Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University

Introduction to Strongly Correlated Electron Materials, Dynamical Mean Field Theory (DMFT) and its extensions. Application to the Mott Transition.

Gabriel Kotliar

Physics Department and

Center for Materials Theory

Rutgers University

•Strongly Correlated Electrons: diverse examples and unifying themes. Cargese August 8-20 (2005).

THE STATE UNIVERSITY OF NEW JERSEY

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Plan of the course. Lecture I.

Motivation: Strongly Correlated Electron Systems require a new starting point or (non-Gaussian) reference system for their description.

DMFT provides such a reference frame, mapping the full many body problem on the lattice to a much simpler system, a quantum impurity model in a self consistent medium. DMFT a first stab at correlated electron materials. Pedagogical derivations of mean field theories, Weiss theory, density functional theory, DMFT.


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Plan of the course. Lecture II.

Motivation: The temperature and pressure driven Mott transition. The most basic competition: kinetic energy vs Coulomb, itineracy vs localization.

Single site DMFT in action. Some results on the frustrated Hubbard model. Comparison with earlier theories. Comparison with experiments.

General lessons, and system specific extensions. [LDA+DMFT]. Mott transition across the actinide series.


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Plan of the course:Lecture III.

The plaquette as a reference frame. Cluster DMFT studies of the doped Mott

insulator and the problem of high temperature superconductivity.

Connection with the d-wave RVB approach

and with some experiments. Correlated superconductivity in Am ?


RUTGERS

References http://www.physics.rutgers.edu/~kotliar/publications.html 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).

The Mott transition. G. Kotliar in 50 years of Condensed Matter Physics. P. Ong and R. Bahtt editors. Princeton University Press .

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


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Weakly correlated electrons:band theory.

Fermi Liquid Theory. Simple conceptual picture of the ground state, excitation spectra, transport properties of many systems (simple metals, semiconductors,….).

In a certain low energy regime, adiabatic Continuity to a Reference Systen of Free Fermions with renormalized parameters. Rigid bands , optical transitions , thermodynamics, transport………


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Standard Model of Solids Qualitative predictions: low temperature

dependence of thermodynamics and transport.

Optical response, transition between the bands.

Filled bands give rise to insulting behavior. Compounds with odd number of electrons are metals.

Kinetic Boltzman equations for QP. scattering off phonons or disorder, ee. int etc.

( ) 1Fk l

~H constR~ const S T ~VC T

2 ( )F Fe k k l

h


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Quantitative Tools of Electronic Structure. 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

Static mean field theory. Derived from a functional which gives the total energy. Excellent starting point for computation of spectra in perturbation theory in screened Coulomb interaction GW.

= W

= [ - ]-11CV

= G

- [ - ]KS crystV V10KSG 1G

GW approximation (Hedin )


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LDA+GW: semiconducting gaps


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DFT +GW approaches do not always work. …. Solid State Physics Chapter 2.


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The electron in a solid: particle picture.Ba

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

Mott: correlations localize the electron

e_ e_ e_ e_

Superexchange

Ba

Think in real space , solid collection of atoms

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

1

T


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Mott : Correlations localize the electron

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

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

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

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


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Localization vs Delocalization Strong Correlation Problem

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


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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.


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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


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Strongly correlated systems are usually treated with model Hamiltonians

Conceptually one wants to restrict the number of degrees of freedom by eliminating high energy degrees of freedom.

In practice other methods (eg constrained LDA , GW, etc. are used)


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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


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

( , )k

A k G kk

Photoemission and the Theory of Electronic Structure

Limiting case itinerant electrons( ) ( )k

k

A

( ) ( , )k

A A k

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

Hubbard bands

Local Spectral Function

A BU


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Strong Correlation effects appear in 3d- 4f (and sometimes 5f) systems. Because their wave functions are more localized. Many compounds. Also p electron in organic materials with large volumes can be strongly correlated.


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

Breakdown of the Standard Model. Strong Correlation Anomalies cannot be understood within the Breakdown of standard model of solids. Metallic “resistivities beyond the Mott limit.

2 ( )F Fe k k l

h


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Optical conductivity of a periodic Anderson model with two electrons per site,U=3, V=.25,t=1,T=.001,.005,.001,.02,.03, weak disorder =.05


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Failure of the StandardModel: Anomalous Spectral Weight Transfer as a function of T.

Optical Conductivity Schlesinger et.al (1993)

0( )d

Neff depends on T


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Non local transfer of spectral weight in photoemission. For a review Damascelli et. al. RMP. Figure from Norman et. al. cond-mat/0507031


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Correlated Materials do big things

Huge resistivity changes. Mott transition. 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.


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Strongly Correlated Materials.

Large thermoelectric response in CeFe4 P12 (H. Sato et al. cond-mat 0010017). Ando et.al.

NaCo2-xCuxO4 Phys. Rev. B 60, 10580 (1999). Gigantic Volume Collapses. Lanthanide and

actinides. Large and ultrafast optical nonlinearities Sr2CuO3

(T Ogasawara et.a Phys. Rev. Lett. 85, 2204 (2000) )

……………….


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Strong correlation anomalies

Metals with resistivities which exceed the Mott Ioffe Reggel limit.

Transfer of spectral weight which is non local in frequency.

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


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Basic competition between kinetic energy and Coulomb interactions.

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

The approach should allow to incorporate material specific information.

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

DMFT!


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Philosophy of the Approach

Study correlated materials by using a small number of correlated sites in a medium as a reference frame. [one site, a link, a plaquette etc…]

Goal is to understand what aspects of the experimental data can be understood from this very simple framework. After this is done, we can see whether if what is left out, longer wavelenght non Gaussian physics is important. How to incorporate that, is one of the greatest theoretical challenges in the field. A lot of the physics of correlated materials can be understood within DMFT!


RUTGERS

DMFT


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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 ki i

Neglect k dependence of self energy Muller-Hartmann 89

3 1~ [ ] ij ij ijG t

d


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DMFT mapping (Georges Kotliar 1992)

1( )[ ]

[ ]( )imp nn k imp nk

G ii t i

ww m w

é ùê úD = ê ú- + - S Dê úë ûå

Notice that if the self energy is local it is the self energy of an Anderson impurity model. Determine the bath of the impurity model from:


RUTGERS

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

1 1( ) [ ( )]

( )n loc nloc n

i R G iG i

w ww

-D = + 1[ ]

[ ]k k

R zz e

=-å

†

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 Weiss field

2( ) ( )n loc ni t G iw wD =Semicircular density of states. Behte lattice.


RUTGERS

DMFT mapping (Georges Kotliar 1992)

1( )[ ]

[ ]( )imp nn k imp nk

G ii t i

ww m w

é ùê úD = ê ú- + - S Dê úë ûå

Notice that if the self energy is local it is the self energy of an Anderson impurity model. Determine the bath of the impurity model from:


RUTGERS

Technical Intermission : Construction on Mean Field Theories.


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Effective Action point of view.

Identify observable, A. Construct a free energy functional of <A>=a, [a] which is stationary at the physical value of a.

Example, density in DFT theory. (Fukuda et. al.). DMFT (R. Chitra and G.K (2000) (2001). H=H0+ H1. [a,J0]=F0[J0 ]–a J0 _ + hxc [a]

Functional of two variables, a ,J0.

H0 + A J0 Reference system to think about H.

J0 [a] Is the functional of a with the property <A>0 =a < >0 computed with H0 + A J0

Many choices for H0 and for A


RUTGERS

Constructing mean field functionals

Extremize a to get [J0]= exta [a,J0]. Functional of Weiss field J0 only.

While the focus is on observable a, for example spin, the formalism leads to approximations to correlation functions. Orenstein-Zernicke form.


RUTGERS

Mean-Field Classical Systems. Cavity construction and functional approach.

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

,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


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DFT: effective action construction

( )( )

Wr

j r


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DFT: Kohn Sham formulation

=


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Mean-Field : Classical vs Quantum

Classical case Quantum case

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

†

0 0 0

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

s st m t t tt ¯

¶+ - D - +

¶òò ò

( )wD

†( )( ) ( )

MFo n o n SG c i c is sw w D=- á ñ

1( )

1( )

( )[ ][ ]

nk

n kn

G ii

G i

ww e

w

=D - -

D

å

,ij i j i

i j i

J S S h S- -å å

MF eff oH h S=-

effh

0 0 ( )MF effH hm S=á ñ

eff ij jj

h J m h= +å

† †

, ,


t c c c c U n n


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Solving the DMFT equations

G 0 G

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

S . C . C .

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

G0 G

Im p u rityS o lver

S .C .C .


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Ex: Baym Kadanoff functional,a= G, H0 = free electrons J0[a]=

1 1 10[ ] [ ] [( ) ] [ ]BK G TrLn G Tr G G G G

10[ , ] [ ] [ ] [ ]G TrLn G Tr G G

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

10[ ] [ ] ( [ ] [ ])self GTrLn G ext Tr G G

Viewing it as a functional of J0, Self Energy functional(Potthoff)


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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



RUTGERS

CDMFT and NCS as truncations of the Baym Kadanoff functional

[ , , , 0, 0, ]CDMFT Gij ij ij C Gij ij ij C

10[ , ] [ ] [ ] [ ]G TrLn G Tr G G


[ , ,| | , 0, 0,| | ]ncs Gij ij i j r Gij ij i j r


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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)


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

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

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

cond-matt 0307587 (2003)


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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) ]


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Strongly Correlated Electrons and DMFT. The challenges. Learn to solve the DMFT

equations more accurately or more explicitly. Incrase the accuracy of the CDMFT itself.

Identify which strong correlation phenomena can be capture from a local DMFT perspective using sites, linkes, plaquettes, etc as reference systems, and which aspects involve non local and non Gaussian fluctuations.

MANY INTERESTING PROBLEMS WHERE THIS TECHNIQUE CAN BE USED. Space of all materials is very large….suprises are hidden in each corner….


RUTGERS

Extracting Lattice quantities from Cluster Quantities.

1

1( , )

( , ) ( )lattlatt

G kM k t k

SLk,k+i-j

1( , ) W ( )latt k SL

s

W i jN

1( , )

( ) ( , )lattlatt

G kt k k

W can be either G, or M (cumulant, irreducible with respect to t ) so there is no uniqueness in the reconstruction of lattice quantities,e.g


RUTGERS

Self Energy and Cumulant Periodization (Stanescu and Kotliar 2005)

( ) ( ) ( ) ( )latt A A B B C Ck S k S k S k

A="(0, ),( ,0)" ( ) .5(1 cos( )cos( ))

B=( , )" ( ) .25(1 cos( )-cos( )+cos( )cos( ))

"(0,0)" ( ) .25(1 cos( )+cos( )+ cos( )cos( ))

A x y

B x y x y

C x y x y

S k k k

S k k k k k

C S k k k k k

1 1 -1 1( ( ) ) ( ( )) ( ) ( ( )) ( ) ( ( )) ( )latt A A B B C Ck S k S k S k

( ) ( ) ( ) ( ) ( ) ( ) ( )latt A A B B C Ck S k S k S k

11(0, ) + ( ) ( ) ( )A A A B B C CA S k S k S k


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Comparison of 2 and 4 sites (Tudor Stanescu)


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Main Omission of this Course

Techniques for solving quantitatively the Anderson Impurity Model. G[G0]See Reviews.

Qualitative behavior of the solution of the Anderson Impurity Model. Kondo Physics.

Extension to describe ordered phases. Superconductivity. Antiferromagnetism. Etc…

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Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University

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