Strongly Correlated Electron Systems: a DMFT Perspective

Gabriel Kotliar

Physics Department and

Center for Materials Theory

Rutgers University• Brookhaven National Lab February 17th 2005

• Introduction to Dynamical Mean Field Theory (DMFT) ideas.

• Application 1 : Optical Conductivity in Cerium. Mott transition or volume collapse. [K. Haule et. al. PRL 2005]

• Application 2 :The approach to the Mott transition in Kappa Organics and Cuprates. [O. Parcollet G. Biroli and GK PRL ]Work with M. Civelli M. Capone O. Parcollet V. Sarma B. Kyung A.M. Tremblay and D. Senechal]

cluster cluster exterior exteriorH H H H

H clusterH

Simpler "medium" Hamiltonian

cluster exterior exteriorH H

2

4 3

1

A. Georges and G. Kotliar PRB 45, 6479 (1992). G. Kotliar,S. Savrasov, G. Palsson and G. Biroli, PRL 87, 186401 (2001) .

Site Cell. Cellular DMFT. C-DMFT. G. Kotliar,S.. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001)

tˆ(K) hopping expressed in the superlattice notations.

•Other cluster extensions (DCA Jarrell Krishnamurthy, Katsnelson and Lichtenstein periodized scheme, Nested

Cluster Schemes Schiller Ingersent ), causality issues, O. Parcollet, G. Biroli and GK cond-matt 0307587 (2003)

Dynamical Mean Field Theory• Captures the non local Gaussian physics of

model and the local non Gaussian effects. • Assigns to each lattice model a quantum

impurity problem, (local degrees of freedom in a Gaussian bath) which describes its local physics.

• Spectral density functional. Impurity model generates the “exact spectra” of a system.

Good starting point for doing perturbation theory in correlated materials. [ Analogy with Kohn Sham

System].

Testing CDMFT (G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001) ) with two sites in the Hubbard model in one dimension. [V. Kancharla C. Bolech and GK PRB 67, 075110 (2003)][[M.CaponeM.Civelli V Kancharla C.Castellani and GK P. R B 69,195105 (2004)

U/t=4.]

T/W

Phase diagram of a Hubbard model with partial frustration at integer filling. Thinking about the Mott transition in

single site DMFT.

M. Rozenberg et. al. Phys. Rev. Lett. 75, 105 (1995)

Mott transition in layered organic conductors S Lefebvre et al.

Single site DMFT and kappa organics

. ARPES measurements on NiS2-xSex

Matsuura et. Al Phys. Rev B 58 (1998) 3690. Doniaach and Watanabe Phys. Rev. B 57, 3829 (1998) Mo et al., Phys. Rev.Lett. 90, 186403 (2003).

Two paths for calculation of electronic structure of

strongly correlated materials

Correlation Functions Total Energies etc.

Model Hamiltonian

Crystal structure +Atomic positions

DMFT ideas can be used in both cases.

Electronic Structure and EDMFT Phys. Rev. B 62, 12715 (2000) 

1 †1( ) ( , ') ( ') ( ) ( ) ( )

2Cx V x x x i x x xff f y y-+ +òò ò

†( ') ( )G x xy y=- < > ( ') ( ) ( ') ( )x x x x Wff ff< >- < >< >=

Ex. Ir>=|R, > Gloc=G(R, R ’) R,R’’

1 10

1 1[ , , , ] [ ] [ ] [ ] [ ] [ , ]

2 2C hartreeG W M P TrLn G M Tr G TrLn V P Tr P W E G W

Introduce Notion of Local Greens functions, Wloc, Gloc G=Gloc+Gnonloc .

Sum of 2PI graphs[ , ] [ , , 0, 0]EDMFT loc loc nonloc nonlocG W G W G W

One can also view as an approximation to an exact Spectral Density Functional of Gloc and Wloc) Approximations……………...

Convergence in R space of first self energy correction for Si in a.u. (1 a.u.= 27.2 eV) N. Zein

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0 1 2

Sigma s GW

Sigma p GW

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0 1 2

D Sigma s

D Sigma p

Lowest order graph in the screened coulomb interaction (GW approximation) treated self consistently reproduces the gap of silicon. [Exp : 1.17 ev, GW 1.24 ] W. Ku, A. Eguiluz, PRL 89,126401 (2002)

GW self energy Self energy correction beyond GW

Coordination Sphere Coordination Sphere

DMFT Phonons in fcc DMFT Phonons in fcc -Pu-Pu

  C11 (GPa) C44 (GPa) C12 (GPa) C'(GPa)

Theory 34.56 33.03 26.81 3.88

Experiment 36.28 33.59 26.73 4.78

( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003)

(experiments from Wong et.al, Science, 22 August 2003)

Case study Elemental Cerium.

• Study the alpha to gamma transition.

• Test the approach, in a well studied setting.

• Differentiate between the Kondo volume collapse picture and the Mott transition picture.

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 (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

-phase (delocalized:Kondo-

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

Qualitative Ideas.

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

• Mathematical implementation, “itinerant phase” treat spdf electrons by LDA, “localized phase” put f electron in the core.

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

Qualitative Ideas

• “screened moment alpha phase” Kondo effect between spd and f takes place. “unscreend moment gamma 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).

Photoemission&experiment

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

•K. Haule V. Udovenko S. savrasov and GK. (2003)

X.Zhang M. Rozenberg G. Kotliar PRL 70,1666(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!!!!

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 whether a Kondo binding unbinding takes pace (Kondo collapse picture).

Optical Conductivity Temperature dependence.

Origin of the features.

Conclusion• The anomalous temperature dependence and the formation of a pseudogap, suggests

that the Kondo collapse picture is closer to the truth for Cerium.

• Possible experimental verification in Ce(ThLa) alloys.

• Qualitative agreement with experiments, quantitative discrepancies. (see however J.Y. Rhee, X. Wang, B.N. Harmon, and D.W. Lynch, Phys. Rev. B 51, 17390 (1995)) .

Insulatinganion layer

-(ET)2X are across Mott transition

ET =

X-1

[(ET)2]+1conducting ET layer

t’t

modeled to triangular lattice

X- Ground State

U/t t’/t

Cu2(CN)3Mott insulator

8.2 1.06

Cu[N(CN)2]Cl Mott insulator

7.5 0.75

Cu[N(CN)2]Br SC 7.2 0.68

Cu(NCS)2SC 6.8 0.84

Cu(CN)[N(CN)2]

SC 6.8 0.68

Ag(CN)2 H2O SC 6.6 0.60

I3SC 6.5 0.58

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

Evolution of the k resolved Spectral Function at zero frequency. (Kappa organics Parcollet Biroli and GK PRL, 92, 226402. (2004)) )

( 0, )vs k A k

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

U/D=2 U/D=2.25

Mechanism for hot spot formation

Mott transition in cluster

Phenomena Resembles what is seen in cuprate superconductors.

• “Fermi Arcs” M. Norman et. al. Nature Vol 392 (1998).

• Theoretical approaches. M Rice

Umklapp Scattering C.Honerkamp, M.Salmhofer, N.Furukawa, T.M.Rice,%

• cond-mat/9912358. PRB 63, (2001) 035109.

• F. H. L. Essler, A. M. Tsvelik, cond-mat/0409491.

High Temperature Superconductors

RVB phase diagram of the Cuprate Superconductors

• P.W. Anderson. Connection between high Tc and Mott physics. Science 235, 1196 (1987)

• Connection between the anomalous normal state of a doped Mott insulator and high Tc.

• Baskaran Zhou and Anderson Slave boson approach. <b> coherence order parameter.

singlet formation order parameters.

RVB phase diagram of the Cuprate Superconductors. Superexchange.

• The approach to the Mott insulator renormalizes the kinetic energy Trvb increases.

• The proximity to the Mott insulator reduce the charge stiffness , TBE goes to zero.

• Superconducting dome. Pseudogap evolves continously into the superconducting state.

G. Kotliar and J. Liu Phys.Rev. B 38,5412 (1988)

.

• Retains ideas of the slave boson mean field and make many of the results more solid but also removes many difficulties.

• Can treat coherent and incoherent spectra on the same footing.

• Can treat dynamical fluctuations between different singlet order parameters.

• Allows the investigation of broken symmetries. Superconductivity, Antiferromagnetism, etc.

Cellular DMFT and Dynamical RVB

Superconductivity in the Hubbard model role of the Mott transition and influence of the super-

exchange. (M. Capone V. Kancharla. CDMFT+ED, 4+ 8 sites t’=0) . RVB

Superconductivity and Antiferromagnetism t’=0 M. Capone

V. Kancharla (see also CPT Senechal and Tremblay ).

.

• Allows the investigation of the normal state underlying the superconducting state, by forcing a symmetric Weiss function, follow the normal state near the Mott transition.

• Earlier studies (Katsnelson and Lichtenstein, M. Jarrell, M Hettler et. al. Phys. Rev. B 58, 7475 (1998). T. Maier et. al. Phys. Rev. Lett

85, 1524 (2000) ) used QMC as an impurity solver and DCA as cluster scheme.

• We use exact diag ( Krauth Caffarel 1995 with effective temperature 32/t=124/D ) as a solver and Cellular DMFT as the mean field scheme.

CDMFT and Dynamical RVB

Follow the “normal state” with doping. 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

Hole doped case t’=-.3t, U=16 t n=.71 .93 .97

K.M . Shen et. al (2004).

For a review Damascelli et. al. RMP (2003)

Approaching the Mott transition: CDMFT Picture

• Qualitative effect, momentum space differentiation. Formation of hot –cold regions is an unavoidable consequence of the approach to the Mott insulating state!

• D wave gapping of the single particle spectra as the Mott transition is approached.

• Similar scenario was encountered in previous study of the kappa organics. O Parcollet Biroli and Kotliar PRL, 92, 226402. (2004) .

Electron Doped Case t’=.3 t, U=16 tn=.7 ,.93, .96

Approaching the Mott transition: CDMFT picture.

• Qualitative effect, momentum space differentiation. Formation of hot –cold regions is an unavoidable consequence of the approach to the Mott insulating state!

• General phenomena, BUT the location of the cold regions depends on parameters.

• Quasiparticles are now generated from the Mott insulator at (, 0).

• Consistenty with many numerical studies of electron hole asymmetry in t-t’ Hubbard models Tohyama Maewawa Phys. Rev. B 67, 092509 (2003) Senechal and Tremblay. PRL 92 126401 (2004) Kusko et. al. Phys. Rev 66, 140513 (2002)

Consistent with experiments. Armitage et. al. PRL (2001).Momentum dependence of the low-energy Photoemission spectra

of NCCO

To test if the formation of the hot and cold regions is the result of the

proximity to Antiferromagnetism, we studied various values of t’/t, U=16.

Electron doped case t’=.9t U=16tn=.69 .92 .96

Approaching the Mott transition:

• Qualitative effect, momentum space differentiation. Formation of hot –cold regions is an unavoidable consequence of the approach to the Mott insulating state!

• General phenomena, but the location of the cold regions depends on parameters.

• With the present resolution, t’ =.9 and .3 are similar. However it is perfectly possible that at lower energies further refinements and differentiation will result from the proximity to different ordered states.

o Qualitative Difference between the hole doped and the electron doped phase diagram is due to the underlying normal state.” In the hole doped, it has nodal quasiparticles near (,/2) which are ready “to become the superconducting quasiparticles”. Therefore the superconducing state can evolve continuously to the normal state. The superconductivity can appear at very small doping.

o Electron doped case, has in the underlying normal state quasiparticles leave in the ( 0) region, there is no direct road to the superconducting state (or at least the road is tortuous) since the latter has QP at (/2, /2).

Fermi Surface Shape renormalization

• Photoemission determines the renormalized hopping parameters.

• The bare parameters, t and t’ should be independent of doping.

• We find that the renormalized hopping parameters are strongly doping dependent!

• t’eff=t’+• teff=t+Furthermore in the electron doped case t’/t

renormalizes towards perfect nesting, while in the hole doped case it renormalizes away from perfect nesting.

Fermi Surface Shape Renormalization

n= 0.95

U/t=8.0 t’=0.0

U/t=16 ARPES

Evolution of the local spectral function in the superconducting state

Evolution of the spectral function in the superconducting state Anderson and

Ong Cond-matt 0405518

Conjecture, Mott transition with Zcold finite ? Continuity with the

insulator at one point in the zone.

Approaching the Mott transition: CDMFT picture.

• Qualitative effect, momentum space differentiation. Formation of hot –cold regions is an unavoidable consequence of the approach to the Mott insulating state!

• General phenomena, BUT the location of the cold regions depends on parameters.

• Quasiparticles are now generated from the Mott insulator at (, 0).

• Consistenty with many numerical studies of electron hole asymmetry in t-t’ Hubbard models Tohyama Maewawa Phys. Rev. B 67, 092509 (2003) Senechal and Tremblay. PRL 92 126401 (2004) Kusko et. al. Phys. Rev 66, 140513 (2002)

Conclusion• Self consistent QIM models are good reference

frames to describe how the electron evolves from localized to itinerant.

• Allows for a close interaction between theory and experiment.

• Breakup of the Fermi surface is an unavoidable consequence of the approach to the Mott transition.

• Non trivial doping dependence of the Fermi surface of the electron and hole doped cuprates.

Goal of a good mean field theory• Provide a zeroth order picture of a physical phenomena.• Provide a link between a simple system (“mean field

reference frame”) and the physical system of interest.• Formulate the problem in terms of local quantities (which

we can compute better ).• Allows to perform quantitative studies, and predictions .

Focus on the discrepancies between experiments and mean field predictions.

• Generate useful language and concepts. Follow mean field states as a function of parameters.

• Exact in some limit.• Can be made system specific, useful tool for material

exploration and for interacting with experiment.

Collaborators References

• Reviews: A. Georges G. Kotliar W. Krauth and M. Rozenberg RMP68 , 13, (1996).

• Reviews: G. Kotliar S. Savrasov K. Haule V. Oudovenko O. Parcollet and C. Marianetti. Submitted to RMP (2005).

• Gabriel Kotliar and Dieter Vollhardt Physics Today 57,(2004)

CDMFT angle resolved spectran=.96 t’/t=.-.3 U=16 t

• i

Fermi Surface Renormalization CDMFT and expt. Civelli et. al.

Unfortunately the Hubbard model does not capture the trend of supra with t’.

Need augmentation.V. Kancharla

CDMFT and Dynamical RVB n=.96 t’/t=.-.3 U=16 t

• i

Electron doped case t’=.9t U=16tn=.69 .92 .96

Color scale x=.9,.32,.22

Hole doped case t’=-.3t, U=16 t n=.71 .93 .97

Color scale x= .37 .15 .13

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

† †

, ,

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

t c c c c U n n

Energy

Configurational Coordinate in the space of Hamiltonians

T

DMFT and correlated electron materials.

DMFT Cavity Construction. A. Georges and G. Kotliar PRB 45, 6479 (1992). First happy marriage of atomic and band physics.

Reviews: A. Georges G. Kotliar W. Krauth and M. Rozenberg RMP68 , 13, 1996 Gabriel Kotliar and Dieter Vollhardt Physics Today 57,(2004)

= W

= [ - ]-11CV

= G

+ [ - ]KSV10KSG 1G

1 10

1 10

loc

loc

G

W

G M

V P1

1

( )

1

( )

lock

locq C

GH k

Wv q

M

P

0 0G V,,intM PLocal Impurity Model

Input: ,M P

Output: Self-Consistent Solution

Spectral Density Functional Theory withinLocal Dynamical Mean Field Approximation

,loc locG W 1 10

1 10

loc

loc

G

W

G M

V P1

1

( )

1

( )

lock

locq C

GH k

Wv q

M

P

0 0G V,,i n tM PImpurity Solver

Input: ,M P

Output: Self-Consistent Solution

Spectral Density Functional Theory withinLocal Dynamical Mean Field Approximation

Mott transition and d wave superconductivity

Estimates of upper bound for Tc from exact diag. M. Capone. U=16t, t’=0, ( t~.35 ev, Tc ~140)

Problems with the approach.

• Numerous other competing states. Dimer phase, box phase , staggered flux phase . Different decouplings, different answers.

• Neel order• Stability of the pseudogap state at finite temperature.

[Ubbens and Lee] • Missing incoherent spectra . [ fluctuations of slave

bosons ]• Temperature dependence of the penetration depth [Wen

and Lee , Ioffe and Millis ] • Theory:[T]=x-Ta x2 , Exp: [T]= x-T a. • Mean field is too uniform on the Fermi surface, in

contradiction with ARPES.

o Start from functional of G and W (Chitra and Kotliar (2000), Ambladah et. al.

o Make local or cluster approximation on

o FURTHER APPROXIMATIONS:The light, SP (or SPD) electrons are extended, well described by LDA .The heavy, d(or f) electrons are localized treat by DMFT.LDA Kohn Sham Hamiltonian already contains an average interaction of the heavy electrons, subtract this out by shifting the heavy level (double counting term) .

o Truncate the W operator act on the H sector only. i.e.

• Replace W() or V0() by a static U. This quantity can be estimated by a constrained LDA calculation or by a GW calculation with light electrons only. e.g.

M.Springer and F.Aryasetiawan,Phys.Rev.B57,4364(1998) T.Kotani,J.Phys:Condens.Matter12,2413(2000). FAryasetiawan M Imada A Georges G Kotliar S Biermann and A Lichtenstein cond-matt (2004)

( , ', ) ( ') ( ) ( )( ( ) ) ( ')dcxc R H R Rr r r r V r r E rabe a ab bw d f w fS = - - S S -

( , ', ) ( ) ( ) ( ) ( ') ( ')R H R R R RW r r r r W r rabgde a b abgd g dw ff wff=S

1 10

1 1[ , , , ] [ ] [ ] [ ] [ ] [ , ]

2 2C hartreeG W M P TrLn G M Tr G TrLn V P Tr PW E G W

[ , ] [ , , 0, 0]EDMFT loc loc nonloc nonlocG W G W G W

or the U matrix can be adjusted empirically.• At this point, the approximation can be derived from

a functional (Savrasov and Kotliar 2001)

• FURTHER APPROXIMATION, ignore charge self consistency, namely set

LDA+DMFT V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997) See also . A Lichtenstein and M. Katsnelson PRB 57, 6884 (1988).

Reviews:Held, K., I. A. Nekrasov, G. Keller, V. Eyert, N. Blumer, A. K. �McMahan, R. T. Scalettar, T. Pruschke, V. I. Anisimov, and D. Vollhardt, 2003, Psi-k Newsletter #56, 65.

• Lichtenstein, A. I., M. I. Katsnelson, and G. Kotliar, in Electron Correlations and Materials Properties 2, edited by A. Gonis, N. Kioussis, and M. Ciftan (Kluwer Academic, Plenum Publishers, New York), p. 428.

• Georges, A., 2004, Electronic Archive, .lanl.gov, condmat/ 0403123 .

loc[ ]G

[ ] [ ]LDAVxc Vxc

LDA+DMFT Formalism : V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359-7367 (1997). S. Y. Savrasov and G. Kotliar, Phys. Rev. B 69, 245101 (2004). V. Udovenko S. Savrasov K. Haule and G. Kotliar

Cond-mat 0209336

Momentum space differentiation and divergence of Im c at the Mott transition

Notice the similarity in the evolution of the real part of the self energies.

CDMFT vs single site DMFT

RVB states • G. Baskaran Z. Shou and P.W Anderson Solid State

Comm 63, 973 (1987). RVB state with Fermi surface ( 2 d, line of zeros ).

• G. Kotliar Phys. Rev. B37 ,3664 (1998). I Affleck and B. Marston. Phys.Rev. B 37, 3774 (1998). RVB State with four point zeros in 2d. Two states are related by Su(2) symmetry I Affleck Z.Zhou, T. Hsu P.W. Anderson PRB 38,745 (1998).

o G. Kotliar and J. Liu Phys.Rev. B 38,5412 (1988). Doping selects the d –wave superconductor as the most favorable RVB state away from half filling.

o Parallel development of RVG ideas with variational wave functions. C. Gross R. Joynt and T.M.Rice PRB 36, 381 (1987) F. C. Zahng C. Gros T M Rice and H Shiba Supercond. Scie Tech. 1, 36 (1988).

ARPES U/t=8.0 t’=0.0

n= 0.88

n= 0.88

n= 0.95

n= 0.95

Approaching the Mott transition: plaquette CDMFT .

• Qualitative effect, momentum space differentiation. Formation of hot –cold regions is an unavoidable consequence of the approach to the Mott insulating state!

• General phenomena, but the location of the cold regions depends on parameters. Study the “normal state” of the Hubbard model is useful.

• On the hole doped site, slave boson mean field RVB scenario is correct. High Tc superconductivity MAY result follow from doping a Mott insulator phase but it is not necessarily follow from it. One may not be able to connect the Mott insulator to the superconductor if the nodes are in the “wrong place”.

DMFT:Realistic Implementations

• Focus on the “local “ spectral function A() (and of the local screened Coulomb interaction W() ) of the solid.

• Write a functional of the local spectral function such that its stationary point, give the energy of the solid.

• No explicit expression for the exact functional exists, but good approximations are available.

• The spectral function is computed by solving a local impurity model in a medium .Which is a new reference system to think about correlated electrons.

DMFT Cavity Construction. A. Georges and G. Kotliar PRB 45, 6479 (1992). First happy marriage of a technique from atomic physics and a technique

band theory.

Reviews: A. Georges G. Kotliar W. Krauth and M. Rozenberg RMP68 , 13, 1996 Gabriel Kotliar and Dieter Vollhardt Physics Today 57,(2004)

Local Self Energy