gabriel kotliar physics department and center for materials theory rutgers university

Dynamical Mean Field Theory from Model Hamiltonian Studies of the Mott Transition to

Electronic Structure Calculations

Gabriel Kotliar

Physics Department and

Center for Materials Theory

Rutgers University

11 Conference on Recent Progress in Many Body Physics

UMIST July 9-15th 2001

THE STATE UNIVERSITY OF NEW JERSEY

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Outline

What is DMFT, when is it useful and how is it done.

What has been accomplished. Ex. model Hamiltonian studies of the finite temperature Mott transition.

How to combine DMFT and band structure, formal aspects.

Results for some real materials.


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References, Collaborators

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

Finite T Mott endpoint: Kotliar Lange and Rozenberg PRL 84, 5180 (2000))

Realistic Calculations:S. Savrasov and GK cond-mat 0106308. Application to Pu, S.Savrasov GK and E. Abrahams Nature 410, 793 (2001). Fe and Ni A. Lichtenstein M. Katsnelson and GK (PRL in press).


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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 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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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, NRG,ED….)Analytical Methods

G0 G

Im puritySo lver

S .C .C .


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Comments on the DMFT construction Exact in large dimensions [Metzner and Vollhardt 89]

Trick to sum all LOCAL skeleton graphs, [Muller Hartman 89].

Can be used for susceptibilities, ordered states etc.. Non perturbative construction, works even when

skeleton expansion fails.

†

,

1 1~ ~ (1)ij i j

j

t c c d Od d

1

~ d ij ijtd

1( , )

( )k

G k ii i


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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.

Solution of this problem should lead to advances in electronic structure theory (LDA +DMFT)


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A time-honored example: Mott transition in V2O3 under pressure or chemical substitution on V-site


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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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Mott transition in layered organic conductors Ito et al. (1986) Kanoda (1987) Lefebvre et al. (2001)


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


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Insights into the Mott phenomena

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 et.al.PRL 84, 5180 (2000))


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Expt. Ni Se S Matsuura et. Al.


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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 the compressibility ,in particle hole asymmetric situations e.g. Furukawa and Imada


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Phase diagram 1 band model


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Divergent compressibility U=2.4


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Compressibility QMC two band model, U=3


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Mott transition endpoint

Rapid variation has been observed in optical measurements in vanadium oxide (Thomas) and Ni mixtures(Miyasaka and Takgai)

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.

condensation of doubly occupied sites and onset of coherence .


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Optical Conductivty Miyasaka Takagi (2000)


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Insights from DMFT: think in term of spectral functions , the density is not changing!

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


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


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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+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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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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LDA+DMFTConnection with atomic limit

1[ ] [ ] [ ] logat atG W Tr G Tr G TrG 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 [ [ ]]atWG G

Weiss field


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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


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Comments on LDA+DMFT• Static limit of the functional 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 References

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 and GK full self consistent implementation cond-mat 0106308. Application to Pu, S.Savrasov GK and E. Abrahams

Nature 410, 793 (2001)


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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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Case study Fe and Ni

Archetypical itinerant ferromagnets LSDA predicts correct low T moment Band picture holds at low T


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Iron and Nickel: crossover to a real space picture at high T


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Photoemission Spectra and Spin Autocorrelation: Fe (U=2, J=.9ev,T/Tc=.8) (Lichtenstein, Katsenelson,GK prl 2001)


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


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Iron and Nickel:magnetic properties (Lichtenstein, Katsenelson,GK cond-mat 0102297)

0 3( )q

Meff

T Tc

0 3( )q

Meff

T Tc


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Ni and Fe: theory vs exp ( T=.9 Tc)/ ordered moment

Fe 1.5 ( theory) 1.55 (expt) Ni .3 (theory) .35 (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)


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Fe and Ni 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, it is about twice as large in Ni and in Fe

Mean field calculations using measured exchange constants(Kudrnovski Drachl PRB 2001) right Tc for Ni but overestimates Fe , RPA corrections reduce Tc of Ni by 10% and Tc of Fe by 50%.


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


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Small amounts of Ga stabilize the phase


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Delocalization-Localization across the actinide series

o f electrons in Th Pr U Np are itinerant . From Am on they are localized. Pu is at the boundary.

o Pu has a simple cubic fcc structure,the phase which is easily stabilized over a wide region in the T,p phase diagram.

o The phase is non magnetic.o Many LDA , GGA studies ( Soderlind et. Al 1990,

Kollar et.al 1997, Boettger et.al 1998, Wills et.al. 1999) give an equilibrium volume of the an equilibrium volume of the phasephaseIs 35% lower than experimentIs 35% lower than experiment

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


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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% Is 35% lower than experimentlower than experiment

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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Pu: DMFT total energy vs Volume


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Lda vs Exp Spectra


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


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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 simple elements


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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:

DCA (M. Jarrell et.al) , CDMFT ( Savrasov GK Palsson and Biroli )

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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Outlook

Extensions of DMFT implemented on model systems, (e.g. Motome and GK ) carry over to more realistic framework. Better determination of Tcs.

First principles approach: determination of the Hubbard parameters, and the double counting corrections long range coulomb interactions E-DMFT

Improvement in the treatement of multiplet effects in the impurity solvers, phonon entropies, ………


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. ARPES measurements on NiS2-xSex

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

gabriel kotliar physics department and center for materials theory rutgers university

Documents

mott phenomenaevolution

ising mott endpoint

band structure

finite t mott endpoint

band theory

band limit

dmft constructionexact

dmft review