Correlation Effects in Itinerant Magnets, Application of LDA+DMFT(Dynamical Mean Field

Theory) and its static limit the LDA+U method.

Gabriel Kotliar

Physics Department and

Center for Materials Theory

Rutgers University

A. Lichtenstein M. Katsnelson and G. Kotliar Phys. Rev Lett. 87, 067205 (2001)

I. Yang S. Savrasov and G. Kotliar Phys. Rev. Lett. 87, 216405 (2001)

I. Yang Rutgers Ph.D Thesis (Dec-2001)

Supported by the ONR

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The Strong Correlation ProblemTwo limiting cases of the electronic structure of

solids are understood:the high density limit and the limit of well separated atoms.

Many materials have electron states that are in between these two limiting situations and require the development of new electronic structure methods to predict some of its properties (spectra, energy, transport,….)

DMFT simplest many body technique which treats simultaneously the open shell atomic limit and the band limit .

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DMFT

Developed initially to treat correlation effects in model Hamiltonians.

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

Recently extended to perform first principles calculations. [V. Anisimov, A. Poteryaev, M. Korotin, Anokhin and G. Kotliar, J. Phys. Cond. Mat 9, 7359 (1997). S. Savrasov, G. Kotliar and E. Abrahams, Nature 410, 793 (2001). ]

Unlike DFT, DMFT computes both free energies and one electron (photoemission ) spectra and many other physical quantities at finite temperatures.

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

3( )

eff

T Tc

2

3( )

eff

T Tc

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Other aspects that require to treate correlations beyond LDA

Magnetic anisotropy. L.S effect. LDA predicts the incorrect easy axis(100) for Nickel .(instead of the correct one (111) (Savrasov’s talk)

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

High energy features in the photoemission spectra of Ni (6 ev satellite), 30% band narrowing, reduction of exchange splitting.

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LDA+DMFT Spectral Density Functional (Fukuda, Valiev and Fernando , Chitra and GK, Savrasov and GK).

DFT, consider the exact free energy as a functional of an external potential. Express the free energy as a functional of the local density by Legendre transformation.

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 double Legendre transformton

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

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Spectral Density Functional

Formal construction of a functional of the d spectral density

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

ATOM 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

Atom =Sum of local 2PI graphs build with local Coulomb interaction matrix, parametrized by Slater integrals F0, F2 and F4 and local G.

Express in terms of AIM model.

KS [ ( ) G( ) V ( ) ( ) ]LDA DMFT a b abn nr i r i

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Combining LDA and DMFT The light, SP electrons well described by LDA The heavier D electrons treat by model DMFT. LDA already contains an average interaction of the

heavy electrons, subtract this out by shifting the heavy level (double counting term, Lichtenstein et.al.)

Atomic physics parameters . U=F0 cost of double occupancy irrespectively of spin, J=F2+F4, Hunds energy favoring spin polarization .F2/F4=.6

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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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LDA+DMFT and LDA+U • Static limit of the LDA+DMFT functional ,

• with = HF reduces to the LDA+U functional

of Anisimov et.al.

Simple extension to magnetic case.

( )ab abni

( )0( ) iab ab

abi

n T G i ew

w+

= å

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DMFT

If the self energy matrix is weakly k dependent is the physical self energy.

Since is a matrix, DMFT changes the shape of the Fermi surface

DMFT is absolutely necessary in the high temperature “local moment”regime. LDA+U with an effective U is OK at low energy.

DMFT is needed to describe spectra with QP and Hubbard bands or satellites.

( )ni

( )abni

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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 puzzle: at high temperatures has a

Curie Weiss law with a moment much larger than the ordered moment.

Magnetic anisotropy

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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 PRL 01)

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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Future directions

Including short range correlations. Less local physics, C-DMFT.

including the effects of long range and short range Coulomb interactions, E-DMFT and GW.

Applications are just beginning. More complex systems…….. Alloys, systems containing f and d electrons.

Realistic Theories of Correlated Materials

ITP, Santa-Barbara

July 20 – December 20 (2002)

O.K. Andesen, A. Georges,

G. Kotliar, and A. Lichtenstein

http://www.itp.ucsb.edu/activities/future/

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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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DMFT

Construction is easily extended to states with broken translational spin and orbital order.

Large number of techniques for solving DMFT equations for a review see

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

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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

Kohn Sham eigenvalues, auxiliary quantities.

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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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Double counting term (Lichtenstein et.al)

subtracts average correlation

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However not everything in low T phase is OK as far as LDA goes.. Magnetic anisotropy puzzle. LDA predicts

the incorrect easy axis(100) for Nickel .(instead of the correct one (111)

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

Use LDA+ U to tackle these refined issues, ( compare parameters with DMFT results )

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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 ( )( )[ ] ( ) [ ( ) ( ) ]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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Single site DMFT, functional of local Greens function G.

Express in terms of Weiss field (semicircularDOS)

[ , ] log[ ] ( ) ( ) [ ]ijn n nG Tr i t Tr i G i Gw w w-GS =- - S - S +F

† †,

2

2

[ , ] ( ) ( ) ( )†

( )[ ] [ ]

[ ]loc

imp

L f f f i i f i

imp

iF T F

t

F Log df dfe

[ ]DMFT atom ii

i

GF = Få Local self energy (Muller Hartman 89)

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Numerical Details

256 k points 105 - 106 QMC sweeps Analytic continuation via maximum entropy. ASA

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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 Coulomb interaction matrix and local G

1[ ] ( 1)

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

abi

n T G i ew

w+

= å

KS [ ( ) G( ) V ( ) ( ) ]LDA DMFT a b abn nr i r i