Electronic Structure of Strongly Correlated Materials:a Dynamical Mean Field Theory

(DMFT) approach

Gabriel Kotliar

Physics Department and

Center for Materials Theory

Rutgers University•University of Washington Seattle May 10th 2005

Outline• Introduction to strongly correlated

electrons and Dynamical Mean Field Theory (DMFT).

• The Mott transition problem. Theory and experiments.

• More realistic calculations. Pu the Mott transition across the actinide series.

• Conclusions . Current developments and future directions.

C. Urano et. al. PRL 85, 1052 (2000)

Strong Correlation Anomalies cannot be understood within the standard model of solids, based on a RIGID BAND PICTURE,e.g.“Metallic “resistivities that rise without sign of saturation beyond the Mott limit, temperature dependence of the integrated optical weight up to high frequency

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.

Model Hamiltonians: Hubbard model

† †

, ,

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

t c c c c U n n

U/t

Doping or chemical potential

Frustration (t’/t)

T temperature

Limit of large lattice coordination1

~ d ij nearest neighborsijtd

† 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, 89

1( , )

( )k

G k ii i

Muller-Hartmann 89

Mean-Field Classical vs Quantum

Classical case Quantum case

A. Georges, G. Kotliar Phys. Rev. B 45, 6497(1992)Review: G. Kotliar and D. Vollhardt Physics Today 57,(2004)

†

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 t

G i

ww m

w

D =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

10G-

DMFT as an approximation to the Baym Kadanoff functional

[ , , 0, 0, ]

[ ] [ ] [ ]

DMFT

atomij ij i ii ii i ii

Gii ii Gij ij i j

TrLn i t ii Tr G G

[ , ] [ ] [ ] [ ]ij ijG TrLn i t Tr G G

[ ] Sum 2PI graphs with G lines andU G vertices

cluster cluster exterior exteriorH H H H

H clusterH

Simpler "medium" Hamiltonian

cluster exterior exteriorH H

Medium of free electrons :

impurity model.

Solve for the medium using

Self Consistency

G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001)

Other cluster extensions (DCA Jarrell Krishnamurthy, M Hettler et. al. Phys. Rev. B 58, 7475 (1998)Katsnelson and Lichtenstein periodized scheme. Causality issues O. Parcollet, G. Biroli and GK Phys. Rev. B 69, 205108 (2004)

U/t=4.

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.Capone M.Civelli V Kancharla C.Castellani and GK PR B 69,195105 (2004) ]

One Particle Spectral Function and Angle

Integrated Photoemission

• Probability of removing an electron and transfering energy =Ei-Ef, and momentum k

f() A() M2

• Probability of absorbing an electron and transfering energy =Ei-Ef, and momentum k

(1-f()) A() M2

• Theory. Compute one particle greens function and use spectral function.

e

e

1( , ) Im[ ( , )] Im[ ]

( , )k

A k G kk

Photoemission and the Theory of Electronic Structure

Limiting case itinerant electrons( ) ( )kk

A

( ) ( , )k

A A k

( ) ( ) ( )B AA Limiting case localized electrons

Hubbard bands

Local Spectral Function

A BU

Pressure Driven Mott transition

How does the electron go from the localized to

the itinerant limit ?

T/W

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

DMFT. High temperature universality

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

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)

V2O3:Anomalous transfer of spectral weight

Th. Pruschke and D. L. Cox and M. Jarrell, Europhysics Lett. , 21 (1993), 593

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

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

2000]

Anomalous Resistivity and Mott transition Ni Se2-x Sx

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

Single site DMFT and kappa organics

Ising critical endpoint! In V2O3

P. Limelette et.al. Science 302, 89 (2003)

. 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).

Conclusions.

• Three peak structure, quasiparticles and Hubbard bands.

• Non local transfer of spectral weight.• Large metallic resistivities.• The Mott transition is driven by transfer of

spectral weight from low to high energy as we approach the localized phase.

• Coherent and incoherence crossover. Real and momentum space.

• Theory and experiments begin to agree on a broad picture.

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.

• The combination of realistic band theory and many body physics, is a very broad subject.

• Having a practical and tractable non perturbative method for solving many body Hamiltonians, the next step is to bring more realistic descriptions of the materials Orbital degeneracy and realistic band structure.

• LDA+DMFT V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997).

• The light, sp (or spd) electrons are extended, well described by LDA .The heavy, d (or f) electrons are localized treat by DMFT. Use Khon Sham Hamiltonian after substracting average energy already contained in LDA.

• Add to the substracted Kohn Sham Hamiltonian a frequency dependent self energy, from DMFT.

• Determine the density self consistently.(Chitra, Kotliar, PRB 2001, Savrasov, Kotliar, Abrahams, Nature 2001).

LDA+DMFT Self-Consistency loop

G0 G

Im p u rityS o lver

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

Edc

0( , , )HHi

HH

i

n T G r r i e w

w

w+

= å

Functional formulation. Chitra and Kotliar Phys. Rev. B 62, 12715 (2000)

and Phys. Rev.B (2001) . 

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 Spetral Density Functional of Gloc and Wloc.

Next Step: GW+EDMFT S. Savrasov and GK.(2001). P.Sun and GK. (2002). S.

Biermann F. Aersetiwan and A.Georges . (2002). P Sun and G.K (2003)

Implementation in the context of a model Hamiltonian with short range interactions.P Sun and G. Kotliar cond-matt 0312303 or with a static U on heavy electrons, without self consistency. Biermann et.al. PRL 90,086402 (2003)

W

W

Actinies , role of Pu in the periodic table

Pu phases: A. Lawson Los Alamos Science 26, (2000)

LDA underestimates the volume of fcc Pu by 30%.

Within LDA fcc Pu has a negative shear modulus.

LSDA predicts Pu to be magnetic with a 5 b moment. Experimentally it is not.

Treating f electrons as core overestimates the volume by 30 %

Small amounts of Ga stabilize the phase (A. Lawson LANL)

Pu is not MAGNETIC, alpha and delta have comparable

susceptibility and specifi heat.

Total­Energy­as­a­function­of­volume­for­Total­Energy­as­a­function­of­volume­for­Pu­Pu­W (ev) vs (a.u. 27.2 ev)

(Savrasov, Kotliar, Abrahams, Nature ( 2001)Non magnetic correlated state of fcc Pu.

iw

Zein Savrasov and Kotliar (2004)

Double well structure and Pu Qualitative explanation of negative thermal expansion[ G. Kotliar J.Low

Temp. Physvol.126, 1009 27. (2002)]See also A . Lawson et.al.Phil. Mag. B 82, 1837 ]

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.

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

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

E = Ei - EfQ =ki - kf

DMFT­­Phonons­in­fcc­DMFT­­Phonons­in­fcc­-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)

J. Tobin et. al. PHYSICAL REVIEW B 68, 155109 ,2003

K. Haule , Pu- photoemission with DMFT using vertex corrected NCA.

Dynamical Mean Field View of Pu(Savrasov Kotliar and Abrahams, Nature 2001)

• Delta and Alpha Pu are both strongly correlated, the DMFT mean field free energy has a double well structure, for the same value of U. One where the f electron is a bit more localized (delta) than in the other (alpha).

• Is the natural consequence of earlier studies of the Mott transition phase diagram once electronic structure is about to vary.

• Pu strongly correlated element, at the brink of a Mott instability.

• Realistic implementations of DMFT : total energy, photoemission spectra and phonon dispersions of delta Pu.

• Clues to understanding other Pu anomalies.

Outline• Introduction to strongly correlated electrons.• Introduction to Dynamical Mean Field Theory

(DMFT)• The Mott transition problem. Theory and

experiments.• More realistic calculations. Pu the Mott

transition across the actinide series. • Conclusions . Current developments and future

directions.

Conclusion• DMFT. Electronic Structure Method under

development. Local Approach. Cluster extensions.

• Quantitative results , connection between electronic structure, scales and bonding.

• Qualitative understanding by linking real materials to impurity models. Concepts to think about correlated materials.

• Closely tied to experiments. System specific. Many materials to be studied, realistic matrix elements for each spectroscopy. Optics.……

Some 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)

DMFT : What is the dominant atomic configuration ,what is the fate of the atomic 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

• More realistic calculations, (GGA+U),itineracy, crystal fields ML=-3.9 Mtot=1.1. S. Y. Savrasov and G. Kotliar, Phys. Rev. Lett., 84, 3670 (2000)

• This moment is quenched or screened by spd electrons, and other f electrons. (e.g. alpha Ce).

Contrast Am:(5f)6

Anomalous Resistivity

PRL 91,061401 (2003)

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?

• LDA+DMFT functional computes total energies opens the way to the computation of phonon frequencies in correlated materials (S. Savrasov and G. Kotliar 2002). Combine linear response and DMFT.

Epsilon Plutonium.

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 G. Ackland et. al. PRB.65, 184104 (2002); (and neglecting electronic entropy). TC ~ 600 K.

Further Approximations. o 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() 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 w ff=S

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 . A­Lichtenstein­and­M.­Katsnelson­PRB­57,­6884­(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 Self-Consistency loop

G0 G

Im p u rityS o lver

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

Edc

0( , , )HHi

HH

i

n T G r r i e w

w

w+

= å

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

--é ùê ú= +Sê ú- - - Sê úë ûå

kj il ijklU Udd ®

LDA+DMFT functional

2 *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 KS ab [ ( ) ( ) G V ( ) ( ) ]LDA DMFT a br m r r B r

Mott transition into an open (right) and closed (left) shell systems. AmAt room pressure a localised 5f6 system;j=5/2.

S = -L = 3: J = 0 apply pressure ?

S S

U U

TLog[2J+1]

Uc

~1/(Uc-U)

S=0

???

Americium under pressureAmericium under pressure

Density­functional­based­electronic­structure­calculations: Non­magnetic­LDA/GGA­predicts­volume­50%­off.­ Magnetic­GGA­corrects­most­of­error­in­volume­but­gives­m~6B

(Soderlind et.al., PRB 2000). Experimentally,­Am­has non­magnetic­f6­ground­state­with­J=0­(7F0)

Experimental­Equation­of­State­(after Heathman et.al, PRL 2000)

Mott Transition?“Soft”

“Hard”

Mott transition in open (right) and closed (left) shell systems.

S S

U U

TLog[2J+1]

Uc

~1/(Uc-U)

J=0

???

Tc

Am under pressure: J.C. GriveauJ. Rebizant G. Lander and G. Kotliar PRL (2005)

J. C. Griveau et. al. (2004)

Am Equation of State: LDA+DMFT PredictionsAm Equation of State: LDA+DMFT Predictions

LDA+DMFT predictions: Non­magnetic­f6­ground­

state­with­J=0­(7F0) Equilibrium­Volume:­­­­­­Vtheory/Vexp=0.93

Bulk­Modulus:­Btheory=47­GPa

­­­­­Experimentally­B=40-45­GPa

Theoretical­P(V)­using­LDA+DMFT­

Self-consistent­evaluations­of­total­energies­with­LDA+DMFT­using­matrix­Hubbard­I­method.

Accounting­for­full­atomic­multiplet­structure­using­Slater­integrals:F(0)=4.5 eV, F(2)=8 eV, F(4)=5.4 eV, F(6)=4 eVNew­algorithms­allow­studies­of­complex­structures.­

Predictions­for­Am­II

Predictions­for­Am­IV

Predictions­for­Am­III

Predictions­for­Am­I

Photoemission Spectrum from Photoemission Spectrum from 77FF00 Americium Americium

LDA+DMFT­Density­of­States

Experimental­Photoemission­Spectrum(after J. Naegele et.al, PRL 1984)

Matrix­Hubbard­I­Method

F(0)=4.5 eV F(2)=8.0 eVF(4)=5.4 eV F(6)=4.0 eV

Atomic Multiplets in AmericiumAtomic Multiplets in AmericiumLDA+DMFT­Density­of­States

Exact­Diag.­for­atomic­shell

F(0)=4.5 eV F(2)=8.0 eV F(4)=5.4 eV F(6)=4.0 eV

Matrix­Hubbard­I­Method

F(0)=4.5 eV F(2)=8.0 eVF(4)=5.4 eV F(6)=4.0 eV

Anomalous Resistivity

PRL 91,061401 (2003)

The Mott Transiton across the Actinides Series.

Pu phases: A. Lawson Los Alamos Science 26, (2000)

LDA underestimates the volume of fcc Pu by 30%.

Within LDA fcc Pu has a negative shear modulus.

LSDA predicts Pu to be magnetic with a 5 b moment. Experimentally it is not.

Treating f electrons as core overestimates the volume by 30 %

Small amounts of Ga stabilize the phase (A. Lawson LANL)

Pu is not MAGNETIC, alpha and delta have comparable

susceptibility and specifi heat.

Total­Energy­as­a­function­of­volume­for­Total­Energy­as­a­function­of­volume­for­Pu­Pu­W (ev) vs (a.u. 27.2 ev)

(Savrasov, Kotliar, Abrahams, Nature ( 2001)Non magnetic correlated state of fcc Pu.

iw

Zein Savrasov and Kotliar (2004)

Double well structure and Pu Qualitative explanation of negative thermal expansion[ G. Kotliar J.Low

Temp. Physvol.126, 1009 27. (2002)]See also A . Lawson et.al.Phil. Mag. B 82, 1837 ]

Double well structure is immediate consequence of having two solutions to the DMFT equations and allowing the relaxation of the volume. The itinerant

solution expands the localized solution contracts .

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.

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

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

E = Ei - EfQ =ki - kf

DMFT­­Phonons­in­fcc­DMFT­­Phonons­in­fcc­-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)

J. Tobin et. al. PHYSICAL REVIEW B 68, 155109 ,2003

K. Haule , Pu- photoemission with DMFT using vertex corrected NCA.

Dynamical Mean Field View of Pu(Savrasov Kotliar and Abrahams, Nature 2001)

• Delta and Alpha Pu are both strongly correlated, the DMFT mean field free energy has a double well structure, for the same value of U. One where the f electron is a bit more localized (delta) than in the other (alpha).

• Is the natural consequence of earlier studies of the Mott transition phase diagram once electronic structure is about to vary.

• Pu strongly correlated element, at the brink of a Mott instability.

• Realistic implementations of DMFT : total energy, photoemission spectra and phonon dispersions of delta Pu.

• Clues to understanding other Pu anomalies.

Outline• Introduction to strongly correlated electrons.• Introduction to Dynamical Mean Field Theory

(DMFT)• The Mott transition problem. Theory and

experiments.• More realistic calculations. Pu the Mott

transition across the actinide series. • Conclusions . Current developments and future

directions.

Conclusion• DMFT. Electronic Structure Method under

development. Local Approach. Cluster extensions.

• Quantitative results , connection between electronic structure, scales and bonding.

• Qualitative understanding by linking real materials to impurity models. Concepts to think about correlated materials.

• Closely tied to experiments. System specific. Many materials to be studied, realistic matrix elements for each spectroscopy. Optics.……

Some 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)

DMFT : What is the dominant atomic configuration ,what is the fate of the atomic 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

• More realistic calculations, (GGA+U),itineracy, crystal fields ML=-3.9 Mtot=1.1. S. Y. Savrasov and G. Kotliar, Phys. Rev. Lett., 84, 3670 (2000)

• This moment is quenched or screened by spd electrons, and other f electrons. (e.g. alpha Ce).

Contrast Am:(5f)6

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?

• LDA+DMFT functional computes total energies opens the way to the computation of phonon frequencies in correlated materials (S. Savrasov and G. Kotliar 2002). Combine linear response and DMFT.

Epsilon Plutonium.

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 G. Ackland et. al. PRB.65, 184104 (2002); (and neglecting electronic entropy). TC ~ 600 K.

Further Approximations. o 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() 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 w ff=S

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 . A­Lichtenstein­and­M.­Katsnelson­PRB­57,­6884­(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 Self-Consistency loop

G0 G

Im p u rityS o lver

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

Edc

0( , , )HHi

HH

i

n T G r r i e w

w

w+

= å

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

--é ùê ú= +Sê ú- - - Sê úë ûå

kj il ijklU Udd ®

LDA+DMFT functional

2 *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 KS ab [ ( ) ( ) G V ( ) ( ) ]LDA DMFT a br m r r B r

Mott transition into an open (right) and closed (left) shell systems. AmAt room pressure a localised 5f6 system;j=5/2.

S = -L = 3: J = 0 apply pressure ?

S S

U U

TLog[2J+1]

Uc

~1/(Uc-U)

S=0

???

Americium under pressureAmericium under pressure

Density­functional­based­electronic­structure­calculations: Non­magnetic­LDA/GGA­predicts­volume­50%­off.­ Magnetic­GGA­corrects­most­of­error­in­volume­but­gives­m~6B

(Soderlind et.al., PRB 2000). Experimentally,­Am­has non­magnetic­f6­ground­state­with­J=0­(7F0)

Experimental­Equation­of­State­(after Heathman et.al, PRL 2000)

Mott Transition?“Soft”

“Hard”

Mott transition in open (right) and closed (left) shell systems.

S S

U U

TLog[2J+1]

Uc

~1/(Uc-U)

J=0

???

Tc

Am under pressure: J.C. GriveauJ. Rebizant G. Lander and G. Kotliar PRL (2005)

J. C. Griveau et. al. (2004)

Am Equation of State: LDA+DMFT PredictionsAm Equation of State: LDA+DMFT Predictions

LDA+DMFT predictions: Non­magnetic­f6­ground­

state­with­J=0­(7F0) Equilibrium­Volume:­­­­­­Vtheory/Vexp=0.93

Bulk­Modulus:­Btheory=47­GPa

­­­­­Experimentally­B=40-45­GPa

Theoretical­P(V)­using­LDA+DMFT­

Self-consistent­evaluations­of­total­energies­with­LDA+DMFT­using­matrix­Hubbard­I­method.

Accounting­for­full­atomic­multiplet­structure­using­Slater­integrals:F(0)=4.5 eV, F(2)=8 eV, F(4)=5.4 eV, F(6)=4 eVNew­algorithms­allow­studies­of­complex­structures.­

Predictions­for­Am­II

Predictions­for­Am­IV

Predictions­for­Am­III

Predictions­for­Am­I

Photoemission Spectrum from Photoemission Spectrum from 77FF00 Americium Americium

LDA+DMFT­Density­of­States

Experimental­Photoemission­Spectrum(after J. Naegele et.al, PRL 1984)

Matrix­Hubbard­I­Method

F(0)=4.5 eV F(2)=8.0 eVF(4)=5.4 eV F(6)=4.0 eV

Atomic Multiplets in AmericiumAtomic Multiplets in AmericiumLDA+DMFT­Density­of­States

Exact­Diag.­for­atomic­shell

F(0)=4.5 eV F(2)=8.0 eV F(4)=5.4 eV F(6)=4.0 eV

Matrix­Hubbard­I­Method

F(0)=4.5 eV F(2)=8.0 eVF(4)=5.4 eV F(6)=4.0 eV

Pu in the periodic table

actinides

Small amounts of Ga stabilize the phase (A. Lawson LANL)

Total­Energy­as­a­function­of­volume­for­Total­Energy­as­a­function­of­volume­for­Pu­Pu­W (ev) vs (a.u. 27.2 ev)

(Savrasov, Kotliar, Abrahams, Nature ( 2001)Non magnetic correlated state of fcc Pu.

iw

Zein Savrasov and Kotliar (2004)

DMFT­­Phonons­in­fcc­DMFT­­Phonons­in­fcc­-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)

Mott transition into an open (right) and closed (left) shell systems. In single site DMFT, superconductivity must intervene

before reaching the Mott insulating state.[Capone et. al. ] AmAt room pressure a localised 5f6 system;j=5/2. S = -L = 3: J

= 0 apply pressure ?

S S

U U

TLog[2J+1]

Uc

~1/(Uc-U)

S=0

???

H.Q. Yuan et. al. CeCu2(Si2-x Gex). Am under pressure Griveau et. al.

Superconductivity due to valence fluctuations ?

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)

Epilogue, the search for a quasiparticle peak and its demise,

photoemission, transport. Confirmation of the DMFT

predictions ARPES measurements on NiS2-xSex

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

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

Limelette et. al. [Science] G. Kotliar [Science].

. ARPES measurements on NiS2-xSex

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

One Particle Local Spectral Function and

Angle Integrated Photoemission

• Probability of removing an electron and transfering energy =Ei-Ef,

f() A() M2

• Probability of absorbing an electron and transfering energy =Ei-Ef,

(1-f()) A() M2

• Theory. Compute one particle greens function and use spectral function.

e

e

Dynamical Mean Field Theory

• Focus on the local spectral function A() 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. Which is a new reference system to think about correlated electrons.

• Ref: A. Georges G. Kotliar W. Krauth M. Rozenberg. Rev Mod

Phys 68,1 (1996) . Generalizations to realistic electronic structure. (G. Kotliar and S. Savrasov in )

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

[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.

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.

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, M

Hettler et. al. Phys. Rev. B 58, 7475 (1998)Katsnelson and Lichtenstein periodized scheme, Nested Cluster Schemes , causality

issues, O. Parcollet, G. Biroli and GK cond-matt 0307587 .

Searching for a quasiparticle peak

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)

. ARPES measurements on NiS2-xSex

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

QP in V2O3 was recently found Mo et.al

organics• ET = BEDT-TTF=Bisethylene dithio tetrathiafulvalene

(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

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

Expt. Wong et. al.

Two paths for ab-initio 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.

Failure of the standard model : Anomalous Resistivity:LiV2O4

Takagi et.al. PRL 2000

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) .

Mott Transition in ActinidesMott Transition in Actinides

This­regime­is­not­well­described­by­traditional­techniques­of­electronic­structure­techniques­and­require­new­methods­which­take­into­account­the­itinerant­and­the­localized­character­of­the­electron­on­the­same­footing.

after G. Lander, Science (2003).

The­f­electrons­in­Plutonium­­are­close­to­a­localization-delocalization­transition­(Johansson, 1974) .

after J. Lashley et.al, cond-mat (2005).

Mott Transition

Resistivity in AmericiumResistivity in Americium

Resistivity­behavior(after Griveau et.al, PRL 2005)

Superconductivity

•­Under­pressure,­resistivity­of­Am­raises­almost­an­order­of­magnitude­and­reaches­its­value­of­500­m*cm

•­Superconductivity­in­Am­is­observed­with­Tc­~­0.5K

Photoemission in Am, Pu, SmPhotoemission in Am, Pu, Sm

after J. R. Naegele, Phys. Rev. Lett. (1984).

Atomic multiplet structure­emerges­from­measured­­photoemission­spectra­in­Am (5f6),­Sm(4f6)­-­­Signature­for­f­electrons­localization.

Am Equation of State: LDA+DMFT PredictionsAm Equation of State: LDA+DMFT Predictions

LDA+DMFT predictions: Non­magnetic­f6­ground­

state­with­J=0­(7F0) Equilibrium­Volume:­­­­­­Vtheory/Vexp=0.93

Bulk­Modulus:­Btheory=47­GPa

­­­­­Experimentally­B=40-45­GPa

Theoretical­P(V)­using­LDA+DMFT­

Self-consistent­evaluations­of­total­energies­with­LDA+DMFT­using­matrix­Hubbard­I­method.

Accounting­for­full­atomic­multiplet­structure­using­Slater­integrals:F(0)=4.5 eV, F(2)=8 eV, F(4)=5.4 eV, F(6)=4 eVNew­algorithms­allow­studies­of­complex­structures.­

Predictions­for­Am­II

Predictions­for­Am­IV

Predictions­for­Am­III

Predictions­for­Am­I

Photoemission Spectrum from Photoemission Spectrum from 77FF00 Americium Americium

LDA+DMFT­Density­of­States

Experimental­Photoemission­Spectrum(after J. Naegele et.al, PRL 1984)

Matrix­Hubbard­I­Method

F(0)=4.5 eV F(2)=8.0 eVF(4)=5.4 eV F(6)=4.0 eV

Atomic Multiplets in AmericiumAtomic Multiplets in AmericiumLDA+DMFT­Density­of­States

Exact­Diag.­for­atomic­shell

F(0)=4.5 eV F(2)=8.0 eV F(4)=5.4 eV F(6)=4.0 eV

Matrix­Hubbard­I­Method

F(0)=4.5 eV F(2)=8.0 eVF(4)=5.4 eV F(6)=4.0 eV

Alpha and delta Pu

Failure of the StandardModel: Anomalous Spectral Weight Transfer

Optical Conductivity o of FeSi for T=20,40, 200 and 250 K from Schlesinger et.al (1993)

0( )d

Neff depends on T

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

† †

, ,

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

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+

A. C. Lawson et. al. LA UR 04-6008

F(T,V)=Fphonons+Finvar

=125 K =.5 = 1400 KD

Invar model A. C. Lawson et. al. LA UR 04-6008

Small amounts of Ga stabilize the phase (A. Lawson LANL)

Breakdown of standard model

• Large metallic resistivities exceeding the Mott limit.• Breakdown of the rigid band picture.• Anomalous transfer of spectral weight in photoemission

and optics. • LDA+GW loses its predictive power.

• Need new reference frame, to think about and compute the properties of correlated materials.

• Need new starting point to do perturbation theory.

Limit of large lattice coordination1

~ d ij nearest neighborsijtd

† 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, 89

1( , )

( )k

G k ii i

Muller-Hartmann 89

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

1

T

T/W

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

DMFT. High temperature universality

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

Band Theory: electrons as waves.

Landau Fermi Liquid Theory.

Electrons in a Solid:the Standard Model

•Quantitative Tools. Density Functional Theory+Perturbation

Theory. 2 / 2 ( )[ ] KS kj kj kjV r r y e y- Ñ + =

Rigid bands , optical transitions , thermodynamics, transport………

Success­story­:­Density­Functional­Linear­Success­story­:­Density­Functional­Linear­ResponseResponse

Tremendous­progress­in­ab­initio­modelling­of­lattice dynamics&­electron-phonon interactions­has­been­achieved(Review: Baroni et.al, Rev. Mod. Phys, 73, 515, 2001)

Correlated Materials do big things

• Huge resistivity changes V2O3.

• Copper Oxides. (La2-x Bax) CuO4 High Temperature Superconductivity.150 K in the Ca2Ba2Cu3HgO8 .

• Uranium and Cerium Based Compounds. Heavy Fermion Systems,CeCu6,m*/m=1000

• (La1-xSrx)MnO3 Colossal Magneto-resistance.

Strongly Correlated Materials.

• Large thermoelectric response in NaCo2-xCuxO4

• Huge volume collapses, Ce, Pu……

• Large and ultrafast optical nonlinearities Sr2CuO3

• Large Coexistence of Ferroelectricity and Ferromagnetism (multiferroics) YMnO3.

Breakdown of standard model

• Large metallic resistivities exceeding the Mott limit. Maximum metallic resistivity 200 ohm cm

• Breakdown of the rigid band picture. Anomalous transfer of spectral weight in photoemission and optics.

• The quantitative tools of the standard model fail.

2 ( )F Fe k k l

h

Localization vs Delocalization Strong Correlation Problem

• Many interesting compounds do not fit within the “Standard Model”.

• Tend to have elements with partially filled d and f shells. Competition between kinetic and Coulomb interactions.

• Breakdown of the wave picture. Need to incorporate a real space perspective (Mott).

• Non perturbative problem.• Require a framework that combines both

atomic physics and band theory. DMFT.

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)

1( , )

( )k

G k ii i

Next Step: GW+EDMFT S. Savrasov and GK.(2001). P.Sun and GK. (2002). S.

Biermann F. Aersetiwan and A.Georges . (2002). P Sun and G.K (2003)

Implementation in the context of a model Hamiltonian with short range interactions.P Sun and G. Kotliar cond-matt 0312303 or with a static U on heavy electrons, without self consistency. Biermann et.al. PRL 90,086402 (2003)

W

W

Self-Consistency loop. S. Savrasov and G. Kotliar (2001) and cond-

matt 0308053

G0 G

Im p u rityS o lver

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

LDA+U functional

2 *log[ / 2 . ( ) ( )]

( ) ( ) ( ) ( )

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

2 | ' |

[ ]

aR bR

n

KS abn KS

R

KS KS

i

LDAext xc

DC

R

Tr i V B r r

V r r dr B r m r dr Tr n

r rV r r dr drdr E

r r

G

w

w s fl f

r l

r rr r

- +Ñ - - - -

- - - +

+ + +-

F - F

å

åò ò

ò òå

1[ ] ( 1)

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

abi

n T G i ew

w+

= å

[ ( ), ( ), ]LDA U abr m r n

, KS KS ab [ ( ), ( ), V ( ), ( ), ]LDA U a br m r n r B r

1

2 ab abcd cdn U n

LDA+DMFT functional

2 *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 KS ab [ ( ) ( ) G V ( ) ( ) ]LDA DMFT a br m r r B r

cluster cluster exterior exteriorH H H H

H clusterH

Medium of free electrons :

impurity model.

Solve for the medium using

Self Consistency

G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001)

Example: DMFT for lattice model (e.g. single band Hubbard).

• Observable: Local Greens function Gii ().

• Exact functional [Gii ()

• DMFT Approximation to the functional.

[ , ] log[ ] ( ) ( ) [ ]DMFT DMFTij ii iin n niG Tr i t Tr i G i Gw w w-G S =- - S - S +Få

[ ] Sum of 2PI graphs with local UDMFT atom ii

i

GF = Få

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,,intM PLocal Impurity Model

Input: ,M P

Output: Self-Consistent Solution

Spectral Density Functional Theory withinLocal Dynamical Mean Field Approximation

•Full implementation in the context of a a one orbital model. P Sun and G. Kotliar Phys. Rev. B 66, 85120 (2002).

•After finishing the loop treat the graphs involving Gnonloc Wnonloc in perturbation theory. P.Sun and GK PRL (2004). Related work, Biermann Aersetiwan and Georges PRL 90,086402 (2003) .

 

EDMFT loop G. Kotliar and S. Savrasov in New Theoretical Approaches to Strongly Correlated G Systems, A. M. Tsvelik Ed. 2001 Kluwer Academic Publishers. 259-301 . cond-mat/0208241 S. Y. Savrasov, G. Kotliar, Phys. Rev. B 69, 245101 (2004)

Optical transfer of spectral weight , kappa organics. Eldridge, J., Kornelsen, K.,Wang, H.,Williams, J., Crouch, A., and

Watkins, D., Sol. State. Comm., 79, 583 (1991).

Cluster Extensions[ , , C, 0, 0, ]CDMFT BK Gij ij ij Gij ij ij C

lattice cluster