dynamical mean field approach to strongly correlated electrons

Dynamical Mean Field Approach to Strongly Correlated Electrons

Gabriel Kotliar

Physics Department andCenter for Materials Theory

Rutgers University

Field Theory and Statistical Mechanics

Rome 10-15 June (2002)

THE STATE UNIVERSITY OF NEW JERSEY

RUTGERS

Outline Correlated Electrons and the Mott transition

problem. Dynamical Mean Field Theory. Cavity

construction. Effective action construction.[G Jona-Lasinio, Nuovo Cimento 34, (1964),

De Dominicis and Martin, Fukuda ] Model Hamiltonian Studies of the Mott

transition in frustrated systems. Universal aspects.

Application to itinerant ferromagnets: Fe,Ni. Outlook


RUTGERS

Standard model of solid

(Sommerfeld)

(Bloch )Periodic potential, waves form bands , k in Brillouin zone .

(Landau) Interactions renormalize away.

Justification: perturbative RG (Benfatto Gallavotti)

2 ( )F Fe k k l

h

The electron in a solid: wave picture

Consequences: Maximum metallic resistivity 200 ohm cm

2

2k

k

m


RUTGERS

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


RUTGERS

Evolution of the spectra from localized to itinerant Low densities. Electron as particle bound to

atom. High densities. Electrons are waves spread

thru the crystal. Mott transition problem: evolution between

the two limits, in the open shell case. Non perturbative problem. Key to understanding many interesting

solids.


RUTGERS

Mott transition in V2O3 under pressure or chemical substitution on V-site


RUTGERS

Failure of the Standard Model: NiSe2-xSx

Miyasaka and Takagi (2000)


RUTGERS

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.


RUTGERS

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, 89

1( , )

( )k

G k ii i

Muller-Hartmann 89


RUTGERS

Missing in this limit

Short Range Magnetic Correlations without magnetic order. Long wavelength modes.

Trust more in frustrated situations and at high temperatures.

2

1~ 0ij i j

j

J S S dd


RUTGERS

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 cavity construction A. Georges G. Kotliar 92

†

0 0 0

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

s st t t t ¯= +òò ò

† †

, ,


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


RUTGERS

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

†( )( ) ( )

MFL o n o n HG c i c iw 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


RUTGERS

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, many models……….. Review: A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)]

G0 G

Im puritySo lver

S .C .C .


RUTGERS

Different Extensions

Take larger clusters in the cavity construction, e.g. cellular DMFT.[Kotliar Savrasov Palsson and Biroli], DCA[Jarrell and Krishnamurthy]

Take into account approximately the renormalization of the quartic coupling, e.g. extended DMFT. [Sachdev and Ye, Kajueter Kotliar, Si and Smith]


RUTGERS

Single site DMFT, functional formulation. Construct a functional of the local Greens function

Expressed in terms of Weiss field (semicircularDOS) [G. Kotliar EBJB 99]

[ , ] log[ ]

( ) ( ) [ ]

DMFT ijn

n n atom ii

i

G Tr i t

Tr i G i G

w

w w

-G S =- - S

- S + Få

† †,

2

2

[ , ] ( ) ( ) ( )†

( )[ ] [ ]

[ ]loc

imp

L f f f i i f i

imp

iF T F

t

F Log df dfe

2

Ising analgoy

[ ] [ [2 ]]2LG

hF h Log ch h

J


RUTGERS

C-DMFT functional formulation. Construct a functional of the restriction of the Greens function to the cluster and its supercell translations.

[ , ] log[ ]

( ) ( ) [ ]pcluster

CDMFT ijn

n n

i

G Tr i t

Tr i G i G

w

w w

-G S =- - S

- S + FåSigma and G are non zero on the

selected cluster and its supercell translations and are non zero otherwise.

Lattice quantities are inferred or projected out from the local quantities.


RUTGERS

C-DMFT: test in one dimension. (Bolech, Kancharla and Kotliar 2002)

Gap vs U, Exact solution Lieb and Wu, Ovshinikov

Nc=2 CDMFT

vs Nc=1


RUTGERS

Results: Schematic DMFT phase diagram Hubbard model (partial frustration)


RUTGERS

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.


RUTGERS

Kuwamoto Honig and Appell PRB (1980)M. Rozenberg G. Kotliar H. Kajueter G Thomas D. Rapkikne J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995)


RUTGERS

Qualitative phase diagram in the U, T , plane,full frustration ( GK Murthy and Rozenberg 2002)

Shaded regions :the DMFT equations have a metallic-like and an insulating-like solution).


RUTGERS

Evolution of the Spectral Function with Temperature

Anomalous transfer of spectral weight connected to the proximity to the Ising Mott endpoint (Kotliar Lange and Rozenberg Phys.Rev.Lett.84,5180(2000).ForeshadowedbyCastellaniDiCastroFeinbergRanninger(1979).


RUTGERS

Insights from DMFTThe Mott transition is driven by transfer of spectral weight from low to high energy as we approach the localized phaseControl parameters: doping, temperature,pressure…


RUTGERS

. ARPES measurements on NiS2-xSex

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


RUTGERS

Anomalous Spectral Weight Transfer: Optics

0( ) ,eff effd P J

iV

2

0( ) ,

ned P J

iV m

, ,H hamiltonian J electric current P polarization

, ,eff eff effH J PBelow energy


RUTGERS

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)


RUTGERS

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


RUTGERS

Anomalous Resistivity and Mott transition Ni Se2-x Sx

Insights from DMFT: think in term of spectral functions (branch cuts) instead of well defined QP (poles )


RUTGERS

Insights from DMFT Mott transition as a bifurcation of an

effective action

Important role of the incoherent part of the spectral function at finite temperature

Physics is governed by the transfer of spectral weight from the coherent to the incoherent part of the spectra. Real and momentum space.


RUTGERS

Realistic Calculationsof the Electronic Structure of Correlated materials

Combinining DMFT with state of the art electronic structure methods to construct a first principles framework to describe complex materials.

Anisimov Poteryaev Korotin Anhokin and Kotliar J. Phys. Cond. Mat. 35, 7359 (1997)

Savrasov Kotliar and Abrahams Nature 410, 793 (2001))


RUTGERS

Spectral Density Functional : effective action construction ( Chitra and GK PRB 2001).

DFT, exact free energy as a functional of an external potential. Legendre transform to obtain a functional of the density DFT(r)]. [Hohenberg and Kohn, Lieb, Fukuda]

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 (r),G(R,R)(i)]

A useful approximation to the exact functional can be constructed.

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


RUTGERS

Combining LDA and 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, subtract this out by shifting the heavy level (double counting term)

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


RUTGERS

LDA+DMFT Self-Consistency loop

G0 G

Im puritySo 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

EdcU

E

0( , , )HHi

HH

i

n T G r r i e w

w

w+

= å


RUTGERS

Case study Fe and Ni

Band picture holds at low T. LSDA predicts correct low T moment

At high temperatures has a Curie Weiss law with a (fluctuating) moment larger than the T=0 ordered moment.

Localization delocalization crossover as a function of T.


RUTGERS

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


RUTGERS

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

0 3( )q

Meff

T Tc


RUTGERS

Ni and Fe: theory vs exp / ordered moment

Fe 2.5 ( theory) 2.2(expt) Ni .6 (theory) .6(expt)

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)


RUTGERS

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


RUTGERS

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


RUTGERS

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

(more itinerant but 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, twice as large in Ni and in Fe Cluster methods.


RUTGERS

Outlook Many open problems!

Strategy: advancing our understanding scale by scale.

New local physics in plaquettes.

Cluster methods to capture longer range magnetic correlations. New structures in k space. Cellular DMFT

Many applications to real materials.


RUTGERS

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


RUTGERS

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

dynamical mean field approach to strongly correlated electrons

Documents

dmft cavity construction

mott transition problem

kotliar ebjb

kajueter kotliar

electron e

cellular dmft

extended dmft

cdmft functional formulation