Computational Studies of Strongly Correlated Materials Using Dynamical Mean Field Theory Gabriel Kotliar Center for Materials Theory Rutgers University

Computational Studies of Strongly Correlated Materials Using Dynamical Mean Field Theory

Gabriel Kotliar

Center for Materials Theory

Rutgers University

Research Support: NSF DMR 0096462 ,DOE DE-FG02-99ER45761 ,ONR N00014-21-0766

THE STATE UNIVERSITY OF NEW JERSEY

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Outline, Collaborators, References Introduction to extensions of DMFT for applications to

electronic structure. [ S. Savrasov and GK cond-matt0308053]

C-DMFTstudy of the Mott transition. [O. Parcollet G. Biroli and GK cond-mat 0308577 ]

Applications to materials: MIT in Ti2O3[S. Poteryaev S. Lichtenstein and GK cond-mat 0311319 ]

Delta –Epsilon transition in Plutonium [Xi Dai S. Savrasov GK A Migliori H. Ledbetter E. Abrahams Science 300, 953 (2003)]

Outlook.


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DMFT Cavity Construction: A. Georges and G. Kotliar PRB 45, 6479 (1992). Figure adapted from : Gabriel Kotliar and Dieter Vollhardt Physics Today 57,(2004)http://www.physics.rutgers.edu/~kotliar/RI_gen.html

http://www.physics.rutgers.edu/~kotliar/RI_gen.html

http://www.physics.rutgers.edu/~kotliar/RI_gen.html


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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

EDMFT [H. Kajueter Rutgers Ph.D Thesis 1995 Si and Smith PRL77, 3391(1996) R. Chitra and G. Kotliar PRL84,3678 (2000)]

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 oc Go c n n Ub b

s st t t t d t t ¯+òò

† †

, ,

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

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

0 0( , ')Do n nt t+


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Site Cell. Cellular DMFT. C-DMFT. G. Kotliar,S.. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001)

tˆ(K) hopping expressed in the superlattice notations.

•Other cluster extensions (DCA Jarrell Krishnamurthy, Katsnelson and Lichtenstein periodized scheme, Nested

Cluster Schemes Schiller Ingersent ), causality issues, O. Parcollet, G. Biroli and GK cond-matt 0307587 (2003)


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Two paths for ab-initio 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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LDA+DMFT V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997). A Lichtenstein and M. Katsnelson PRB 57, 6884 (1988).

The light, SP (or SPD) electrons are extended, well described by LDA .The heavy, D (or F) electrons are localized treat by DMFT.

LDA Kohn Sham Hamiltonian already contains an average interaction of the heavy electrons, subtract this out by shifting the heavy level (double counting term)

Kinetic energy is provided by the Kohn Sham Hamiltonian (sometimes after downfolding ). The U matrix can be estimated from first principles of viewed as parameters. Solve resulting model using DMFT.


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Functional formulation. Chitra and Kotliar (2001), Ambladah et. al. (1999) Savrasov and Kotliarcond- matt0308053 (2003).

1 †1( ) ( , ') ( ') ( ) ( ) ( )

2Cx V x x x i x x xff f y y-+ +òò ò

†( ') ( ')G R Ry r y r=- < > ( ') ( ) ( ') ( )R R R R Wf r f r f r f r< >- < >< >=Ir>=|R, >

[ , ] [ , , 0, 0]EDMFT loc loc nonloc nonlocG W G W G W

1 1 1 10

1 1[ , ] [ ] [ ] [ , ]

2 2 C hartreeG W TrLnG Tr G G G TrLnW Tr V W W E G W

Double loop in Gloc and Wloc


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Next Step: GW+EDMFT S. Savrasov and GK.(2001). P.Sun and GK. (2002). S. Biermann F. Aersetiwan and A.Georges . (2002). P Sun and G.K (2003)

Implementation in the context of a model Hamiltonian with short range interactions.P Sun and G. Kotliar cond-matt 0312303 or with a static U on heavy electrons, without self consistency. Biermann et.al. PRL 90,086402 (2003)

W

W


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Self-Consistency loop. S. Savrasov and G. Kotliar (2001) and

cond-matt 0308053

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

U

E

0( , , )HHi

HH

i

n T G r r i e w

w

w+

= å


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How good is this approximation ? It becomes exact as the coordination

number increases or in the limit of infinite dimensions introduced by Metzner and Vollhardt. PRL 62,34, (1989).

How good is it in low dimensions ? Promising recent developments from theory and experiments.


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Impurity Solvers.

Hubbard I. Quantum Montecarlo. Rational Approximations to the self energy,

constructed with slave bosons. cond-mat/0401539 V. Oudovenko, K. Haule, S. Savrasov D. Villani and G. Kotliar.

Extensions of NCA. Th. Pruschke and N. Grewe, Z. Phys. B: Condens. Matter 74, 439, 1989. SUNCA K. Haule, S. Kirchner, J. Kroha, and P. W¨olfle, Phys. Rev. B 64, 155111, (2001). K. Haule et. al. (2004)


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One dimensional Hubbard model .Compare 2 site cluster (in exact diag with Nb=8) vs exact Bethe Anzats,[V. Kancharla C. Bolech and GK PRB 67, 075110 (2003)][ [M. Capone M.Civelli V Kancharla C.Castellani and GK cond-mat 0401060] See presentation S20.012 at 16:42.

U/t=4.


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Schematic DMFT phase diagram and DOS of a partially frustrated integer filled Hubbard model and pressure driven Mott transition.

S Lefebvre et al. PRL (2000)


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Recent Experiments support qualitative single site DMFT predictions

Mo et al., Phys. Rev.Lett. 90, 186403 (2003).

Limelette et. al.(2003)

Ito et. al. (1995)


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Theoretical issue: is there a Mott transitionin the integer filled Hubbard model, and is it well described by the single site DMFT ?

Study frustrated t t’ model t’/t=.9

YES! Parcollet Biroli Kotliar cond-matt 0308577


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Evolution of the k resolved Spectral Function at zero frequency.

Uc=2.35+-.05, Tc/D=1/44

U/D=2 U/D=2.25

( 0, )vs k A k

Qualitative effect, formation of hot regions! D wave gapping of the single particle spectra

as the Mott transition is approached. Very strong k dependece near the trasition.


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Ti2O3 : Coulomb or Pauling

C.E.Riceet all, Acta CrystB33,1342(1977) LTS 250 K, HTS 750 K.


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Ti2O3.

Isostructural to V2-xCrxO3. Al lot of the qualitative physics of the high temperature part of the phase diagram of V2O3 can be understood within single site DMFT. Is this true in Ti2O3?

Band Structure Calculations good metal. L.F. Mattheiss, J. Phys.: Condens. Matter 8, 5987 (1996) .Unrestricted Hartree Fock calculations produce large antiferromagnetic gap. M. Cati, et. al. Phys. Rev. B. f55 , 16122 (1997).


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U = 2, J = 0.5, W = 0.5β = 20 eV-1, LT structure

U = 2, J = 0.5, W = 0.5 β = 10 eV-1, HT

structure

2site-Cluster DMFT with intersite Coulomb2site-Cluster DMFT with intersite Coulomb

A. Poteryaev


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Pauling and Coulomb Ti2O3[S. Poteryaev S. Lichtenstein and GK cond-mat 0311319 ]

Dynamical Goodenough-Honing picture


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Conclusion Ti2O3

2 site cluster DMFT describes the MIT in Ti2O3. Different from V2O3 where single site DMFT works

well, and cluster corrections are small [A. Poteryaev] It requires the Coulomb interactions, and a frequency

dependent enhancement of the a1g-a1g hopping, induced by the Coulomb interactions. [Haldane Ph.D thesis, Q Si and GK 1993 ].Dynamical Pauling-Goodenough mechanism is able to trigger the MIT at low enough temperatures.

Coulomb and Pauling synergistically cooperate.


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Pu phases: A. Lawson Los Alamos Science 26, (2000)

LDA underestimates the volume of fcc Pu by 30%

Predicts magnetism in d Pu and gives negative shear

Core-like f electrons overestimates the volume by 30 %


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The delta –epsilon transition

The high temperature phase, (epsilon) is body centered cubic, and has a smaller volume than the (fcc) delta phase.

What drives this phase transition?

Having a functional, that computes total energies opens the way to the computation of phonon frequencies in correlated materials (S. Savrasov and G. Kotliar 2002)

TotalEnergyasafunctionofvolumeforTotalEnergyasafunctionofvolumeforPuPu

(after Savrasov, Kotliar, Abrahams, 2001,410,793, 2001)

DMFTPhononsinfccDMFTPhononsinfcc-Pu-Pu

C11 (GPa) C44 (GPa) C12 (GPa) C'(GPa)

Theory 34.56 33.03 26.81 3.88

Experiment 36.28 33.59 26.73 4.78

(after Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003)

(experiments from Wong et.al, Science, 22 August 2003)

DMFTPhononsinbccDMFTPhononsinbcc-Pu-Pu


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Phonon entropy drives the epsilon delta phase transition

Epsilon is slightly more delocalized than delta, has SMALLER volume and lies at HIGHER energy than delta at T=0. But it has a much larger phonon entropy than delta.

Different from Cerium see Jeong et. al. cond-mat/0308416

At the phase transition the volume shrinks but the phonon entropy increases.

Estimates of the phase transition following Drumont and Ackland et. al. PRB.65, 184104 (2002); (and neglecting electronic entropy). TC ~ 600 K.


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Outlook Dynamical mean field theory,” local reference “ for

correlated electron systems. Analogy to FLT, DFT. The need of simpler

reference frames for thinking about complex problems.

Future directions: downfolding and RG, algorithmic speedups.

While a general method is under construction, the extensions described in this talk, already allow to perform quantitative calculations and obtain quantitative insights.


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Conclusion

Introduction to DMFT and its extensions. Flexibility of a local approach.

DMFT describes well the Mott transitions. Formation of hot and cold regions near FS.

MIT in Ti2O3 cluster DMFT .Dynamical Pauling-Coulomb mechanism.

Delta-Epsilon Plutonium. Correlations and phonon entropy.


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Alpha Gamma Cerium

B. Johansson, Philos. Mag. 30, 469 (1974). Mott transition of the f electrons as a function of pressure. Ce alpha gamma transition. spd electrons are spectators..

J.W. Allen and R.M. Martin, Phys. Rev. Lett. 49, 1106 (1982); Kondo volume collapse picture. The dominant effect is the spd-f hybridization. J.W. Allen and L.Z. Liu, Phys. Rev. B 46, 5047 (1992).


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LDA+DMFT:Ce M..Z¨olfl,I.A.NekrasovTh.Pruschke,V.I.Anisimov J. Keller,Phys.Rev. Lett 87, 276403 (2001). K. Held, A.K. McMahan, and R.T. Scalettar, Phys. Rev.Lett. 87, 276404 (2001) A.K.McMahan,K.Held,andR.T.Scalettar,Phys Rev. B 67, 075108 (2003).

1. Successful calculations of thermodynamics.

2. Mott transition and Kondo collapse give rise to similar spectra and phase diagram.


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To resolve the conflict between the Mott transition and the volume collapse picture : Turn to Optics! Haule et.al.

Qualitative idea. The spd electrons have much larger velocities, so optics will be much more sensitive to their behavior.

See if they are simple spectators (Mott transition picture ) or wether a Kondo binding unbinding takes pace (Kondo collapse picture).

General method, bulk probe.


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Optics formula

double poledouble pole

single pole

One divergence integrated out!


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The volume of alpha is 28.06°A and the temperature 580K. The volume of the gamma phase is34.37°A and T = 1160K. Experiments : alpha at 5 K and gamma phase at 300 K.

Theory: Haule et. al. cond-matt 04Expt: J.W. vanderEb PRL 886,3407 (2001)


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Temperature dependence of the optical conductivity.


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Origin of the features.


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Conclusion: Cerium

Qualitatively good agreement with existing experiment.

Some quantitative disagreement . Experiments should study the temperature

dependence of the optics. Optics + Theory can provide a simple

resolution of the Mott vs K-Collapse conundrum.


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Benchmarking SUNCA, V. Udovenko and K. Haule


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Interpolative scheme with slave bosons.


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Photoemission&experiment

•A. Mc Mahan K Held and R. Scalettar (2002)

•Zoffl et. al (2002)

•K. Haule V. Udovenko S. Savrasov and GK. (2004)


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Application to Materials

Ti2O3-V2O3 Cerium: Alpha to Gamma Transition. Plutonium : Alpha-Delta-Epsilon.


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

Documents

Computational Studies of Strongly Correlated Materials Using Dynamical Mean Field Theory Gabriel Kotliar Center for Materials Theory Rutgers University