Introduction to Dynamical Mean Field Theory (DMFT) and its Applications to the Electronic Structure of Correlated Materials

Zacatecas Mexico PASSI School . Montauk June (2006).

G.Kotliar Physics Department Center for Materials Theory

Rutgers University.

Collaborators: K. Haule (Rutgers), C. Marianetti (Rutgers ) S. Savrasov (UC Davis)

References

• Electronic structure calculations with dynamical mean-field theory: G. Kotliar, S. Savrasov, K. Haule, V. Oudovenko, O. Parcollet, and C. Marianetti, Rev. of Mod. Phys. 78, 000865 (2006).Dynamical Mean Field Theory of Strongly Correlation Fermion Systems and the Limit of Infinite Dimensions:  A. Georges, G. Kotliar, W. Krauth, and M. Rozenberg, Rev. of Mod. Phys. 68, 13-125 (1996).

• Electronic Structure of Strongly Correlated Materials: Insights from Dynamical Mean Field Theory: Gabriel Kotliar and Dieter Vollhardt, Physics Today 57, 53 (2004).

• What is a strongly correlated material ?

Band Theory: electrons as waves.

Landau Fermi Liquid Theory.

Electrons in a Solid:the Standard Model

•Quantitative Tools. Density Functional Theory

•Kohn Sham (1964)2 / 2 ( )[ ] KS kj kj kjV r r y e y- Ñ + =

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

Static Mean Field Theory.

22

[ ]totE r

Kohn Sham Eigenvalues and Eigensates: Excellent starting point for perturbation theory in the screened interactions (Hedin 1965)

[ ]nk E kn band index, e.g. s, p, d,,fn band index, e.g. s, p, d,,f

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)

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.

= W

= [ - ]-1

1CV

= G

- [ - ]

KS crystV V10KSG 1G

GW approximation (Hedin )

Kohn Sham Eigenvalues and Eigensates: Excellent starting point for perturbation theory in the screened interactions (Hedin 1965)

Self Energy Self Energy

VanShilfgaarde (2005)VanShilfgaarde (2005)

33

Strong Correlation Problem:where the standard

model fails

• Fermi Liquid Theory works but parameters can’t be computed in perturbation theory.

• Fermi Liquid Theory does NOT work . Need new concepts to replace of rigid bands !

• 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.

44

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 (fully localized or fully itinerant). •Situation realized by applying a control parameters, e.g. pressure. Metal to Insulator Transition. •Some materials have several species of electrons, some localized (f ‘s d’s ) some itinerant (sp, spd) . OSMT. Heavy Fermions.• Introducing carries (electrons or holes) to a Mott insulator. Doping Driven Mott transition.

• Why is it worthwhile to study correlated electron materials ?

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.

Strongly correlated systems

• Copper Oxides. High Temperature Superconductivity.

• Cobaltates Anomalous thermoelectricity.

• Manganites . Colossal magnetoresistance.• Heavy Fermions. Huge quasiparticle masses.• 2d Electron gases. Metal to insulator transitions. • Lanthanides, Transition Metal Oxides,

Multiferroics………………..

55

Basic Questions

• How does the electron go from being localized to itinerant.

• How do the physical properties evolve.

• How to bridge between the microscopic information (atomic positions) and experimental measurements.

• New concepts, new techniques

• How do we probe SCES experimentally ?

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

Spectral Function Photoemission and correlations

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

f() A() M2

e

Angle integrated spectral Angle integrated spectral function function

( , ) ( )dkA k A 88

a)a) Weak CorrelationWeak Correlation

b)b) Strong CorrelationStrong Correlation

Strong Correlation Problem:where the standard

model fails

• Fermi Liquid Theory works but parameters can’t be computed in perturbation theory.

• Fermi Liquid Theory does NOT work . Need new concepts to replace of rigid bands !

• 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.

44

• How do we approach the problem of

strongly correlated electron stystems ?

Two roads for ab-initio calculation of electronic

structure of strongly correlated materials

Correlation Functions Total Energies etc.

Model Hamiltonian

Crystal structure +Atomic positions

Strongly correlated systems are usually treated with model

Hamiltonians

• Tight binding form. Eliminate the “irrelevant” high energy degrees of freedom

Add effective Coulomb interaction terms.

One Band 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 temperatureMott transition as a function of doping, pressure temperature etc.

• How do we reduce the many body problem to something tractable ?

,ij i j i

i j i

J S S h S- -å å MF eff oH h S=-

DMFT Cavity Construction. 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). G. Kotliar S. Savrasov K. Haule V. Oudovenko O. Parcollet and C. Marianetti Rev. Mod. Phys. 78, 865 (2006) . G. Kotliar and D . Vollhardt Physics 53 Today (2004)

1( , )

( )k

G k ii i

Extremize a functional of the local spectra. Local self energy.

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

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

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)

What is the structure of the DMFT problem ?

• Embedding and truncation

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 p u rityS o lver

S .C .C .

• How do we generalize this construction realistic systems ?

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

Dynamical Mean Field Theory. Cavity Construction.

A. Georges and G. Kotliar PRB 45, 6479 (1992).

†

0 0 0

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

s st m tt

t t ¯

¶+ D-

¶- +òò ò

,ij i j i

i j i

J S S h S- -å å eMF offhH S=-† †

, ,

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

t c c c c U n n

*

( )V Va a

a a

ww e

D =-å

† † † † †Anderson Imp 0 0 0 0 0 0 0

, , ,

( +c.c). H c A A A c c UcV c c c

A(A())

1010

A. Georges, G. Kotliar (1992)

( )wDlatt ( ,

1 G [ ]

( ) [( ) ])

[ ]n impn

n

ik ii

ktw m

ww+ + - S

DD

=

latt( ) G ([ [)] ] ,imp n nk

G i i kw wD D=å

[ ]ijij

jm mJth hb= +å

11

( ( )( )

( [))

][ ]

imp n

imp n

kn

G i

Gti

ik

w

ww -D

D

=+-

å

A(A())

1111

Dynamical Mean Field Theory

• Weiss field is a function. Multiple scales in strongly correlated materials.

• Exact in the limit of large coordination (Metzner and Vollhardt 89) , kinetic and interaction energy compete on equal footing.

• Immediate extension to real materials

, ,

, 22

[ ] [ ]( )

[ ] [ ]spd sps spd f

f spd ff

H k H kt k

H k H k

æ ö÷ç ÷ç ÷ç ÷çè ø®

| 0 ,| , | , | | ... JLSJM g> > ¯> ¯> >®

DFT+DMFTDFT+DMFT1212

Evolution of the DOS. Theory and experiments

( )A 1313

DMFT Qualitative Phase diagram of a frustrated Hubbard model at integer

filling

T/W

1414

Interaction with Experiments. Photoemission Three peak strucure. V2O3:Anomalous

transfer of spectral weight

M. Rozenberg G. Kotliar H. Kajueter G Thomas D. Rapkine J Honig and P

Metcalf Phys. Rev. Lett. 75, 105 (1995)

T=170T=170

T=300T=300

1515

. Photoemission measurements and TheoryV2O3 V2O3 Mo, Denlinger, Kim, Park, Allen, Sekiyama, Yamasaki, Mo, Denlinger, Kim, Park, Allen, Sekiyama, Yamasaki,

Kadono, Suga, Saitoh, Muro, Metcalf, Keller, Held, Eyert, Anisimov, Kadono, Suga, Saitoh, Muro, Metcalf, Keller, Held, Eyert, Anisimov, Vollhardt PRL . (2003Vollhardt PRL . (2003))

NiSxSeNiSxSe1-x1-xMatsuura Watanabe Kim Doniach Shen Thio Bennett (1998)Matsuura Watanabe Kim Doniach Shen Thio Bennett (1998)

Poteryaev et.al. (to be published)Poteryaev et.al. (to be published)1616

How do we solve the impurity model

?

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)

• 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.

How good is DMFT ?

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

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

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)]

• How do we incorporate the Long Range Coulomb Interactions

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+

How do we merge band theory and DMFT ?

• How do we extract total energies ?

observable of interestobservable of interest is the "local“is the "local“ Green's functionsGreen's functions (spectral (spectral function)function)

Currently feasible approximations: LDA+DMFT:

Spectral density functional theory

(G. Kotliar et.al., RMP 2006).

Variation gives st. eq.:Generalized Q. impurity problem!

General impurity problem

Diagrammatic expansion in terms of hybridization +Metropolis sampling over the diagrams

•Exact method: samples all diagrams!•Allows correct treatment of multiplets

k

K.H. Phys. Rev. B 75, 155113 (2007)

Exact “QMC” impurity solver, expansion in terms of hybridization

P. Werner, Phys. Rev. Lett. 97, 076405 (2006)

• What are the characteristics of the spectra of a correlated system , in the simplest model ?

Pressure Driven Mott transition

T/W

Phase diagram of a Hubbard model with partial frustration at integer filling. [Rozenberg et. al. PRL 1995] Evolution of the Local Spectra as a function of U,and T. Mott transition driven by transfer of spectral weight Zhang Rozenberg Kotliar PRL (1993)..

Mott transition in one band model. Review Georges et.al. RMP 96

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

Spectral Evolution at T=0 half filling full frustration

Parallel development: Fujimori et.al

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)

n=1

Order in Perturbation Theory

Order in PT

Range of the clusters

Basis set size. DMFT

GW

r site CDMFT

l=1

l=2

l=lmax

r=1

r=2

n=2

GW+ first vertex correction

• How do we go directly from structure to physical observables ?

1

( , )( )k

G k ii i

Can the various approaches (DMFT, DFT, DFT+U be unified )?

[ ] [ ( ) ]F J S JAZ e d d e y yy y+- + - += =ò

[ ]F

J A aJ

dd

=< >=

[ ] [ [ ] ] [ ]a F J a aJ aG = -

Spectral density functional. Effective action construction.e.g Fukuda et.al

hartree xcDG=G +G

0 intS S Sl= +

0 1J J Jl= + +L

[ ]aG = 0G1l+ G +L[ ]a+DG0 0 0[ ]F J aJ-

1

int

0

[ ] ( , ( , ))a d S J al l lDG = < >ò

0[ ]J aa

ddDG

=00

[ ]F

J aJ

dd

=

In practice we need good approximations to the exchange correlation, in DFT LDA. In spectral density functional theory, DMFT. Review: Kotliar et.al. Rev. Mod. Phys. 78, 865 (2006)

Kohn Sham equations

Different methods differ by the choice of variable a used.

• DFT

• Spin and Density FT

*a=G ( , ', ) ( ') ( ) ( )loc Rb Ra ababRr r r r G

a= (r)

a= (r), (r)

Spectral Density Functional R. Chitra and G.K Phys. Rev. B 62, 12715 (2000). S. Savrasov and G.K PRB (2005)

C DMFT extend the notion of “locality”to several unit cells

a= (r),

lda+dmft

( ), correlated orbitals (f or d)

[ ( )]

ab

lda at ab dc

G ab

G

U (and form of dc) are input parameters.

EDMFT a=“ Gloc Wloc” Cluster Greens Function and Screened interaction, No input parameters. Recently impelemented and tested for sp systems. Si C ….N. Zein et.al.PRL 96, (2006) 226403 Zein and Antropov PRL 89,126402

Review: Kotliar et.al. Rev. Mod. Phys. 78, 865 (2006)

DFT+DMFT

• What is the ultimate theory , without any external parameters ?

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.

Classical case Quantum case

A. Georges, G. Kotliar (1992)

Mean-Field : Classical vs Quantum

†

0 0 0

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

s st m tt

t t ¯

¶+ D-

¶- +òò ò

( )wD

†( )( ( ) )) (

MFo n oo n n Sc i c iG i s ss ww w D=- á ñ

( )

(()

)

11

([ ]

)[ ]n

n

kn

G i

G it ki m

w

wwD

D

=- - +

å

,ij i j i

i j i

J S S h S- -å å

eMF offhH S=-

effh

00 ( )MF effH hm S=á ñ

ijff jj

e mh J h= +å

† †

, ,

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

t c c c c U n n

Easy!!!

0 [ ]S th heffbá ñ=

Hard!!!QMC: J. Hirsch R. Fye (1986)NCA : T. Pruschke and N. Grewe (1989)PT : Yoshida and Yamada (1970)NRG: Wilson (1980)

• Pruschke et. al Adv. Phys. (1995) • Georges et. al RMP (1996)

IPT: Georges Kotliar (1992). .QMC: M. Jarrell, (1992), NCA T.Pruschke D. Cox and M. Jarrell

(1993),ED:Caffarel Krauth and Rozenberg (1994)Projective method: G Moeller (1995). NRG: R. Bulla et. al. PRL 83, 136 (1999),……………………………………...

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)

Conclusion • DMFT, method under very active development.

But there is now a clear formulation (and to large extent implementation) as a fully self consistent, controlled many body approach to solids.

• It gives good quantitative results for total energies, phonon and photoemission spectra, and transport of materials. Many examples…all over the periodic table.

• Helpful in developing intuition and qualitative insights in correlated electron materials.

• With advances in implementation, we will be able to focus on deviations from (cluster) dynamical mean field theory.

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.

V2O3 under pressure or

Schematic DMFT phase diagram of a partially frustrated integered filled

Hubbard model.

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

.

Schematic DMFT phase Implications for transport.

Material Properties: total energy and phonon spectra