Correlation Effects in Itinerant Magnets : Towards a realistic Dynamical Mean Field Approach Gabriel...

Correlation Effects in Itinerant Magnets : Towards a realistic

Dynamical Mean Field Approach

Gabriel Kotliar

Physics Department

Rutgers University

In Electronic Structure and Computational Magnetism

July 15-17 (2002)

THE STATE UNIVERSITY OF NEW JERSEY

RUTGERS

Outline

Dynamical Mean Field Theory: a tool for treating correlations in model Hamiltonians.

Towards Realistic implementations of DMFT.

Applications to Fe and Ni.

Conclusions and outlook.


RUTGERS

Acknowledgements Collaborators and References:

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).

Useful Discussions K. Hathaway and G. Lonzarich

Support NSF and ONR


RUTGERS

Strong Correlation Problem

•Two limiting cases of the electronic structure problem are well understood. The high density limit ( spectrum of one particle excitations forms bands) and the low density limit (spectrum of atomic like excitations, Hubbard bands).•Correlated compounds: electrons in partially filled shells. Not close to the well understood limits . Non perturbative regime. Standard approaches (LDA, HF ) do not work well.


RUTGERS

Motivations for going beyond density functional theory. DFT is a theory for ground state properties. Its

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

This does not work for strongly correlated systems, eg oxides containing 3d, 4f, 5f elements. Character of the spectra (QP bands + Hubbard bands ) is not captured by LDA.

LDA –GGA is less accurate in determining some ground state properties in correlated materials.


RUTGERS

DMFT DMFT simplest many body technique which

describes correctly the open shell atomic limit and the band limit . Exact in the limit of large lattice coordination.

Band physics (i.e. kinetic energy) survive in the atomic limit (superexchange). Some aspects of atomic physics survive even in itinerant systems (J, U, Hubbard bands, satellites, L)

Computations of one electron spectra, transport properties…

Spectral density functional. Connects the one electron spectral function and the total energy.


RUTGERS

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

† †

, ,

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

t c c c c U n n


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 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 oc Go c n n Ub 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

( , ')o o oD n nt t ¯+


RUTGERS

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®

Alternative approaches DCA (Jarrell et.al.) Periodic clusters (Lichtenstein and Katsnelson)


RUTGERS

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


RUTGERS

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

G0 G

Im puritySo lver

S .C .C .


RUTGERS

From model hamiltonians to realistic calculations. DMFT as a method to be incorporated in electronic

structure calculations. Important in regimes where local moments are

present (e.g. NiO above its Neel temperature) Incorporate realistic structure and orbital

degeneracy information in many body studies. Combination of electronic structure(LDA,GGA,GW)

and many body methods (DMFT)


RUTGERS

Interface with electronic structure.

Derive model hamiltonians, solve by DMFT

(or cluster extensions). Total energy? Full many body aproach, treat light electrons byt

GW or screend HF, heavy electrons by DMFT [GK and Chitra, GK and S. Savrasov, P.Sun and GK cond-matt 0205522]

Treat correlated electrons with DMFT and light electrons with DFT (LDA, GGA +DMFT)


RUTGERS

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

Calculations of U, Edc, study as a function of these parameters.


RUTGERS


RUTGERS

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.


RUTGERS

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

Correlation Functions Total Energies etc.

Model Hamiltonian

Crystal structure +Atomic positions


RUTGERS

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)


RUTGERS

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 and Anisimov, Mc Mahan et. Al. Hybertsen et.al) or viewed as parameters


RUTGERS

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.


RUTGERS

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


RUTGERS

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 &


RUTGERS

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


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

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) ………………..


RUTGERS

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)]

Extension to realistic setting [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). ] Lichtenstein and Katsnelson [Phys.Rev. B 57, 6884(1998) ]

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


RUTGERS

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


RUTGERS

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.


RUTGERS


( ) ( ) ( ) ( )

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 all 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

( ) ( )G i iw w¾¾®D


RUTGERS

Outer loop relax

0( ) ( , , ) i

i

r T G r r i e w

w

r w+

= å


U

Edc

0( , , )HHi

HH

i

n T G r r i e w

w

w+

= å

ff &

Impurity Solver

SCC

G,G0

DMFTLDA+U

Hartree-Fock


RUTGERS

Outer loop relax

0( ) ( , , ) i

i

r T G r r i e w

w

r w+

= å


U

Edc

0( , , )HHi

HH

i

n T G r r i e w

w

w+

= å

ff &

Impurity Solver

SCC

G,G0

DMFTLDA+U

Imp. Solver: Hartree-Fock


RUTGERS

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


DMFT

U

E

0( , , )HHi

HH

i

n T G r r i e w

w

w+

= å


RUTGERS

LDA+DMFT and LDA+U

• Static limit of the LDA+DMFT functional ,

• with = HF reduces to the LDA+U functionalof Anisimov et.al.

• Crude approximation. Reasonable in ordered situations.

, ( ) nab ab cd cdni U

( )0( ) iab ab

abi

n T G i ew

w+

= å


RUTGERS

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


RUTGERS

Applications of LDA+DMFT

Organics Alpha-Gamma Cerium V2O3 Volume collapse in Pu Photoemission of ruthenates Doping driven Mott transition in LaSrTiO3 Itinerant Ferromagnetism Bucky Balls


RUTGERS


RUTGERS

Applications: Itinerant Ferromagnetism, Ni Fe

Compromise in the resources used for the solution of the one electron problem, and the many body problem.

Goal: obtain an overall approximate but consistent picture of how correlations affect physical properties. Estimate sensitivity on parameters.

Tc, spectra, susceptibility, [QMC- impurity solver] [ASA, relatively small number of k points]

Magnetic anisotropy [HF-impurity solver][full potential LMTO, large number of k points, non collinear magnetization]


RUTGERS

Case study Fe and Ni

Archetypical itinerant ferromagnets

LSDA predicts correct low T moment

Band picture holds at low T


RUTGERS

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


RUTGERS

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) )

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

Photoemission spectra of Ni : 6 ev satellite 30% band narrowing, reduction of exchange splitting.


RUTGERS

DMFT-QMC: Numerical Details

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


RUTGERS

Photoemission Spectra and Spin Autocorrelation: Fe (U=2, J=.9ev,T/Tc=.8) (Lichtenstein, Katsenelson,GK prl 2001)


RUTGERS

Photoemission and T/Tc=.8 Spin Autocorrelation: Ni (U=3, J=.9 ev)


RUTGERS

Iron and Nickel:magnetic properties (Lichtenstein, Katsenelson,GK PRL 01)

2

0 3( )q

Meff

T Tc

c

T

T


RUTGERS

Ni and Fe: theory vs exp

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)


RUTGERS


RUTGERS

MAEissmall(1eV/Atom)

Ni:2.8eV/Atomeasyaxis111Fe:1.4eV/Atomeasyaxis100LongstandingproblemEarlypapers

•VanVleck(PR1937)•Brooks(PR1940)

Magnetic anisotropy


RUTGERS

Trygget.al(1995);SCFTotalenergywithlarge#ofk-points;WrongeasyaxisforNi.

Otherrelatedworks:

Halilovetal.(1998)G.Schneideretal.(1997)Wangetal.(1993)Beidenet.al.(1998)

LDA calculations


RUTGERS

Full-potentialmultiplekappaLMTOmethod.

Paulitreatmentofrelativisticeffects.

Non-collinearintraatomicmagnetismincluded.

ExploredifferentEdc.(DetailsIYangPh.Dthesis)GeneralizedrelativisticLDA+Uwithoccupanciesn’

MethodMethod


RUTGERS

WorkofTrygget.al.provesequivalenceofspecialpointsandtetrahedra.Confirmed.(broadening0.15Ry.)ConvergentEtotneeds15000k’s.Weuse28000k’s.Convergencycheckedto100000k’s.SUNE10Kwith64processorsused.LDAresultsofTrygget.al.reproduced:Ni0.5eV001,exp.2.8eV111,Fe0.5eV001,exp.1.4eV001.

Numerical Considerations


RUTGERS

StudiesofMAEasfunctionofUandJ.

BothUandJinfluencemagneticmomentwhichisOKinLDA:0.6BforNiand2.2BforFe.

HowtofixmomentinLDA+U:

FindM(U,J)andtracepathforwhichmomentdoesnotchange.

LDA+U ResultsLDA+U Results


RUTGERS

MagneticmomentasfunctionofUandJforNi

NN i - M(U,J)i - M(U,J)


RUTGERS

MagneticmomentasfunctionofUandJforFe

Fe - M(U,J)Fe - M(U,J)


RUTGERS

MAEasafunctionofU(J)

U=1.9eV,J=1.2eV

U=1.2eV,J=0.8eV

Ni

Fd


RUTGERS

egformingX2pocket

eg

LDA vs LDA+U for NiLDA vs LDA+U for Ni


RUTGERS

Ni U=2,J=.1 PT (Katsenelson and Lichtenstein)cond-matt 2002


RUTGERS

Conclusions Satellite in majority band at 6 ev, 30 % reduction

of bandwidth, exchange splitting reduction from band theory value (.6ev) to .3 ev

Spin wave stiffness controls the effects of spatial flucuations, it is about twice as large in Ni and in Fe. Single site should work for Ni, and overestimate Tc for 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%.


RUTGERS

OverallconsistentpictureoftheeffectsofcorrelationsonitinerantmagnetsusingDMFT.CanreproducecorrecteasyaxisandMAEofFeandNi.CancorrecttheFermisurfaceofNi.

Conclusions


RUTGERS

Work in progress

With existing techniques, derive practical formulae for the magnetic anisotropy of systems containing partially localized and itinerant electrons.

Further tests of DMFT on interesting materials.

Incorporate extensions of DMFT to incorporate frequency dependent interations (GW+DMFT) and to larger clusters.


RUTGERS


RUTGERS

NochangesofFermisurfacefound

LDA and LDA+U bands for Fe


RUTGERS

LDAelectronicstructureforNiE(k) for NiE(k) for Ni


RUTGERS

CalculatedFermisurfaceforNiusingLDA+U.NoartificialX2pocket

Fermi Surface for Ni

Impurity Solver

SCC

G,G0

DMFTLDA+U

Hartree-Fock


RUTGERS

GW+DMFT functional.

S. Savrasov and GK. P. Sun and GK. (cond matt).

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/


RUTGERS

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 .


RUTGERS

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)]


RUTGERS

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.


RUTGERS

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)


RUTGERS

Double counting term (Lichtenstein et.al)

subtracts average correlation


RUTGERS

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 )


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 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

† †

, ,


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


RUTGERS

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)


RUTGERS


( ) ( ) ( ) ( )

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

Calculated MAE for Ni and FeCalculated MAE for Ni and Feusing LDA+U methodusing LDA+U method

S. Y. SavrasovNew Jersey Institute of Technology

In collaboration with:Imseok Yang (Ph.D Thesis, RU)Gabriel Kotliar (RU)

Sponsored by Office of Naval Research

Grant No: ONR 4-2650Phys. Rev. Lett. 87, 216405 (2001)


RUTGERS

TotalEnergyDFTjobwithhugek-pointsummationproblem.

Eckardet.al(1987);Rightorder;WrongeasyaxisforFe.

Daalderlopet.al(1990);Forcetheorem;WrongeasyaxisforNi.

VaryingpositionofFermilevel,artificialX2pocketinfluenceseasyaxis.

Calculations


RUTGERS

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 5 ( theory) 1043(expt) Ni 700 (theory) 631 (expt)

CorrelationsCorrelations

Many-bodyHubbardinteractionsareimportant(notcapturedbyLDA)

DMFT:onsitecorrelationsaretreatedexactly,bothatomicandbandlimitareOK.

StaticlimitofDMFT:LDA+Umethod:

Self-energy()->(static)

Solutionofimpuritymodelcollapsestodeterminationofn

Problemcanbesolvednow.


RUTGERS

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 .


RUTGERS

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

Correlation Effects in Itinerant Magnets : Towards a realistic Dynamical Mean Field Approach Gabriel...

Documents

Transcript of Correlation Effects in Itinerant Magnets : Towards a realistic Dynamical Mean Field Approach Gabriel...