Dynamical Mean Field Theory for Electronic Structure Calculations Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University

Dynamical Mean Field Theory for Electronic Structure Calculations

Gabriel Kotliar

Physics Department and

Center for Materials Theory

Rutgers University

CECAM workshop on realistic studies of correlations

Lyon July 25-29th 2001

THE STATE UNIVERSITY OF NEW JERSEY

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Outline Choice of Basis. Realistic self consistency

condition Integration with LDA. Effective

action formulation. Comparison with LDA and LDA+U

Some examples in real materials, transition metals and actinides.


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Acknowledgements: Collaborators, Colleagues, Support …..

S. Lichtenstein (Nijmeigen), E Abrahams (Rutgers)

G. Biroli (Rutgers), R. Chitra (Rutgers-Jussieux), V. Udovenko (Rutgers), S. Savrasov (Rutgers-NJIT)

G. Palsson, I. Yang (Rutgers) NSF, DOE and ONR


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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 construction: A. Georges, G. Kotliar, PRB, (1991)]

†

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 C-DMFT.

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 and G. Palsson)•Improved estimators for the lattice self energy are available (Biroli and Kotliar)•In simple cases C-DMFT converges faster than other causal cluster schemes.


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Realistic DMFT self consistency 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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Solving the DMFT equations

G0 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….)•Semi-analytical Methods

G0 G

Im puritySo lver

S .C .C .


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DMFT+QMC (A. Lichtenstein, M. Rozenberg)


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


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


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Case study: IPT half filled Hubbard one band

(Uc1)exact = 2.1 (Exact diag, Rozenberg, Kajueter, Kotliar 1995) , (Uc1)IPT =2.4

(Uc2)exact =2.95 (Projective self consistent method, Moeller Si Rozenberg Kotliar PRL 1995 ) (Uc2)IPT =3.3

(TMIT ) exact =.026+_ .004 (QMC Rozenberg Chitra and Kotliar PRL 1999), (TMIT )IPT =.5

(UMIT )exact =2.38 +- .03 (QMC Rozenberg Chitra and Kotliar PRL 1991), (UMIT )IPT =2.5 For realistic studies errors due to other sources (for example the value of U, are at least of the same order of magnitude).


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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


•H and S, 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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Solving the impurity Multiorbital situation and several

atoms per unit cell considerably increase the size of the space H (of heavy electrons).

QMC scales as [N(N-1)/2]^3 N dimension of H

Fast interpolation schemes (Slave Boson at low frequency, Roth method at high frequency, + 1st mode coupling correction), match at intermediate frequencies. (Savrasov et.al 2001)


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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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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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LDA+U functional

2 *log[ / 2 . ( ) ( )]

( ) ( ) ( ) ( )

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

2 | ' |

[ ]

aR bR

n

KS abn KS

R

KS KS

i

LDAext xc

DC

R

Tr i V B r r

V r r dr B r m r dr Tr n

r rV r r dr drdr E

r r

G

w

w s fl f

r l

r rr r

- +Ñ - - - -

- - - +

+ + +-

F - F

å

åò ò

ò òå

1[ ] ( 1)

2DC G Un nF = - ( )0( ) iab

abi

n T G i ew

w+

= å

[ ( ), ( ), ]LDA U abr m r n

, KS KS ab [ ( ), ( ), V ( ), ( ), ]LDA U a br m r n r B r

1

2 ab abcd cdn U n


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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 functional

2 *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

G1

[ ] ( 1)2DC G Un nF = - ( )0( ) i

ab

abi

n T G i ew

w+

= å

KS KS ab [ ( ) ( ) G V ( ) ( ) ]LDA DMFT a br m r r B 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+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 [ [ ]]atW

G G

Weiss field


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LDA+DMFT Self-Consistency loop

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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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



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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(1988).

S.SavrasovandG.Kotliar,funcionalformulationforfullselfconsistentimplementation(2001)


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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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Pu: Anomalous thermal expansion (J. Smith LANL)


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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 Is 35% lower than experimentexperiment

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% lower than Is 35% lower than experimentexperiment

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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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.


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Problems with the conventional viewpoint of Pu

The specific heat of delta Pu, is only twice as big as that of alpha Pu.

The susceptibility of alpha Pu is in fact larger than that of delta Pu.

The resistivity of alpha Pu is comparable to that of delta Pu.

Only the structural and elastic properties are completely different.


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Pu Specific Heat


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Anomalous ResistivityJ. Smith LANL


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MAGNETIC SUSCEPTIBILITY


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Dynamical Mean Field View of Pu(Savrasov Kotliar and Abrahams, Nature 2001)

Delta and Alpha Pu are both strongly correlated, the DMFT mean field free energy has a double well structure, for the same value of U. One where the f electron is a bit more localized (delta) than in the other (alpha).

Is the natural consequence of the model hamiltonian phase diagram once electronic structure is about to vary.

This result resolves one of the basic paradoxes in the physics of Pu.


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Pu: DMFT total energy vs Volume


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Lda vs Exp Spectra

DO

S, s

t./[e

V*c

ell]


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


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Earlier Studies of Magnetic Anisotropy

Erickson Daalderop


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Case study Fe and Ni Archetypical itinerant ferromagnets LSDA predicts correct low T

moment Band picture holds at low T Main challenge, finite T properties

(Lichtenstein’s talk). Magnetic anisotropy puzzle. LDA

predicts the incorrect easy axis for Nickel .

LDA Fermi surface has features which are not seen in DeHaas Van Alphen ( Lonzarich)


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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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Ni moment


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Fe moment\


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Magnetic anisotropy Ni


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Magnetic anisotropy Fe


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Magnetic anisotropy


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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


RUTGERS

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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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 and 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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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, ………

Documents

Dynamical Mean Field Theory for Electronic Structure Calculations Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University