 1+ 2

Gabriel Kotliar

Center for Materials Theory

Rutgers University


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The Mott transition in f electron materials:a dynamical mean field theory (DMFT) perspective. Part 3.

Gabriel Kotliar


Rutgers University


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Outline Some comments on the role of DMFT in solid state

physics problems and the strong correlation problem. (Part I)

Introduction to DMFT: cavity construction E-DMFT and cluster methods. (Part I)

Introduction to DMFT: functional method. (Part III) Interfaces with electronic structure. DMFT as a first

principles method. first principles approach GW-U method. LDA+DMFT. (Part III)


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Outline

The Mott transition problem. (Part I) Predictions of single site DMFT, and

experimental verification. Phase Diagram, Optics, Photoemission, Transport. (Part I)

Conclusions of Part I. System specific studies of materials.

LDA+DMFT. Some case studies.


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Outline SYSTEM SPECIFIC STUDIES. The Mott transition in kappa organics. A

CDMFT study. (Part II) The metal to insulator transition in Ti2O3.A

CMDFT study. (Part II) Itinerant Magnetism in Fe and Ni. (Part II). Conclusions of Part II.


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Outline The Mott transition in f electron systems. The role of

the spd band. The coupling to the structure, volume collapse transitions. (Part III)

Case study: alpha-gamma transition in Cerium. Photoemission and Optical Spectroscopy. (Part III)

Case study: the phases of plutonium, photoemission, total energy and lattice vibrations. (Part III)

Case study: Americium under pressure. (Part III).


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Weakly correlated electrons:band theory. Simple conceptual picture of the ground

state, excitation spectra, transport properties of many systems (simple metals, semiconductors,….).

A methods for performing quantitative calculations. (Density functional theory, in various approximations+ perturbation theory in the Coulomb interactions).


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Momentum Space , bands, k in Brillouin zone is good quantum number.

Landau Fermi liquid theory interactions renormalize away at low energy, simple band picture in effective field holds.

2 ( )F Fe k k l

h

The electron in a solid: wave picture

Maximum metallic resistivity 200 ohm cm


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Standard Model of Solids Qualitative predictions: low temperature dependence of

thermodynamics and transport.

Optical response, transition between the bands. Qualitative predictions: filled bands give rise to

insulting behavior. Compounds with odd number of electrons are metals.

Quantitative tools: Density Functional Theory with approximations suggested by the Kohn Sham formulation, (LDA GGA) is a successful computational tool for the total energy. Good starting point for perturbative calculation of spectra,eg. GW. Kinetic equations yield transport coefficients.

~H constR~ const S T ~VC T


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Kohn Sham reference system

2 / 2 ( ) KS kj kj kjV r y e y- Ñ + =

( ')( )[ ( )] ( ) ' [ ]

| ' | ( )

LDAxc

KS ext

ErV r r V r dr

r r r

drr r

dr= + +

-ò

2( ) ( ) | ( ) |kj

kj kjr f rr e y=å

Excellent starting point for computation of spectra in perturbation theory in screened Coulomb interaction GW.

Success story : Density Functional Linear Success story : Density Functional Linear ResponseResponse

Tremendous progress in ab initio modelling of lattice dynamics& electron-phonon interactions has been achieved(Review: Baroni et.al, Rev. Mod. Phys, 73, 515, 2001)

(Savrasov, PRB 1996)


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LDA+GW: semiconducting gaps


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The electron in a solid: particle picture.Ba

Array of hydrogen atoms is insulating if a>>aB.

Mott: correlations localize the electron

e_ e_ e_ e_

Superexchange

Ba

Think in real space , solid collection of atoms

High T : local moments, Low T spin-orbital order ,RVB.

1

T


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Mott : Correlations localize the electron

Low densities, electron behaves as a particle,use atomic physics, real space

One particle excitations: Hubbard Atoms: sharp excitation lines corresponding to adding or removing electrons. In solids they broaden by their incoherent motion, Hubbard bands (eg. bandsNiO, CoO MnO….)H H H+ H H H motion of H+ forms the lower Hubbard band

H H H H- H H motion of H_ forms the upper Hubbard band

Quantitative calculations of Hubbard bands and exchange constants, LDA+ U, Hartree Fock. Atomic Physics.


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Localization vs Delocalization Strong Correlation Problem

•A large number of compounds with electrons in partially filled shells, are not close to the well understood limits (localized or itinerant). Non perturbative problem.•These systems display anomalous behavior (departure from the standard model of solids).•Neither LDA or LDA+U or Hartree Fock work well.•Dynamical Mean Field Theory: Simplest approach to electronic structure, which interpolates correctly between atoms and bands. Treats QP bands and Hubbard bands.


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Strong correlation anomalies

Metals with resistivities which exceed the Mott Ioffe Reggel limit.

Transfer of spectral weight which is non local in frequency.

Dramatic failure of DFT based approximations (say DFT-GW) in predicting physical properties.


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Failure of the standard model : Anomalous Resistivity:LiV2O4

Takagi et.al. PRL 2000


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Failure of the StandardModel: Anomalous Spectral Weight Transfer

Optical Conductivity Schlesinger et.al (1993)

0( )d

Neff depends on T


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The competition of kinetic energy and Coulomb interactions, is a central issue that needs to be resolved.

One needs a tool that treats quasiparticle bands and Hubbard bands on the same footing to contain the band and atomic limit.

The approach should allow to incorporate material specific information.

When the neither the band or the atomic description applies, a new reference point for thinking about correlated electrons is needed.

DMFT!


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Outline Some comments on the role of DMFT in

solid state physics problems and the strong correlation problem.

Introduction to DMFT: cavity construction E-DMFT and cluster methods.

Introduction to DMFT: functional method. Interfaces with electronic structure. A truly

first principles approach GW-U method. LDA+DMFT.


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DMFT


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1

10

1( ) ( )

( )n nn k nk

G i ii t i

w ww m w

-

-é ùê ú= +Sê ú- + - Sê úë ûå

Single site DMFT Impurity cavity construction: A. Georges, G. Kotliar, PRB, (1992)]

†

0 0 0

[ ] ( )[ ( , ')] ( ')o o o oS Go c Go c U n nb b b

s st t t t ¯= +òò ò

† †

, ,

( )( )ij ij i j j i i ii j i

t c c c c U n n

0

†( )( ) ( ) ( )L n o n o n S GG i c i c iw w w=- á ñ

10 ( ) ( )n n nG i i iw w m w- = + - D

0

1 † 10 0 ( )( )[ ] ( ) [ ( ) ( ) ]n n n n S Gi G G i c i c ia bw w w w- -S = + á ñ

Weiss field


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Extension to clusters. Cellular DMFT. C-DMFT. G. Kotliar,S.Y. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001)

tˆ(K) is the hopping expressed in the superlattice notations.

•Other cluster extensions (DCA, nested cluster schemes, PCMDFT ), causality issues, O. Parcollet, G. Biroli and GK

cond-matt 0307587 (2003)


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More general DMFT loop

( )k LMTOt H k E® - LMTO

LL LH

HL HH

H HH

H H

é ùê ú=ê úë û

ki i Ow w®1

0 n HHiG i Ow e- = + - D

0 0

0 HH

é ùê úS =ê úSë û

0 0

0 HH

é ùê úD =ê úDë û

0

1 †0 0 ( )( )[ ] ( ) [ ( ) ( )HH n n n n S Gi G G i c i c ia bw w w w-S = + á ñ

110

1( ) ( )

( ) ( ) HH

LMTO HH

n nn k nk

G i ii O H k E i

w ww w

--é ùê ú= +Sê ú- - - Sê úë ûå

† † †0

0 0

( ) ( , ') ( ') a ab b abdc a b c dc G c U c c c cb b

t t t t +òò


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Mean-Field : Classical vs Quantum

Classical case Quantum case

Phys. Rev. B 45, 6497 A. Georges, G. Kotliar (1992)

†

0 0 0

( )[ ( ')] ( ')o o o oc c U n nb b b

s st m t t tt ¯

¶+ - D - +

¶òò ò

( )wD

†( )( ) ( )

MFo n o n SG c i c is sw w D=- á ñ

1( )

1( )

( )[ ][ ]

nk

n kn

G ii

G i

ww e

w

=D - -

D

å

,ij i j i

i j i

J S S h S- -å å

MF eff oH h S=-

effh

0 0 ( )MF effH hm S=á ñ

eff ij jj

h J m h= +å

† †

, ,


t c c c c U n n


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Solving the DMFT equations

G 0 G

I m p u r i t yS o l v e r

S . C .C .

•Wide variety of computational tools (QMC,ED….)Analytical Methods•Extension to ordered states. Review: A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)]

G0 G

Im puritySo lver

S .C .C .


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Methods of solution : some examples

Iterative perturbation theory. A Georges and G Kotliar PRB 45, 6479 (1992). H Kajueter and G. Kotliar PRL (1996). Interpolative schemes (Oudovenko et.al.)Exact diag schemes Rozenberg et. al. PRL 72, 2761 (1994)Krauth and Caffarel. PRL 72, 1545 (1994)Projective method G Moeller et. al. PRL 74 2082 (1995). NRG R. Bulla PRL 83, 136 (1999)


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QMC M. Jarrell, PRL 69 (1992) 168, Rozenberg Zhang Kotliar PRL 69, 1236 (1992) ,A Georges and W Krauth PRL 69, 1240 (1992) M. Rozenberg PRB 55, 4855 (1987).

NCA Prushke et. al. (1993) . SUNCA K. Haule (2003).

Analytic approaches, slave bosons. Analytic treatment near special points.


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How good is DMFT ?


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Single site DMFT is exact in the Limit of large lattice coordination

1~ d ij nearest neighborsijt

d

† 1~i jc c

d

†

,

1 1~ ~ (1)ij i j

j

t c c d Od d

~O(1)i i

Un n

Metzner Vollhardt, 891

( , )( )k

G k ii i

Muller-Hartmann 89


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C-DMFT: test in one dimension. (Bolech, Kancharla GK PRB 2003)

Gap vs U, Exact solution Lieb and Wu, Ovshinikov

Nc=2 CDMFT

vs Nc=1


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N vs mu in one dimensional Hubbard model .Compare 2 site cluster (in exact diag with Nb=8) vs exact Bethe Anzats, [M. Capone C. Castellani M.Civelli and GK (2003)]


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Comments on DMFT.

Review of DMFT, technical tools for solving DMFT eqs. A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)]

CDMFT , instead of studying finite systems with open or periodic boundary conditions, study a system in a medium. Connection with DMRG, infer the density matrix by using a Gaussian anzats, and the periodicity of the system.


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1

10

1( ) ( )

( )n nn k nk

G i ii t i

w ww m w

-

-é ùê ú= +Sê ú- + - Sê úë ûå

DMFT Impurity cavity construction

1

10

1( ) ( )

V ( )n nk nk

D i ii

w ww

-

-é ùê ú= +Pê ú- Pê úë ûå

0

1 † 10 0 ( )( )[ ] ( ) [ ( ) ( ) ]n n n n S Gi G G i c i c ia bw w w w- -S = + á ñ

†

0 0

( ) ( , ') ( ') ( , ') o o o oc Go c n n Ub b

s st t t t d t t ¯+òò

† †

, ,


t c c c c U n n

()

1 100 0 0( )[ ] ( ) [ ( ) ( ) ]n n n n Si G D i n i n iw w w w- -P = + á ñ

,ij i j

i j

V n n

0 0( , ')Do n nt t+


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Outline

The Mott transition problem. Predictions of DMFT, and experimental

verification. Phase Diagram, Optics, Photoemission, Transport.

System specific studies of materials. LDA+DMFT. Some case studies.


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The Mott transition problem

Universal and non universal aspects. Frustration and the success of DMFT. In the

phases without long range order, DMFT is valid if T > Jeff. Need frustration to supress it. When T < Jeff LRO sets in. If Tneel is to high it oblitarates the Mott phenomena.

t vs U fundamental competition and secondary instabilities.


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V2O3 under pressure or


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NiSe2-xSx


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Mott transition in layered organic conductors S Lefebvre et al. cond-mat/0004455, Phys. Rev. Lett. 85, 5420 (2000)


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Single site DMFT and expt.

Single site DMFT study of the Mott transition, based on a study of the Hubbard model on frustrated lattices made several interesting qualitative predictions.

New experiments and reexamination of old ones give credence to that the local picture is quite good.


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Pressure Driven Mott transition


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Insights from DMFT Low temperature Ordered phases . Stability depends on chemistry and crystal structureHigh temperature behavior around Mott endpoint, more universal regime, captured by simple models treated within DMFT. Role of magnetic frustration.


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Schematic DMFT phase diagram of a partially frustrated integered filled Hubbard model.


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Qualitative single site DMFT predictions. Spectra of the strongly correlated metallic

regime contains both quasiparticle-like and Hubbard band-like features.

Mott transition is drive by transfer of spectral weight.


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X.Zhang M. Rozenberg G. Kotliar (PRL 1993)

Spectral Evolution at T=0 half filling full frustration


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Parallel development: Fujimori et.al


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Evolution of the Spectral Function with Temperature

Anomalous transfer of spectral weight connected to the proximity to the Ising Mott endpoint (Kotliar Lange nd Rozenberg Phys. Rev. Lett. 84, 5180 (2000)


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Qualitative single site DMFT predictions: Optics Spectra of the strongly correlated metallic

regime contains both quasiparticle-like and Hubbard band-like features.

Mott transition is drive by transfer of spectral weight. Consequences for optics.


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Schematic DMFT phase diagram Hubbard model (partial frustration). Evidence for QP peak in V2O3 from optics.

M. Rozenberg G. Kotliar H. Kajueter G Thomas D. Rapkine J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995)


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Anomalous transfer of spectral weight in v2O3


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Anomalous transfer of optical spectral weight V2O3

:M Rozenberg G. Kotliar and H. Kajuter Phys. Rev. B 54, 8452 (1996).

M. Rozenberg G. Kotliar H. Kajueter G Tahomas D. Rapkikne J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995)


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Anomalous transfer of optical spectral weight, NiSeS. [Miyasaka and Takagi 2000]


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Anomalous Spectral Weight Transfer: Optics

0( ) ,eff effd P J

iV

Schlesinger et.al (FeSi) PRL 71 ,1748 , (1993) B Bucher et.al. Ce2Bi4Pt3PRL 72, 522 (1994), Rozenberg et.al. PRB 54, 8452, (1996).

2

0( ) ,

ned P J

iV m

ApreciableT dependence found.

, ,H hamiltonian J electric current P polarization

, ,eff eff effH J PBelow energy


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Searching for a quasiparticle peak


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. ARPES measurements on NiS2-xSex

Matsuura et. Al Phys. Rev B 58 (1998) 3690. Doniaach and Watanabe Phys. Rev. B 57, 3829 (1998)


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1998); S.-K. Mo etal., Phys. Rev. Lett. 90,

186403 (2003).16. K. Maiti et al., Europhys.

Lett. 55, 246 (2001); A. Sekiyama

et al., http://arXiv.org/abs/cond-mat/0312429.

17. P. Limelette et al., Science 302, 89 (2003); F. Kagawa et al.,

http://arXiv.org/abs/cond-mat/0307304.


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S.-K. Mo et al., Phys. Rev. Lett. 90, 186403 (2003)..


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Schematic DMFT phase Implications for transport.


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Anomalous Resistivity and Mott transition Ni Se2-x Sx

Crossover from Fermi liquid to bad metal to semiconductor to paramagnetic insulator.


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Ising endpoint finally found

P. Limelette et al., Science 302, 89 (2003) F. Kagawa et al.,http://arXiv.org/abs/cond-

mat/0307304


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Evolution of the Spectral Function with Temperature

Anomalous transfer of spectral weight connected to the proximity to the Ising Mott endpoint (Kotliar Lange nd Rozenberg Phys. Rev. Lett. 84, 5180 (2000)


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V2-xCrx O3


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Ising critical endpoint in (VCr)2O3 Limlette et. al. Science 302, 89 (2003)


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Conclusion G. Kotliar Science 89 (2003)

An electronic model accounts for all the qualitative features of the finite temperature of a frustrated system at integer occupancy.

The observation of the spinodal lines and the wide classical critical region where mean field holds indicate the coupling to the lattice is quantitatively very important.


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Conclusion

Single site DMFT describes the main features of the experiments at high temperatures using a simple model.

Made non trivial predictions. Finite temperature conclusions are robust.

At low temperatures clusters will bring refinements of this picture.

System specific studies including electron phonon coupling needed to obtain quantitative estimates of say T_MIT.


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The Mott transition in transition metal oxides and in organic materials:a dynamical mean field theory (DMFT) perspective. Part 1+ 2

Gabriel Kotliar


Rutgers University


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Schematic Single Site DMFT phase diagram.


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Single site DMFT has provides a first glimpse of what happens to the electronic structure as a function of correlation strength, and temperature.

Neglects the effects of magnetic exchange J on the quasiparticle properties in the paramagnetic phase. To take these into account we need to go beyond single site DMFT.

Test and refine the single site DMFT picture.


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O. Parcollet G. Biroli and G. Kotliar (kappa organics) cond-mat 0308577

A. Poteryaev A. Lichtenstein and G. Kotliar (Ti2O3 ) cond-matt 0311319

Fe and Ni. A. Lichtenstein M. Katsnelson and G. Kotliar. Phys Rev. Lett 87, 67205 , (2001).

Cerium. K. Haule et. al. unpublished.


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Two roads for ab-initio calculation of electronic structure of strongly correlated materials

Correlation Functions Total Energies etc.

Model Hamiltonian

Crystal structure +Atomic positions


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Mott transition in layered organic conductors S Lefebvre et al. cond-mat/0004455, Phys. Rev. Lett. 85, 5420 (2000)


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organics

ET = BEDT-TTF=Bisethylene dithio tetrathiafulvalene

organics = (ET)2 X

Increasing pressure ----- increasing t’ ------------

X0 X1 X2 X3 (Cu)(2CN)3 Cu(NCN)2 Cl Cu(NCN2)2Br Cu(NCS)2 Spin liquid Mott transition


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Theoretical issue: is there a Mott transition

in the integer filled Hubbard model, and is it

well described by the single site DMFT ?


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Controversy on the unfrustrated case. Comment on "Absence of a Slater Transition in the Two-Dimensional Hubbard Model"

B. Kyung, J.S. Landry, D. Poulin, A.-M.S. Tremblay Phys. Rev. Lett. 90, 099702-1 (2003)


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Double Occupancy vs U

CDMFT Parcollet, Biroli GK (2003)

Study frustrated t t’ model t’/t=.9


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Compare with single site results Rozenberg Chitra Kotliar PRL 2002


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Mott transition in cluster (QMC)


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Evolution of the spectral function at low frequency.

( 0, )vs k A k

If the k dependence of the self energy is weak, we expect to see contour lines corresponding to Ek = const and a height increasing as we approach the Fermi surface.

k

k2 2

k

Ek=t(k)+Re ( , 0)

= Im ( , 0)

( , 0)Ek

k

k

A k


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Evolution of the k resolved Spectral Function at zero frequency. (Parcollet Biroli and GK)

Uc=2.35+-.05, Tc/D=1/44

U/D=2 U/D=2.25

( 0, )vs k A k


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Near the transition k dependence is strong. Qualitative effect, formation of hot regions! D wave gapping of the single particle

spectra as the Mott transition is approached.

Square symmetry is restored as we approched the insulator.


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Deviations from single site DMFT


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Lattice and cluster self energies


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Mechanism for hot spot formation


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Conclusion.

Mott transition survives in the cluster setting. Role of magnetic frustration.

Surprising result: formation of hot and cold regions as a result of an approach to the Mott transition. General result ?

Unexpected role of the next nearest neighbor self energy.

CDMFT a new window to extend DMFT to lower temperatures.


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Outline SYSTEM SPECIFIC STUDIES. The Mott transition in kappa organics. A

CDMFT study. The metal to insulator transition in Ti2O3.A

CMDFT study. Itinerant Magnetism in Fe and Ni. Cerium.


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Metal to insulator transition in Ti2O3

Isostructural to V2O3. All the qualitative physics of the high temperature part of the phase diagram of V2O3 can be understood within single site DMFT. Computations with a realistic density of states, and multiorbital impurity model (Mo et. al. ) quantitative improvement.

Is the same thing true in Ti2O3?


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Ti2O3 V2O3 : Resistivities


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Ti2O3 Structure


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Relevant Orbitals: Goodenough picture


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Physical parameters U= 2ev. Nearest neighbor Coulomb repulsions,

V, W .5 =.5 ev.


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Ti2O3 vs V2O3

As a function of temperature, there is no magnetic transition in Ti2O3, unlike V2O3

As a function of temperature, there is no structural change, unlike V2O3 which becomes monoclinic at low temperatures.

In V2O3 the distance between the Vanadium pairs increases as the temperature decreases. In Ti2O3 the distance between the Vanadium pairs decreases as one lowers the temperature.

LTS 250 K, HTS 750 K.


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Earlier work.

Band Structure Calculations always produce a good metal. L.F. Mattheiss, J. Phys.: Condens. Matter 8, 5987 (1996)

Unrestricted Hartree Fock calculations produce large antiferromagnetic gap. M. Cati, G. Sandrone, and R. Dovesi, Phys. Rev. B. f55 , 16122 (1997).


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Ti2O3 LDA-DOS

LTS HTS


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Methodology:1 and 2 site CDMFT

Impurity solver. Multiband QMC. Derivation of the effective Hamiltonian.

Massive downfolding with O Andersen’s new Nth order LMTOS. Coulomb interactions estimated using dielectric constant W=.5 ev. U on titanium 2 ev. J= .2 ev.


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Single site DMFT LTS


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Two-site CDMFT for

beta=20, and beta=10

(T=500,1000)

Poteryaev Lichtenstein

and GK


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Single site DMFT vs CDMFT


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Important role played by the Coulomb nn repulsion.


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Physical Origin of the renormalization of the hopping. Generalized Anderson impurity model.

A Coulomb repulsion between the local orbital and the bath of conduction electrons

Renormalizes down the hybridization between the local orbital and the conduction electron.

D. Haldane PRL 40, 416 (1978) and Cambridge Unive. Ph.D thesis.

Q. Si and G. Kotliar PRL 70, 3143 (1993).


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CDMFT

Single site DMFT, Basic Unit single site. Titanium Oxide Basic Unit: Titanium pair. Kappa Organics, Basic Unit 4 site plaquette. Minimal reference frames.


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Case study Fe and Ni

Archetypical itinerant ferromagnets LSDA predicts correct low T moment Band picture holds at low T Main puzzle: at high temperatures has a

Curie Weiss law with a moment larger than the ordered moment.


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Iron and Nickel: crossover to a real space picture at high T (Lichtenstein, Katsnelson and Kotliar Phys Rev. Lett 87, 67205 , 2001)


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Photoemission Spectra and Spin Autocorrelation: Fe (U=2, J=.9ev,T/Tc=.8) (Lichtenstein, Katsenelson,Kotliar Phys Rev. Lett 87, 67205 , 2001)


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Photoemission and T/Tc=.8 Spin Autocorrelation: Ni (U=3, J=.9 ev)


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Iron and Nickel:magnetic properties (Lichtenstein, Katsenelson,GK PRL 01)

0 3( )q

Meff

T Tc


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Ni and Fe: theory vs exp eff high T moment

Fe 3.1 (theory) 3.12 (expt)

Ni 1.5 (theory) 1.62 (expt)

Curie Temperature Tc

Fe 1900 ( theory) 1043(expt) Ni 700 (theory) 631 (expt)


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Fe and Ni Consistent picture of Fe (more localized) and Ni

(more correlated) Satellite in minority band at 6 ev, 30 % reduction

of bandwidth, exchange splitting reduction .3 ev Spin wave stiffness controls the effects of spatial

flucuations, it is about twice as large in Ni and in Fe

Needed cluster extensions to improve on the critical temperature, nested cluster schemes and causality. Biroli Parcollet and Kotliar cond-mat 0307587.


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Case study Cerium.

Study the alpha to gamma transition.

Test accuracy of the approach, in a well study setting. Differentiate between the Kondo volume collapse picture and the Mott transition picture.


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Overview

Various phases :

isostructural phase transition (T=298K, P=0.7GPa)

(fcc) phase

[ magnetic moment

(Curie-Wiess law) ]

(fcc) phase

[ loss of magnetic

moment (Pauli-para) ]

with large

volume collapse

v/v 15

( -phase a 5.16 Å

-phase a 4.8 Å)

volumes exp. LDA LDA+U

28Å3 24.7Å3

34.4Å3 35.2Å3

-phase (localized): High T phaseCurie-Weiss law (localized magnetic moment),Large lattice constantTk around 60-80K


-phase (delocalized:Kondo-

physics): Low T phaseLoss of Magnetism (Fermi liquid Pauli susceptibility) - completely screened magnetic momentsmaller lattice constantTk around 1000-2000K




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Qualitative Ideas.

Johanssen, Mott transition of the f electrons as a function of pressure. Ce alpha gamma transition. spd electrons are spectators.

Mathematical implementation, “metallic phase” treat spdf electrons by LDA, “insulating phase” put f electron in the core.

Allen and Martin. Kondo volume collapse picture. The dominant effect is the spd-f hybridization.


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Qualitative Ideas

“metallic alpha phase” Kondo effect between spd and f takes place. “insulating phase” no Kondo effect (low Kondo temperature).

Mathematical implementation, Anderson impurity model in the Kondo limit suplemented with elastic terms. (precursor of DMFT ideas, but without self consistency condition).


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Photoemission&experiment

•A. Mc Mahan K Held and R. Scalettar (2002)

•K. Haule V. Udovenko and GK. (2003)


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X.Zhang M. Rozenberg G. Kotliar (PRL 1993) A. Georges, G. Kotliar (1992) Phys. Rev. B 45, 6497

Unfortunately photoemission cannot decide between the Kondo collapse picture and the Mott transition picture.Evolution of the spectra as a function of U , half filling full frustration, Hubbard model!!!!


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Resolution: Turn to Optics!

Qualitative idea. The spd electrons have much larger velocities, so optics will be much more senstive to their behavior.

See if they are simple spectators (Mott transition picture ) or wether a Kondo binding unbinding takes pace (Kondo collapse picture).


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Optical conductivity of Ce (expt. Van Der Eb et.al. theory Haule et.al)

experiment

LDA+DMFT •K. Haule V. Udovenko and G Kotliar (2003)


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Origin of the features.


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Conclusion

The anomalous temperature dependence

and the formation of a pseudogap, suggests that the Kondo collapse picture is closer to the truth for Cerium.

Qualitative agreement with experiments, quantitative discrepancies.


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Conclusions.

Kappa Organics. Ti2O3. Fe-Ni Ce ………… Localization - Delocalization as a central theme in

the electronic structure of solids. Next step: Part III. Self consistent determination of

lattice and electrons using DMFT.


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Conclusion ( Part II).C-DMFT studies of the Mott transition reveal the

development of a gap with a d wave symmmetry as the transition is approached.

Plaquette as a minimal reference frame.

Bond as a reference frame, and important role of the Coulomb interactions to trigger the MIT in Ti2O3.

Need cluster treatments of itinerant magnets to obtain accurate critical temperatures. (Nested Cluster Schemes).


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Haule et. al.


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La1-xSrx O3

Adding holes to a Mott insulator in three dimensions.

For very small doping,(x<.07) interesting spin and orbital order takes place, non universal physics and lattice distortions are important. Small energy scales, larger dopings more robust universal behavior.

Canonical Brinkman Rice behavior, good system to test ab-initio DMFT.


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(Tokura et. Al. 1993)A doped Mott insulator:LaxSr1-xO3


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DMFT calculation U near the Mott transition, Rozenberg et.al 94


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Hall Coefficient, electron like.


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La1-xSrxTiO3 photoemission


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Evolution of spectra with doping U=4


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Optical spectral weight


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Optical conductivity


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Realistic Computation of Optical Properties : La1-xSrxTiO3


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The Mott transition in f electron materials:a dynamical mean field theory (DMFT) perspective. Part 3.

Gabriel Kotliar


Rutgers University


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Outline The Mott transition in f electron systems.

Differences with d electron systems. Functional approach to DMFT and

integration with electronic structure methods. Case study: alpha-gamma transition in

Cerium. Photoemission and Optical Spectroscopy.

Case study: the phases of plutonium, photoemission, total energy and lattice vibrations.

Case study: Americium under pressure.


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Mott transition in f electron systems. Volume collapse in Cerium, Plutonium and

Americium. Strong coupling of the electronic structure to

the lattice. Presence of light spd electrons near the

Fermi level. Heavy +light electrons.


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Overview

Various phases :


(fcc) phase

[ magnetic moment

(Curie-Wiess law) ]

(fcc) phase

[ loss of magnetic


with large

volume collapse

v/v 15

( -phase a 5.16 Å

-phase a 4.8 Å)


28Å3 24.7Å3

34.4Å3 35.2Å3








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Phases of Pu (A. Lawson LANL) Los Alamos Science 26, (2000)


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Case study in f electrons, Mott transition in the actinide series. B. Johanssen 1974 Smith and Kmetko Phase Diagram 1984.


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Americium under pressure (Lindbaum et. al. PRB 2003)


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Qualitative Ideas.





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Qualitative Ideas




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Schematic DMFT phase diagram one band Hubbard model. Introduce coupling to the lattice will cause a volume jump across the first order transition. (Majumdar and Krishnamurthy ).


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Cerium


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Qualitative phase diagram in the U, T , plane (two band Kotliar Murthy Rozenberg PRL (2002).

Coexistence regions between localized and delocalized spectral functions.

k diverges at generic Mott endpoints


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Minimum in melting curve and divergence of the compressibility at the Mott endpoint

Vsol

Vliq

mdT V

dP S

é ùDê ú=ê úDë û


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Cerium


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Case study in f electrons, Mott transition in the actinide series. B. Johanssen 1974 Smith and Kmetko Phase Diagram 1984.


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Generalized phase diagram

T

U/WStructure, bands,

orbitals


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Outline Some comments on the role of DMFT in

solid state physics problems and the strong correlation problem.

Introduction to DMFT: cavity construction E-DMFT and cluster methods.

Introduction to DMFT: functional method. Interfaces with electronic structure. A truly

first principles approach GW-U method. LDA+DMFT.


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Model Hamiltonian



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A more general perspective on DMFT.

DMFT as an exact theory. (Chitra and Kotliar PRB 2001 Savrasov and GK cond-matt 2003) analogy to DFT.

DMFT as an approximation (Chitra and Kotliar PRB2002)

DMFT as a new starting point for perturbative expansions. ( P. Sun and G.K PRB 2002)


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DMFT as an exact theory , analogy with DFT Start with TOE


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DFT: effective action construction(Fukuda et.al. ) Chitra and GK

( )( )

Wr

j r


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Exchange and correlation energy Exact formal expressions can be given in

terms of a coupling constant integration.[Harris-Jones, adiabatic connection]

DFT is useful because practical accurate expressions for Exc, exist.

LDA, GGA, hybrids,


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Kohn Sham reference system

2 / 2 ( ) KS kj kj kjV r y e y- Ñ + =

( ')( )[ ( )] ( ) ' [ ]

| ' | ( )xc

KS ext

ErV r r V r dr

r r r

drr r

dr= + +

-ò

2( ) ( ) | ( ) |kj

kj kjr f rr e y=å


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Kohn Sham Greens function is an good point to compute spectra in perturbation theory in screenedCoulomb interaction GW,G0W0

Practical implementations, introduce a finite basis set.

Division into valence (active ) degrees of freedom and core.


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DMFT

Functional derivation.


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DMFT Model Hamiltonian.

Exact functional of the

local Greens function A

+


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DMFT for model Hamiltonians. Kohn Sham formulation.

ij ii ijd=S S

[ , ] log[ ] ( ) ( )

[ ]

ijn n n

xc ii

A Tr i t Tr i A i

A

w w w-GS =- - S - S

+F

Introduce auxiliary field

1( )

( )ii n

xck

n k nii

A ii t i

A

wd

w wd

é ùê úê ú= ê úFê ú- -ê úë û

åExact “local self energy”


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About XC functional. One can derive a coupling constant integration

formulae (Harris Jones formula) for

Generate approximations.

The exact formalism generates the local Greens function and ii is NOT the self energy. However one can use the approach as starting point for computing other quantities.

[ ]xc iiAF

[ ]xcDMFT atom ii

i

AF = Få


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Mapping onto impurity models.

The local Greens function A, and the auxilliary quantity can be computed from the solution of an impurity model that describes the effects of the rest of the lattice on the on a selected central site.


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Comments on functional construction

Atoms as a reference point. Expansion in t. Locality does not necessarily mean a single point.

Extension to clusters. Jii --- Jii Ji i+ Aii --- Ai i+ ii --- i i+ Exact functionalAii ,Ai i+ he lattice self energy and other non local

quantities extending beyond the cluster are OUTSIDE the formalism and need to be inferred.


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Comments on funct. construction.

Construction of approximations in the cluster case requires care to maintain causality.

One good approximate construction is the cellular DMFT: a) take a supercell of the desired range,b)

[ ]xcCDMFT scells

scells

AF = Få

c) obtain estimate of the lattice self energy by restoring translational symmetry. Many other cluster approximations (eg. DCA, the use of lattice self energy in

self consitency condition, restrictions of BK functional, etc. exist). Causality and classical limit of these methods has recently been clarified [ G Biroli O Parcollet and GK]


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Lattice and cluster self energies


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LDA+DMFT

The light, SP (or SPD) electrons are extended, well described by LDA

The heavy, D (or F) electrons are localized,treat by DMFT.

LDA already contains an average interaction of the heavy electrons, substract this out by shifting the heavy level (double counting term)

The U matrix can be estimated from first principles of viewed as parameters


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Spectral Density Functional

The exact functional can be built in perturbation theory in the interaction (well defined diagrammatic rules )The functional can also be constructed from the atomic limit, but no explicit expression exists.

DFT is useful because good approximations to the exact density functional DFT(r)] exist, e.g. LDA, GGA

A useful approximation to the exact functional can be constructed, the DMFT +LDA functional.


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Spectral Density Functional : effective action construction (Fukuda, Valiev and Fernando , Chitra and GK).

DFT, consider the exact free energy as a functional of an external potential. Express the free energy as a functional of the density by Legendre transformation. DFT(r)]

Introduce local orbitals, R(r-R)orbitals, and local GF G(R,R)(i ) =

The exact free energy can be expressed as a functional of the local Greens function and of the density by introducing sources for (r) and G and performing a Legendre transformation, (r),G(R,R)(i)]

' ( )* ( , ')( ) ( ')R Rdr dr r G r r i r


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LDA+DMFT functional2 *log[ / 2 ( ) ( )]

( ) ( ) ( ) ( )

1 ( ) ( ')( ) ( ) ' [ ]

2 | ' |

[ ]

R R

n

n KS

KS n n

i

LDAext xc

DC

R

Tr i V r r

V r r dr Tr i G i

r rV r r dr drdr E

r r

G

a b ba

w

w c c

r w w

r rr r

- +Ñ - - S -

- S +

+ + +-

F - F

åò

ò òå

Sum of local 2PI graphs with local U matrix and local G

1[ ] ( 1)

2DC G Un nF = - ( )0( ) iab

abi

n T G i ew

w+

= å

KS ab [ ( ) G V ( ) ]LDA DMFT a br r


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Comments on LDA+DMFT

• Static limit of the LDA+DMFT functional , with = HF reduces to LDA+U

• Removes inconsistencies of this approach,• Only in the orbitally ordered Hartree Fock

limit, the Greens function of the heavy electrons is fully coherent

• Gives the local spectra and the total energy simultaneously, treating QP and H bands on the same footing.


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LDA+DMFT Self-Consistency loop

G0 G

Im puritySolver

S .C .C .

0( ) ( , , ) i

i

r T G r r i e w

w

r w+

= å

2| ( ) | ( )k xc k LMTOV H ka ac r c- Ñ + =

DMFT

U

E

0( , , )HHi

HH

i

n T G r r i e w

w

w+

= å


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Realistic DMFT loop

( )k LMTOt H k E® -LMTO

LL LH

HL HH

H HH

H H

é ùê ú=ê úë û

ki i Ow w®

10 niG i Ow e- = + - D

0 0

0 HH

é ùê úS =ê úSë û

0 0

0 HH

é ùê úD =ê úDë û

0

1 †0 0 ( )( )[ ] ( ) [ ( ) ( )HH n n n n S Gi G G i c i c ia bw w w w-S = + á ñ

110

1( ) ( )

( ) ( ) HH

LMTO HH

n nn k nk

G i ii O H k E i

w ww w



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Realistic implementation of the self consistency condition

110

1( ) ( )

( ) ( ) HH

LMTO HH

n nn k nk

G i ii O H k E i

w ww w


•H and , do not commute•Need to do k sum for each frequency •DMFT implementation of Lambin Vigneron tetrahedron integration (Poteryaev et.al 1987)


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LDA+DMFT References

V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359-7367 (1997).

A Lichtenstein and M. Katsenelson Phys. Rev. B 57, 6884 (1988).

S. Savrasov and G.Kotliar, funcional formulation for full self consistent implementation. Savasov Kotliar and Abrahams . Application to delta Pu Nature (2001)


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Case study Cerium.

Study the alpha to gamma transition.

Test accuracy of the approach, in a well study setting. Differentiate between the Kondo volume collapse picture and the Mott transition picture.


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Overview

Various phases :


(fcc) phase

[ magnetic moment

(Curie-Wiess law) ]

(fcc) phase

[ loss of magnetic


with large

volume collapse

v/v 15

( -phase a 5.16 Å

-phase a 4.8 Å)


28Å3 24.7Å3

34.4Å3 35.2Å3








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Qualitative Ideas.





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Qualitative Ideas




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Photoemission&experiment

•A. Mc Mahan K Held and R. Scalettar (2002)

•K. Haule V. Udovenko and GK. (2003)


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X.Zhang M. Rozenberg G. Kotliar (PRL 1993) A. Georges, G. Kotliar (1992) Phys. Rev. B 45, 6497

Unfortunately photoemission cannot decide between the Kondo collapse picture and the Mott transition picture.Evolution of the spectra as a function of U , half filling full frustration, Hubbard model!!!!


RUTGERS

Resolution: Turn to Optics!

Qualitative idea. The spd electrons have much larger velocities, so optics will be much more senstive to their behavior.

See if they are simple spectators (Mott transition picture ) or wether a Kondo binding unbinding takes pace (Kondo collapse picture).


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Optical conductivity of Ce (expt. Van Der Eb et.al. theory Haule et.al)

experiment

LDA+DMFT •K. Haule V. Udovenko and G Kotliar (2003)


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Haule et. al.


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Origin of the features.


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Conclusion

The anomalous temperature dependence

and the formation of a pseudogap, suggests that the Kondo collapse picture is closer to the truth for Cerium.

Qualitative agreement with experiments, quantitative discrepancies.


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DMFT studies of elemental Plutonium


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What is the dominant atomic configuration? Local moment?

Snapshots of the f electron Dominant configuration:(5f)5

Naïve view Lz=-3,-2,-1,0,1 ML=-5 B S=5/2 Ms=5 B Mtot=0 L=5, S=5/2, J=5/2, Mtot=Ms=B gJ =.7 B Crystal fields

GGA+U estimate ML=-3.9 Mtot=1.1 (Savrasov GK 2000) This bit is quenches by the f and spd electrons Neutron Scattering in a field (Lander)


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Pu: DMFT total energy vs Volume (Savrasov Kotliar and Abrahams 2001)


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Double well structure and Pu Qualitative explanation

of negative thermal expansion

Sensitivity to impurities which easily raise the energy of the -like minimum.


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Photoemission Technique

Density of states for removing (adding ) a particle to the sample.

Delocalized picture, it should resemble the density of states, (perhaps with some satellites).

Localized picture. Two peaks at the ionization

and affinity energy of the atom.


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Lda vs Exp Spectra


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Pu Spectra DMFT(Savrasov) EXP (Arko Joyce Morales Wills Jashley PRB 62, 1773 (2000)


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Alpha and delta Pu


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Alpha phase is also a correlated metal. It differs from delta in the relative weight of

the resonance and the Hubbard band. Consistent with resistivity and specific heat

measurements.


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Phonon Spectra

Electrons are the glue that hold the atoms together. Vibration spectra (phonons) probe the electronic structure.

Phonon spectra reveals instablities, via soft modes.

Phonon spectrum of Pu had not been measured.

Short distance behavior of the elastic moduli.


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Phonon freq (THz) vs q in delta Pu X. Dai et. al. Science vol 300, 953, 2003


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Inelastic X Ray. Phonon energy 10 mev, photon energy 10 Kev.

E = Ei - EfQ =ki - kf


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Expt. Wong et. al.


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Expts’ Wong et. al. Science 301. 1078 (2003) Theory Dai et. al. Science 300, 953, (2003)


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Shear anisotropy. Expt. vs Theory

C’=(C11-C12)/2 = 4.78 GPa C’=3.9 GPa

C44= 33.59 GPa C44=33.0 GPa

C44/C’ ~ 7 Largest shear anisotropy in any element!

C44/C’ ~ 8.4


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The delta –epsilon transition

The high temperature phase, (epsilon) is body centered cubic, and has a smaller volume than the (fcc) delta phase.

What drives this phase transition?

Having a functional, that computes total energies opens the way to the computation of phonon frequencies in correlated materials (S. Savrasov and G. Kotliar 2002)


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Phases of Pu (A. Lawson LANL) Los Alamos Science 26, (2000)


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Epsilon Plutonium.


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Phonon frequency (Thz ) vs q in epsilon Pu.


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Phonon entropy drives the epsilon delta phase transition

Epsilon is slightly more delocalized than delta, has SMALLER volume and lies at HIGHER energy than delta at T=0. But it has a much larger phonon entropy than delta.

At the phase transition the volume shrinks but the phonon entropy increases.

Estimates of the phase transition following Drumont and Ackland et. al. PRB.65, 184104 (2002); (and neglecting electronic entropy). TC ~ 600 K.


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Apply Realistic DMFT to the study of Am metal. S Murthy and GK Fully self consistent LDA+DMFT

calculations. Use Hubbard 1 as impurity solver. Ingnore multiple splittings but take large

spin orbit coupling Su(6)XSu(8) symmetry.


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LDA+DMFT calculations for fcc Americium S. Murthy and G. K(2003)


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S. Murthy Rutgers Ph.D ThesisP vs V for fcc Am


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Am photoemission spectra. Expt (Negele ) DMFT Theory (S. Murthy)


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Recent experimental studies of Am. J.C Griveaux et. al. ITU Non trivial evolution of the electronic

structure with pressure. Superconductivity near the Mott transition ?


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Griveaux et. al.


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Griveaux et. al.


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Griveaux et. al.


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Conclusion Developing DMFT to face the challenges of

a first principles theory of correlated materials is a very exciting project.

There are many difficulties to surmount, but

there is clear evidence that we are making significant progress. The goal is in sight, and we are getting exciting results along the way. Hope you join us!


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Model Hamiltonian



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Start with the TOE


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Rewrite the TOE as an electron boson problem.

1 †1( ) ( , ') ( ') ( ) ( ) ( )

2Cx V x x x i x x xff f y y-+ +òò ò


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Build effective action for the local greens functions of the fermion and Bose field

r=R+ R unit cell vector position within the unit cell. Ir>=|R, Couple sources to

† ( ) ( ') R Ry r y r( ') ( )R Rf r f r( )Rf r


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Legendre transfor the sources, eliminating the field

Build exact functional of the correlation functionsW(r R,r’ R)

and G (r R,r’ R)

( ') ( ) ( ') ( )R R R R Wf r f r f r f r< >- < >< >= †( ') ( ')G R Ry r y r=- < >


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“Kohn Sham “ decomposition.

[ ] [ , ]HE xc G Wr y+ +


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(E)DMFT pproximation to [ , ]xc G Wy

Sum over all LOCAL 2PI graphs (integrations are restricted over the unit cell ) built with W and G

Map into impurity model to generate G and W

Go beyond this approximation by returning to many body theory and adding the first non local correction.


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Test on extended Hubbard model V/U=.25, P Sun and GK


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EDMFT functional.


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Returning to many body physics.


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Take the solution of the EDMFT equations as an approximation for the TRUE local self energy, and add the leading NON LOCAL corrections to the self energy G_NL W_NL, as a correction.

Do it self consistently and as a one shot iteration G0_NL W0_NL and compare the results.

Documents

Gabriel Kotliar Center for Materials Theory Rutgers University