excitation spectra

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Excitation spectra

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Comments on realistic calculations using DMFGT

Gabriel Kotliar Rutgers University

Trieste 2002

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X.Zhang M. Rozenberg G. Kotliar (PRL 1993)Joo and Udovenko (20010)

Spectral Evolution at T=0 half filling full frustration

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Summary

Basis set LMTO (Savrasov) Materials Information and Design Lab.

(Savrasov’s MINDLAB) Computations of U (Anisimov) Derivation of model hamiltonian Solution via DMFT: mapping onto degenerate

Anderson model in a self consistent bath. Solution of the multiorbital anderson model

Using QMC (Rozenber and Lichtenstein).

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Basis set, bands , DOS

2| ( ) | ( )k xc k LMTOV H k aba bc r c- Ñ + =

|| ( )k k O k aba bc c =

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Computation of U’s

Un n

Jn n

de ded d

de ded d

¯

¯

­ +

­ + ¯

­

­ ¯

=

-=

-

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Comments U is a basis dependent concept. Dynamical mean field theory is a basis

dependent technique.

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Unitary transformation

k

LL LH

HL HH

H HH

H H

é ùê ú=ê úë û

0 0

0 HH

é ùê úS =ê úSë û

† k k k kH U H U®

† k k kU H US ® K dependent!

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Two Roads for calculations of the electronic structure of correlated materials

Crystal Structure +atomic positions

Correlation functions Total energies etc.

Model Hamiltonian

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LDA functional

2log[ / 2 ] ( ) ( )

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

2 | ' |

n KS KS

LDAext xc

Tr i V V r r dr

r rV r r dr drdr E

r r

w r

r rr r

- +Ñ - -

+ +-

ò

ò ò

[ ( )]LDA r

[ ( ), ( )]LDA KSr V r

Conjugate field, VKS(r)

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Minimize LDA functional

[ ]( )( ) ( ) '

| ' | ( )

LDAxc

KS ext

ErV r V r dr

r r r

d rrdr

= + +-ò

0*2

( ) { )[ / 2 ]

( ) ( ) n

n

ikj kj kj

n KSkj

r f tri V

r r ew

w

r e yw

y +=

+Ñ -=å å

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Spectral Density Functional : effective action construction (Fukuda, Valiev and Fernando , Chitra and GK).

DFT, consider the exact free energy as a functional of an external potential. Express the free energy as a functional of the density by Legendre transformation. DFT(r)]

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 sources for (r) and G and performing a Legendre transformation, (r),G(R,R)(i)]

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

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Spectral Density Functional

The exact functional can be built in perturbation theory in the interaction (well defined diagrammatic rules )The functional can also be constructed from the atomic limit, but no explicit expression exists.

DFT is useful because good approximations to the exact density functional DFT(r)] exist, e.g. LDA, GGA

A useful approximation to the exact functional can be constructed, the DMFT +LDA functional.

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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

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Comments on LDA+DMFT• Static limit of the LDA+DMFT functional , with

= HF reduces to LDA+U• Removes inconsistencies of this approach,• Only in the orbitally ordered Hartree Fock limit,

the Greens function of the heavy electrons is fully coherent

• Gives the local spectra and the total energy simultaneously, treating QP and H bands on the same footing.

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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

U

E

0( , , )HHi

HH

i

n T G r r i e w

w

w+

= å

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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

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LDA+DMFT References V. Anisimov, A. Poteryaev, M. Korotin, A.

Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359-7367 (1997).

A­Lichtenstein­and­M.­Katsenelson­Phys.­Rev.­B­57,­6884­(1998).

S.­Savrasov­­­and­G.Kotliar,­funcional­formulation­for­full­self­consistent­implementation­Nature­(2001)

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Applications Look for situations which

Are in between atomic and band behavior.

Many Many Many Compounds Oxides.

BUT ALSO SOME ELEMENTS!

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Mott transition in the actinide series. B. Johanssen 1974 Smith and Kmetko Phase Diagram 1984.

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Pu: DMFT total energy vs Volume(S. Savrasov 2001)

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Lda vs Exp Spectra

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

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Iron and Nickel: crossover to a real space picture at high T(Lichtenstein,Katsnelson andGK)

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Conclusion

The character of the localization delocalization in simple( Hubbard) models within DMFT is now fully understood, nice qualitative insights.

This has lead to extensions to more realistic models, and a beginning of a first principles approach interpolating between atoms and band, encouraging results for many systems

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Outlook Systematic improvements, short range correlations. Take a cluster of sites, include the effect of the rest

in a G0 (renormalization of the quadratic part of the effective action). What to take for G0:

Cluster DMFT, periodic clusters (Lichtenstein and Katsnelson)DCA (M. Jarrell et.al) , CDMFT ( GK )

include the effects of the electrons to renormalize the quartic part of the action (spin spin , charge charge correlations) E. DMFT (Kajueter and GK, Si et.al)

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C-DMFT: test in one dimension. (Bolech, Kancharla and Gk2002)

Gap vs U, Exact solution Lieb and Wu, Ovshinikov PRL 20,1445 (1968)

Nc=2 CDMFT

vs Nc=1

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A (non comprehensive )list of extensions of DMFT

Two impurity method. [A. Georges and G. Kotliar, A. Schiller PRL75, 113 (1995)]

M. Jarrell Dynamical Cluster Approximation [Phys. Rev. B 7475 1998]

Continuous version [periodic cluster] M. Katsenelson and A. Lichtenstein PRB 62, 9283 (2000).

Extended DMFT [H. Kajueter and G. KotliarRutgers Ph.D thesis 2001, Q. Si and J L Smith PRL 77

(1996)3391 ] Coulomb interactions R . Chitra Cellular DMFT GK Savrasov Palsson and Biroli

[PRL87, 186401 2001]

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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

†

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

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Elements of the Dynamical Mean Field Construction and Cellular DMFT, G. Kotliar S. Savrasov G. Palsson and G. Biroli PRL 2001

Definition of the local degrees of freedom Expression of the Weiss field in terms of the

local variables (I.e. the self consistency condition)

Expression of the lattice self energy in terms of the cluster self energy.

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Cellular DMFT : Basis selection

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Lattice action

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Elimination of the medium variables

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Determination of the effective medium.

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Connection between cluster and lattice self energy.

The estimation of the lattice self energy in terms of the cluster energy has to be done using additional

information Ex. Translation invariance

•C-DMFT is manifestly causal: causal impurity solvers result in causal self energies and Green functions (GK S. Savrasov G. Palsson and G. Biroli PRL 2001)•In simple cases C-DMFT converges faster than other causal cluster schemes.

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Improved estimators

• Improved estimators for the lattice self energy are available (Biroli and Kotliar)

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Real Space Formulation of the DCA approximation of Jarrell et.al.

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Affleck Marston model.

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Convergence test in the Affleck Marston

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Convergence of the self energy

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Recent application to high Tc

A. Perali et.al. cond-mat 2001, two patch model, phenomenological fit of the functional form of the vertex function of C-DMFT to experiments in optimally doped and overdoped cuprates

Flexibility in the choice of basis seems important.

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Extended DMFT electron phonon

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Extended DMFT e.ph. Problem

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E-DMFT classical case, soft spins

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E-DMFT classical case Ising limit

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E-DMFT test in the classical case[Bethe Lattice, S. Pankov 2001]

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Advantage and Difficulties of E-DMFT

The transition is first order at finite temperatures for d< 4

No finite temperature transition for d less than 2 (like spherical approximation)

Improved values of the critical temperature

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Conclusion

For “first principles work” there are several many body tools waiting to be used, once the one electron aspects of the problem are clarified.

E-DMFT or C-DMFT for Ni, and Fe ? Promising problem: Qualitative aspects of

the Mott transition within C-DMFT ?? Cuprates?

Realistic Theories of Correlated Materials

ITP, Santa-Barbara

July 27 – December 13 (2002)

Conference November15-19 (2002)

O.K. Andesen, A. Georges,

G. Kotliar, and A. Lichtenstein

http://www.itp.ucsb.edu/activities/future/

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Recent phase diagram of the frustrated Half filled Hubbard model with semicircular DOS (QMC Joo and Udovenko PRB2001).

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Realistic implementation of the self consistency condition

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

•H and , do not commute•Need to do k sum for each frequency •DMFT implementation of Lambin Vigneron tetrahedron integration (Poteryaev et.al 1987)

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Good method to 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. Some unorthodox examples

Fe, Ni, Pu …………….

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Functional Approach The functional approach offers a direct

connection to the atomic energies. One is free to add terms which vanish quadratically at the saddle point.

Allows us to study states away from the saddle points,

All the qualitative features of the phase diagram, are simple consequences of the non analytic nature of the functional.

Mott transitions and bifurcations of the functional .

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Functional Approach

† †,

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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Case study in f electrons, Mott transition in the actinide series

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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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Conventional viewpoint Alpha Pu is a simple metal, it can be described with LDA +

correction. In contrast delta Pu is strongly correlated. Constrained LDA approach (Erickson, Wills, Balatzki, Becker). In

Alpha Pu, all the 5f electrons are treated as band like, while in Delta Pu, 4 5f electrons are band-like while one 5f electron is deloclized.

Same situation in LDA + U (Savrasov and Kotliar, Bouchet et. Al. ) Delta Pu has U=4,

Alpha Pu has U =0.

The character of the localization delocalization in simple( Hubbard) models within DMFT is now fully understood, nice qualitative insights.

This has lead to extensions to more realistic models, and a beginning of a first principles approach interpolating between atoms and band, encouraging results for simple elements

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

†

0 0 0

[ ] ( )[ ( , ')] ( ')o o o oS Go c Go c 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

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DMFTConnection with atomic limit

1[ ] [ ] [ ] logat atG W Tr G Tr G Tr G G-F = D - D - +

10

10[ ] ( ) ( ') (( , ') ) ( ) ( ) ( )at a a abcd a b c d

ab

GS G c c U c c c c

1 10 atG G [ ] atS

atW Log e [ [ ]]atW

G G

Weiss field