theoretical treatments of correlation effects

Theoretical Treatments of Correlation Effects

Gabriel Kotliar

Physics Department and

Center for Materials Theory

Rutgers University

Workshop on Chemical Physics of Emerging Materials

Schloss Rinberg May 29th 2001

THE STATE UNIVERSITY OF NEW JERSEY

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What can theory contribute to materials research ?

Some universal aspects can be gleaned from simple models. Example, recent DMFT study of the Mott transition endpoint.

Non universal physics requires detailed modeling. Case study Recent LDA+DMFT study of Pu.

Summary


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Why study the Mott phenomena?

Evolution of the electronic structure between the atomic limit and the band limit. Basic solid state problem. Solved by band theory when the atoms have a closed shell. Mott’s problem: Open shell situation.

The “”in between regime”” is ubiquitous central them in strongly correlated systems.


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Mott transition in layered organic conductors S Lefebvre et al. cond-mat/0004455


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A time-honored example: Mott transition in V2O3 under pressure

or chemical substitution on V-site


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Kuwamoto Honig and AppellPRB (1980)


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Phase Diag: Ni Se2-x Sx

G. Czek et. al. J. Mag. Mag. Mat. 3, 58 (1976)


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Theoretical Approach Mean field approach to quantum many

body systems, constructing equivalent impurity models embedded in a bath to be determined self consistently. Use exact numerical techniques as well as semianalytical approaches to study this problem. (DMFT).

Exact in infinite dimensions (Metzner and Vollhardt ) , can be improved systematically using cluster methods (DCA, CDMFT).

Study simple model Hamiltonians (such as the one band model on simple lattices)

Understand the results physically in terms of a Landau theory :certain high temperature aspects are independent of the details of the model and the approximations used. Other results are approximate, and very sensitive on solid state aspects.


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Reviews of DMFT

Prushke T. Jarrell M. and Freericks J. Adv. Phys. 44,187 (1995)

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


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Schematic DMFT phase diagram one band Hubbard model (half filling, semicircular DOS, partial frustration) Rozenberg et.al PRL (1995)


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Insights from DMFT

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


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Evolution of the Spectral Function with Temperature

Anomalous transfer of spectral weight connected to the proximity to an Ising Mott endpoint (Kotliar Lange and Rozenberg PRL 84, 5180 (2000))


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

Resistivity near the metal insulator endpoint ( Rozenberg et. Al 1995) exceeds the Mott limit


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Anomalous Resistivity and Mott transition Ni Se2-x Sx

Miyasaka and Tagaki (2000)


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. ARPES measurements on NiS2-xSex

Matsuura et. Al Phys. Rev B 58 (1998) 3690


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Ising character of Mott endpoint

Singular part of the Weiss field is proportional toMax{ (p-pc) (T- Tc)}1/in mean field and 5 in 3d

couples to all physical quantities which then exhibit a kink at the Mott endpoint. Resistivity, double occupancy,photoemission intensity, integrated optical spectral weight, etc.

Divergence of the specific heat.


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Mott transition endpoint

Rapid variation has been observed in optical measurements in vanadium oxide and nises mixtures

Experimental questions: width of the critical region. Ising exponents or classical exponents, validity of mean field theory

Building of coherence in other strongly correlated electron systems.

Unify concepts from different theoretical approaches, condensation of d and onset of coherence .


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Insights from DMFT Low temperatures several competing phases . Their relative stability depends on chemistry and crystal structureHigh temperature behavior around Mott endpoint, more universal regime, captured by simple models treated within DMFT


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Delocalization Localization across the actinide series


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Small amounts of Ga stabilize the phase


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Problems with LDA

o DFT in the LDA or GGA is a well established tool for the calculation of ground state properties.

o Many studies (Freeman, Koelling 1972)APW methods

o ASA and FP-LMTO Soderlind et. Al 1990, Kollar et.al 1997, Boettger et.al 1998, Wills et.al. 1999) give

o an equilibrium volume of the an equilibrium volume of the phasephaseIs 35% lower than Is 35% lower than experimentexperiment

o This is the largest discrepancy ever known in DFT based calculations.


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Problems with LDA LSDA predicts magnetic long range

order which is not observed experimentally (Solovyev et.al.)

If one treats the f electrons as part of the core LDA overestimates the volume by 30%

LDA predicts correctly the volume of the phase of Pu, when full potential LMTO (Soderlind and Wills). This is usually taken as an indication that Pu is a weakly correlated system


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LDA+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, substract this out by shifting the heavy level (double counting term)

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


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effective action construction (Fukuda, Valiev and Fernando , Chitra and GK).

Select a set of local orbitals. Define a frequency dependent, local

Greens function by projecting onto the local orbitals.

The exact free energy can be expressed as a functional of the local Greens function and of the density

A useful approximation to the exact functional road to total energy calculations.


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LDA+DMFT

V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359-7367 (1997).

A Lichtenstein and M. Katsenelson Phys. Rev. B 57, 6884 (1988).

S. Savrasov G. Kotliar and E. Abrahams full self consistent implementation ( Nature, 2001)


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LDA+DMFT Self-Consistency loop

G0 G

Im puritySolver

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


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Pu: DMFT total energy vs Volume (S. Savrasov )


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Lda vs Exp SpectraD

OS

, st./

[eV

*cel

l]


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Pu Spectra DMFT(Savrasov et. al ) EXP (Arko et. al)


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Outlook Some universal aspects can

be gleaned from simple models. Recent DMFT study of the Mott transition endpoint.

Many more simple qualitative pictures of little corners in the space of all materials, are still to be found.

Non universal physics requires detailed modeling. Recent LDA+DMFT study of Pu.

New developments in many body and electronic structure methods, predictions of new compounds? More interactions with chemical physics and material science.


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Mean-Field : Classical vs Quantum

Classical case Quantum case

†

0 0 0

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

s st t t t ¯= +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

† †

, ,

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

t c c c c U n n


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Landau Functional

† †,

2

2

[ , ] ( ) ( ) ( )†

† † † †

0

†

Mettalic Order Para

( )[ ] [ ]

mete

[ ]

[ , ] [ [ ] ]

( )( )

r: ( )

( ) 2 ( )[ ]( )

loc

LG imp

L f f f i i f i

imp

loc f

imp

iF T F

t

F Log df dfe

dL f f f e f Uf f f f d

d

F iT f i f i TG i

i

i

2

2

Spin Model An

[ ] [[ ]2 ]

alogy:

2LG

t

hF h Log ch h

J

G. Kotliar EPJB (1999)


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Double counting correction

Simplest case F0 only. Generalization

Lichtenstein et.al in

The context of LDA+U

theoretical treatments of correlation effects

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