Electronic Structure of Correlated Materials : a DMFT Perspective Gabriel Kotliar Physics Department...

Electronic Structure of Correlated Materials : a DMFT Perspective

Gabriel Kotliar

Physics Department and

Center for Materials Theory

Rutgers University

In Realistic Theories , GRC on Correlated Electrons.

June 29-July 3rd 2002

Supported by the NSF DMR 0096462

THE STATE UNIVERSITY OF NEW JERSEY

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Outline

Introduction Basic ideas of Dynamical Mean Field Theory and

its extensions. Qualitative successes of DMFT. Realistic implementation of DMFT. Illustrations: NiO (with S. Savrasov) Fe and Ni (with Lichtenstein and Katsnelson) Outlook.


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Weakly correlated electrons:band theory. Simple conceptual picture of the ground

state, excitation spectra, transport properties of many systems (simple metals, semiconductors,….).

A methods for performing quantitative calculations. (Density functional theory, in various approximations).

Success story : Density Functional Linear Success story : Density Functional Linear ResponseResponse

Tremendous progress in ab initio modelling of lattice dynamics& electron-phonon interactions has been achieved(Review: Baroni et.al, Rev. Mod. Phys, 73, 515, 2001)

(Savrasov, PRB 1996)


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Limitations of LDA

LDA spectra can be taken a starting point for perturbative (eg. GW ) calculations of excitation spectra and transport.

THIS DOES NOT WORK for strongly correlated systems, eg oxides containing 3d, 4f, 5f elements. CHARACTER OF THE EXCITATION SPECTRA is not captured by LDA.

LDA does not have good predicted power for ground state properties in this system either.


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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). Standard approaches (LDA, HF ) do not work well.•Dynamical Mean Field Theory. Treats atoms and bands. Treats QP bands and Hubbard bands. Exact in large dimensionality.


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Strongly correlated electrons

Large degeneracy. Low energy scales. Many Competing forms of long range order (various

forms of charge-spin-orbital and even currents) Quasidegenerate ground states, with different

forms of magnetic order. Competition between different possible states,

frustration, phase separation. Tunability. Intricate microsctrucure. Mesoscale.


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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 Impurity cavity construction: A. Georges, G. Kotliar, PRB, (1992)]

†

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 0 0 ( )( )[ ] ( ) [ ( ) ( ) ]n n n n S Gi G G i c i c iw w w w- -S = + á ñ

Weiss field


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1

10

1( ) ( )

( )n nn k nk

G i ii t i

w ww m w

-


DMFT Impurity cavity construction

1

10

1( ) ( )

V ( )n nk nk

D i ii

w ww

-

-é ùê ú= +Pê ú- Pê úë ûå 0

1 † 10 0 ( )( )[ ] ( ) [ ( ) ( ) ]n n n n S Gi G G i c i c ia bw w w w- -S = + á ñ

†

0 0

( ) ( , ') ( ') ( , ') o o o o o oc Go c n n U n nb b

s st t t t d t t ¯ ¯+òò

† †

, ,


t c c c c U n n

()

1 100 0 0( )[ ] ( ) [ ( ) ( ) ]n n n n Si G D i n i n iw w w w- -P = + á ñ

,ij i j

i j

V n n

( , ')Do t t+


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C-DMFT

C:DMFT The lattice self energy is inferred from the cluster self energy.

0 0cG G ab¾¾® c

abS ¾¾®Sij ijt tab¾¾®


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C-DMFT: test in one dimension. (Bolech, Kancharla GK cond-mat 2002)

Gap vs U, Exact solution Lieb and Wu, Ovshinikov

Nc=2 CDMFT

vs Nc=1


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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 puritySo lver

S .C .C .


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DMFT: Methods of Solution


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A (non comprehensive )list of extensions of DMFT Two impurity method. [A. Georges and G. Kotliar,

A. Schiller K Ingersent ] Bethe Peirels [ A. Georges and G. Kotliar] Dynamical Cluster Approximation [M. Jarrell et.

Al. Phys. Rev. B 7475 1998] Periodic cluster [Lichtenstein and Katsnelson]. Cellular Dynamical Mean Field Theory [G. Kotliar

et.al] Extended DMFT [Sachdev and Ye, Parcollet and

Georges, H. Kajueter and G. Kotliar, Q. Si and J L Smith , R . Chitra and G. Kotliar]

Combination with lowest order Perturbation theory for the light orbitals [Savrasov Kotliar, Ping Sun]


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Insights from model DMFT studies Canonical phase diagram at integer occupation. Low temperature Ordered phases . Stability depends on details (chemistry and crystal structure)High temperature behavior around Mott endpoint, more universal regime, captured by simple models treated within DMFT. Role of magnetic frustration.


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Schematic DMFT phase diagram Hubbard model (partial frustration)

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


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Kuwamoto Honig and AppellPRB (1980)


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Mott transition in pyrites: NiSe2-xSx Miyasaka and

Takagi (2000)


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


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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…


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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)


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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)


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


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General Lessons

Coherence –incoherence crossover. Mott transition in one band model. Transfer of spectral weight. Coexistence of atomic like and band like

excitations at finite temperatures. Anomalous transport. Simple laws for transfer of spectral weight

around special points.


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Realistic DMFT methods. Spectral functions. Finite temperatue. Excitations. Ground state properties are a byproduct of spectra. Can be computed more reliably being less sensitive

on long distance details. High temperature. NON PERTURBATIVE, using the Weiss field as a

variable one can cross the barrier where skeleton PT theory breaks down.


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Combine Dynamical Mean Field Theory with Electronic structure methods. Single site DMFT made correct qualitative

predictions. Make realistic by: Incorporating all the electrons. Add realistic orbital structure. U, J. Add realistic crystal structure. Allow the atoms to move.


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Two roads for ab-initio calculation of electronic structure of strongly correlated materials

Correlation Functions Total Energies etc.

Model Hamiltonian

Crystal structure +Atomic positions


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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).

Lichtenstein and Katsenelson PRB (1998) Savrasov Kotliar and Abrahams Nature 410, 793

(2001)) Kotliar, Savrasov, in Kotliar, Savrasov, in New Theoretical New Theoretical approaches to strongly correlated systemsapproaches to strongly correlated systems , , Edited by A. Tsvelik, Kluwer Publishers, 2001)Edited by A. Tsvelik, Kluwer Publishers, 2001)


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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 (Gunnarson Anisimov Hybertsen et.al) or viewed as parameters


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Density functional theory and Dynamical Mean Field Theory DFT: Static mean field, electrons in an

effective potential. Functional of the density.

DMFT: Promote the local (or a few quasilocal Greens functions or observables) to the basic quantities of the theory.

Express the free energy as a functional of those quasilocal quantities.


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

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 GFG(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, the DMFT +LDA functional. Savrasov Kotliar and Abrahams Nature 410, 793 (2001))

Full self consistent implementation.

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


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LDA+DMFT-outer loop relax

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

Edc

0( , , )HHi

HH

i

n T G r r i e w

w

w+

= å

ff &


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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 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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Very Partial list of application of realistic DMFT to materials QP bands in ruthenides: A. Liebsch et al (PRL 2000) phase of Pu: S. Savrasov et al (Nature 2001) MIT in V2O3: K. Held et al (PRL 2001) Magnetism of Fe, Ni: A. Lichtenstein et al PRL (2001) transition in Ce: K. Held et al (PRL 2000); M. Zolfl et al

PRL (2000). 3d doped Mott insulator La1-xSrxTiO3 (Anisimov et.al

1997, Nekrasov et.al. 1999, Udovenko et.al 2002) ………………..

Failures of lda Failures of lda

NiO dielectric constant. LSDA:35.7 Exp:5.7

Lattice dynamics cannot be predicted:

• Optical G-phonon in MnO within LSDA: 3.04 THz, Experimentally: 7.86 THz (Massidda, et.al, PRL 1999)

• Bulk modulus for metallic Plutonium is one order of magnitude too large within LDA (214 GPa vs. 30 GPa) Also elastic constants are off.(Bouchet, et.al, J.Phys.C, 2001)


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Functional approach allows computation of linear response.(S. Savrasov and GK 2002

Apply to NiO, canonical Mott insulator.

U= J=.9

Simple Impurity solver Hubbard 1.

Results for NiO: PhononsResults for NiO: Phonons

Solid circles – theory, open circles – exp. (Roy et.al, 1976)


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NiO U=8ev, J=1ev, Savrasov and GK (2002)


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Case study Fe and Ni

Archetypical itinerant ferromagnets LSDA predicts correct low T moment Band picture holds at low T Curie behavior at high temperatures. Crossover between a real space and a

momentum space prediction.


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


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Photoemission Spectra and Spin Autocorrelation: Fe (U=2, J=.9ev,T/Tc=.8) (Lichtenstein, Katsenelson,Kotliar Phys Rev. Lett 87, 67205 , 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 PRL 01)

2

0 3( )eff

q

M

T Tc


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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)


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Role of theory

Orient the thinking about materials. Visualization. Generate

Refine questions, ask about deviation from

DMFT.


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Outlook. Many qualitative aspects of the Mott transition in

clusters need to be understood. The notions and the calculations of U’s need to be

refined a revisited. (E-DMFT). Replacing LDA part by simple low order diagrammatic scheme (local GW)

RG techniques and cluster impurity solvers. Small clusters may be needed for accurate

computations of critical temperatures. Role of long wavelength fluctuations?

Many materials to be studied, and insights to be gained.


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Acknowledgements: Development of DMFT

Collaborators: V. Anisimov, C. Bolech, G. Biroli, R. Chitra, V. Dobrosavlevic, D. Fisher, A. Georges, H. Kajueter, V. Kancharla, W.Krauth, E. Lange, A. Lichtenstein, G. Moeller, Y. Motome, G. Palsson, O. Parcollet, M. Rozenberg, S. Savrasov, Q. Si, V. Udovenko ,X.Y. Zhang

Support: National Science Foundation.

.


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Future Challenges

Develop user friendly interfaces, for first principles calculations of realistic DMFT, for visualization of spectra, resolved in real space, momentum space and orbital space. FAT DMFT. [Done for LDA, S. Savrasov, Material Information and DESIGN Laboratory] and for further code development.


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Outlook

The Strong Correlation Problem:How to deal with a multiplicity of competing low temperature phases and infrared trajectories which diverge in the IR

Strategy: advancing our understanding scale by scale

Generalized cluster methods to capture longer range magnetic correlations

New structures in k space. Cellular DMFT


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. ARPES measurements on NiS2-xSex

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


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Photoemission V2O3 Held et.al. PRL 2001


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V2O3 LDA+DMFT Held et.al. PRL 2001


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Outlook

Extensions of DMFT implemented on model systems, 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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Hubbard model

† †

, ,


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.


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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 k ii i

Muller-Hartmann 89


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Dynamical Mean Field Theory, cavity constructionA. Georges G. Kotliar 92

†

0 0 0

( )[ ( ')] ( ')eff o o o oc c U n nSb b b

s st m t t tt ¯

¶= + - D - +

¶òò ò ( )wD

†( )( ) ( )

MFo n o n SG c i c iw w D=- á ñ 1( )

1( )

( )[ ][ ]

nk

n kn

G ii

G i

ww e

w

=D - -

D

å

† †

, ,


t c c c c U n n


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Extended DMFT electron phonon


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Extended DMFT e.ph. Problem


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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

†( )( ) ( )

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


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1

10

1( ) ( )

( )n nn k nk

G i ii t i

w ww m w

-


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

† †

, ,


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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X.Zhang M. Rozenberg G. Kotliar (PRL 1993)

Spectral Evolution at T=0 half filling full frustration


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Anomalous Spectral Weight Transfer: Optics

0( ) ,eff effd P J

iV

Schlesinger et.al (FeSi) PRL 71 ,1748 , (1993) B Bucher et.al. Ce2Bi4Pt3PRL 72, 522 (1994), Rozenberg et.al. PRB 54, 8452, (1996).

2

0( ) ,

ned P J

iV m

ApreciableT dependence found.

, ,H hamiltonian J electric current P polarization

, ,eff eff effH J PBelow energy


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

reduces to LDA+U• DMFT retain correlations effects in the

absence of orbital ordering. • (for example treating the impurity model in the

Hubbard 1 approximation).• Functional formulation allows calculation of total

energies and linear response.• Gives the local spectra and the total energy

simultaneously, treating QP and H bands on the same footing.

Electronic Structure of Correlated Materials : a DMFT Perspective Gabriel Kotliar Physics Department...

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