dynamical mean field theory and electronic structure calculations gabriel kotliar center for...

Dynamical Mean Field Theory and Electronic Structure Calculations

Gabriel Kotliar

Center for Materials Theory

Rutgers University

THE STATE UNIVERSITY OF NEW JERSEY

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Outline

Incorporating electronic structure methods in DMFT. C-DMFT. [M. Capone, M. Civelli ]

Why do we need k-sum to do optics. Cerium puzzles. [K. Haule V. Udovenko ] Why do we need functionals to do total energies. Phonons and plutonium puzzles.

[X. Dai S. Savrasov ]

Physics Today Vol 57, 53 (2004) Gabriel Kotliar and Dieter Vollhardt


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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

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

The U matrix can be estimated from first principles of viewed as parameters. Solve resulting model using DMFT.


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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

Single site DMFT Impurity cavity construction: A. Georges, G. Kotliar, PRB 45, 6497 (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 ( )( )[ ] ( ) [ ( ) ( ) ]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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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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Realistic DMFT loop: matrix inversion-tetrahedron method

( )k LMTOt H k E® - LMTO

LL LH

HL HH

H HH

H H

é ùê ú=ê úë û

ki i Ow w®1

0 n HHiG 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

† † †0

0 0

( ) ( , ') ( ') a ab b abdc a b c dc G c U c c c cb b

t t t t +òò


RUTGERS

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

tˆ(K) is the hopping expressed in the superlattice notations.

•Other cluster extensions (DCA, Katsnelson and Lichtenstein periodized scheme, nested cluster schemes, PCMDFT ),

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


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N vs mu in one dimensional Hubbard model .Compare 2 site cluster (in exact diag with Nb=8) vs exact Bethe Anzats, [M. Capone M.Civelli C. Castellani V Kancharla and GK 2004]


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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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Spectral Density Functional : Effective action construction R. Chitra G. Kotliar PRB 62,12715. Kotliar Savrasov in New Theoretical Approaches to Strongly Correlated Systems, A. M. Tsvelik ed. (2001) Kluwer Academic Publishers. 259-301; cond-mat/0208241. S Savrasov G Kotliar cond-mat0308053.

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)]

Allows computation of total energy, phonons!!!!

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


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LDA+DMFT Self-Consistency loop. See also S. Savrasov and G.

Kotliar 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


RUTGERS

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).


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Application to Materials

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


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Overview

Various phases :

isostructural phase transition (T=298K, P=0.7GPa)

(fcc) phase

[ magnetic moment

(Curie-Wiess law) ]

(fcc) phase

[ loss of magnetic

moment (Pauli-para) ]

with large

volume collapse

v/v 15

( -phase a 5.16 Å

-phase a 4.8 Å)

volumes exp. LDA LDA+U

28Å3 24.7Å3

34.4Å3 35.2Å3

-phase (localized): High T phaseCurie-Weiss law (localized magnetic moment),Large lattice constantTk around 60-80K

-phase (localized): High T phaseCurie-Weiss law (localized magnetic moment),Large lattice constantTk around 60-80K

-phase (delocalized:Kondo-

physics): Low T phaseLoss of Magnetism (Fermi liquid Pauli susceptibility) - completely screened magnetic momentsmaller lattice constantTk around 1000-2000K

-phase (delocalized:Kondo-

physics): Low T phaseLoss of Magnetism (Fermi liquid Pauli susceptibility) - completely screened magnetic momentsmaller lattice constantTk around 1000-2000K


RUTGERS

Qualitative Ideas.

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.

Mathematical implementation, “metallic phase” treat spdf electrons by LDA, “insulating phase” put f electron in the core.

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.


RUTGERS

Qualitative Ideas

alpha phase Kondo effect between spd and f takes place. “insulating phase” no Kondo effect (low Kondo temperature).

Mathematical implementation, Anderson impurity model in the suplemented with elastic terms. (precursor of realistic DMFT ideas, but without self consistency condition). J.W. Allen and L.Z. Liu, Phys. Rev. B 46, 5047 (1992).


RUTGERS

LDA+DMFT:Ce spectra

M.B.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).

Successful calculations of thermodynamics.


RUTGERS

X.Zhang M. Rozenberg G. Kotliar (PRL 70, 1666(1993)).

Unfortunately photoemission cannot decide between the Kondo collapse picture and the Mott transition picture.Evolution of the spectra as a function of U , half filling full frustration, Hubbard model!!!!


RUTGERS

The schematic phase diagram of cannot distinguish between the two scenarios. J.W. Allen and L.Z. Liu, Phys. Rev.

B 46, 5047 (1992). Kondo impurity model + elastic terms.

DMFT phase diagram of a Hubbard model at integer filling, has a region between Uc1(T) and Uc2(T) where two solutions coexist. A. Georges G. Kotliar W. Krauth and M Rozenberg RMP 68,13,(1996).

Coupling the two solutions to the lattice gives a phase diagram akin to alpha gamma cerium. Majumdar and Krishnamurthy PRL 73 (1994).


RUTGERS

Photoemission&experiment

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

•Zoffl et. al (2002)

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


RUTGERS

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 senstive 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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Temperature dependence of the optical conductivity.


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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)


RUTGERS

Optical conductivity of Ce (expt. Van Der Eb et.al. theory Haule et.al)

experiment

LDA+DMFT •K. Haule et.al.


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Origin of the features.


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Conclusion: Cerium

Qualitatively good agreement with existing experiment.

Some quantitative disagreement, see however .

Experiments should study the temperature dependence of the optics.

Optics + Theory can provide a simple resolution of the Mott vs K-Collapse conundrum.


RUTGERS

Phases of Pu (A. Lawson LANL) Los Alamos Science 26, (2000)


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Delta phase of Plutonium: Problems with LDA

o Many studies and implementations.(Freeman, Koelling 1972)APW methods, ASA and FP-LMTO Soderlind et. Al 1990, Kollar et.al 1997, Boettger et.al 1998, Wills et.al. 1999).all give an equilibrium an equilibrium volume of the volume of the phasephaseIs 35% lower than Is 35% lower than experiment experiment this is the largest discrepancy ever known in DFT based calculations.

LSDA predicts magnetic long range (Solovyev et.al.) Experimentally Pu is not magnetic.

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


RUTGERS

Pu: DMFT total energy vs Volume (Savrasov Kotliar and Abrahams 2001)


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DMFT studies of Pu.

Savrasov, S. Y., and G. Kotliar, 2003, Phys. Rev. Lett. 90(5), 056401/1.

Savrasov, S. Y., and G. Kotliar, 2003, cond-mat/0308053 .

Savrasov, S. Y., G. Kotliar, and E. Abrahams, 2001, Nature 410, 793

Dai X. Savrasov S.Y. Kotliar G. Migliori A. Letbetter H, Abrahams A. Science 300, 953, (2003)


RUTGERS

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

The shear moduli in the delta phase were calculated within LDA and GGA by Bouchet et. al. (2000) and c’ is negative!

.


RUTGERS

X.Zhang M. Rozenberg G. Kotliar (PRL 70, 1666(1993)).

Evolution of the spectra as a function of U , half filling full frustration.


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Alpha and delta Pu : Expt. Arko et.al. PRB 62, 1773 (2000). DMFT: Savrasov and Kotliar


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Phonon freq (THz) vs q in delta Pu X. Dai et. al. Science vol 300, 953, 2003


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Expts’ Wong et. al. Science 301. 1078 (2003) Theory Dai et. al. Science 300, 953, (2003)


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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)


RUTGERS

Phases of Pu (A. Lawson LANL) Los Alamos Science 26, (2000)


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Epsilon Plutonium.


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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.

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.


RUTGERS

Phonons epsilon


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Summary

Incorporating electronic structure methods in DMFT. C-DMFT. [M. Capone, M. Civelli ]

Why do we need k-sum to do optics. Cerium puzzles. [K. Haule V. Udovenko ] Why do we need functionals to do total energies. Phonons and plutonium puzzles.

[X. Dai S. Savrasov ]


RUTGERS


RUTGERS

Why is optics calculation not completely trivial?

Analytic tetrahedron method:

Integral is analytic and simple

(combination of logarithms)

Energies linearly interpolated no simple analytic expression

Product of two energies linearly interpolated ATM applicable

but numerically very unstable because of quadratic pole

1D example: Parabola has 2 zeros (2poles)

Line has no zeros (no poles)


RUTGERS

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


RUTGERS

Schematic DMFT phase diagram one band Hubbard model. Rozenberg et. al. 1996. Introduce coupling to the lattice will cause a volume jump across the first order transition. (Majumdar and Krishnamurthy ).


RUTGERS

Shear anisotropy. Expt. vs Theory

C’=(C11-C12)/2 = 4.78 GPa C’=3.9 GPa

C44= 33.59 GPa C44=33.0 GPa

C44/C’ ~ 7 Largest shear anisotropy in any element!

C44/C’ ~ 8.4


RUTGERS

Benchmarking SUNCA, V. Udovenko and K. Haule

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