gabriel kotliar physics department and center for materials theory rutgers university

Towards Realistic Electronic Structure Calculations of Correlated

Materials Exhibiting a Mott Transition.

Gabriel Kotliar

Physics Department and

Center for Materials Theory

Rutgers University

March Meeting of the American Physical Society Seattle 2001

Collaborators: S. Savrasov, V. Udovenko(Rutgers), R.Chitra(Jussieux) S. Lichtenstein (Nijmeigen) M. Rozenberg (UBA) E Lange (McKinsey)

THE STATE UNIVERSITY OF NEW JERSEY

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Outline Introduction to the correlation

driven localization delocalization transition ( Mott transition).

Some lessons from very simple models DMFT study of a one band Hubbard with a semicircular density of states.

Extensions to more realistic situations . Case studies in d and f electrons

Outlook for further developments


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The Mott Hubbard problem

Ba

Array of H atoms, e is localized a>>aB, extended if a<<aB

e_ e_ e_ e_

Momentum space, bands Real space, atoms

Ba

High T : local moments

Low T: spin orbital order

1

T

1

T

const


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The Mott Hubbard problem

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.

Stimulates the development of new electronic structure methods (LDA+DMFT).


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Mott transition in layered organic conductors S Lefebvre et al. cond-mat/0004455


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A time-honored example: Mott transition in V2O3 under pressure

or chemical substitution on V-site


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Kuwamoto Honig and AppellPRB (1980)


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Phase Diag: Ni Se2-x Sx

G. Czek et. al. J. Mag. Mag. Mat. 3, 58 (1976)


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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: Model Calculation

†

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 ( )( )[ ] ( ) [ ( ) ( ) ]HH 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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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….)

•Analytical Methods

G0 G

Im puritySo lver

S .C .C .


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DMFT: Methods of solution

Prushke T. Jarrell M. and Freericks J. Adv. Phys. 44,187 (1995)

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

Iterative perturbation theory. A Georges and G Kotliar PRB 45, 6479 (1992). H Kajueter and G. Kotliar PRL (1996). S. Savrasov et.al (2001).Projective method G Moeller et. al. PRL 74 2082 (1995). NRG R. Bulla PRL 83, 136 (1999)QMC M. Jarrell, PRL 69 (1992) 168, Rozenberg Zhang Kotliar PRL 69, 1236 (1992) ,A Georges and W Karuth PRL 69, 1240 (1992) M. Rozenberg PRB 55, 4855 (1987).


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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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Mott transition in model system The qualitative features of the Rutgers-

ENS results for the Mott transition were challenged in a series of publications: S Kehrein Phys. Rev Lett. 3192 (1998),R. Noack and F. Gebhardt, Phys. Rev. Lett. 82, 1915 (1999), J. Schlipf et. al. Phys. Rev. Lett 82, 4890 (1999).

These works missed subtle aspects of the and non perturbative character of the region near the metal to insulator transition such as the singular behavior of the self energy


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Landau Functional

† †,

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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Functional Approach

The Landau functional offers a direct connection to the atomic energies

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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Insights from DMFT

The Mott transition is driven by transfer of spectral weight from low to high energy as we approach the localized phaseControl parameters: doping, temperature,pressure…


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Evolution of the Spectral Function with Temperature

Anomalous transfer of spectral weight connected to the proximity to an Ising Mott endpoint (Kotliar Lange and Rozenberg PRL 84, 5180 (2000))


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Insights from DMFT: think in term of spectral functions (branch cuts) instead of well defined QP (poles )

Resistivity near the metal insulator endpoint ( Rozenberg et. Al 1995) exceeds the Mott limit


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Anomalous Resistivity and Mott transition Ni Se2-x Sx

Miyasaka and Tagaki (2000)


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. ARPES measurements on NiS2-xSex

Matsuura et. Al Phys. Rev B 58 (1998) 3690


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Insights from DMFT Low temperatures several competing phases . Their relative stability depends on chemistry and crystal structureHigh temperature behavior around Mott endpoint, more universal regime, captured by simple models treated within DMFT


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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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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


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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

--é ùê ú= +Sê ú- - - Sê úë ûå


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DMFT +LDA : effective action construction (Fukuda, Valiev and Fernando ,Argaman and Makov, Chitra and GK).

Select a set of local orbitals. Define a frequency dependent, local

Greens function by projecting onto the local orbitals.

The exact free energy can be expressed as a functional of the local Greens function and of the density

The functional can be built in perturbation theory in the interaction (well defined diagrammatic rules )

The functional can also be constructed from the atomic limit.

A useful approximation to the exact functional can be constructed, the DMFT +LDA functional.


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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.Savrasovfullselfconsistentimplementation(2001)


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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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Pu: DMFT total energy vs Volume (S. Savrasov )


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Lda vs Exp SpectraD

OS

, st./

[eV

*cel

l]


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


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Iron and Nickel: band picture at low T, crossover to real space picture at high T


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Photoemission Spectra and Spin Autocorrelation: Fe(Lichtenstein, Katsenelson,GK cond-mat 0102297)


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Photoemission and Spin Autocorrelation: Ni


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Iron and Nickel:mgnetic properties (Lichtenstein, Katsenelson,GK cond-mat 0102297)


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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 3.09 (theory) 3.12 (expt)

Ni 1.50 (theory) 1.62 (expt)

Curie Temperature Tc

Fe 1900 ( theory) 1043(expt) Ni 700 (theory) 631 (expt)


RUTGERS

Conclusion The delocalization delocalization

transition is a very relevant problem to the electronic structure of solids.

The character of the localization delocalization in the Hubbard model within DMFT is now fully understood.

This has lead to extensions to more realistic models, and a beginning of a first principles approach interpolating

between atoms and bands.


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Outlook Need more experience in the estimates

of the double counting term and the Coulomb interaction parameters.

Combinations of DMFT and GW. Incorporate effects of long range Coulomb interactions.

E-DMFT Model calculation. Mott transtion at T=0

Is first order. R. Chitra and G. Kotliar PRL 84, 3678

(2000). Extension to multiple site clusters(DCA M. Jarrell et. al., Two impurity

DMFT Schiller, Ingersent Georges Kotliar, C-Dmft , E-DMFT …..)


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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 oS Go c Go c n nb b b

s st t t t ¯= +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

Lda+dmft functional dynamical Kohn Sham field

[ ] 1( ) ( ) ( , )n n k LMTO n

k

A i i O H k E i Aabab

w w c w-

= - + -å

Weiss field

† †[ ] log ( , ') ( ) ( ') 'Sat a b

ab

W dc dce c c d dt t t t t t-D =- - Dåò ò

†

0( ) ( ') ( , ')

( , ')at

j

Wc c Aaba b

ab

dt t t t

d t t=- =

D

DMFTfunctional ( , , , , )LDA DMFT KSV Ar c+G D

1 1 *'

, , ' , , ,

1( , , ')( ) ( ) ( ) ( ) ( ) ( )

p l q

n p p l ac cc n l q c qs c c R R R

i r r r R O R R i O R R r RNab a b b

a b

w c c c w c- -S = - - - -å

0

,

1, ( )

2n

n

iDC n

i

nUn n T e A iw

aa

w a

w+æ ö- ÷çF = =÷ç ÷÷çè ø å


RUTGERS

LDA+DMFT functional

2

1

log[ / 2 ( , , ')( )]

( ) ( ) ( ) ( )

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

2 | ' |

[ ] [ ] log

n

n KS n

KS n n

i

LDAext xc

DC at at

Tr i V i r r

V r r dr Tr i A i

r rV r r dr drdr E

r r

W Tr A Tr A Tr A A

w

w w c

r w w

r rr r

-

- +Ñ - - S -

- S +

+ + +-

F + D - D - +

åò

ò ò


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LDA+DMFT loop (in a tight binding basis, e.g. LMTO’s U, interaction matrix

0) Guess (r), G(i 1) Form Vxc , Solve AIM to

get and local Greens function of heavy orbitals.

Form LMTO Matrix , overlap matrix and heavy level shift E , form G(k, i

3) Recompute the density and Weiss function G(i to go back to 1.


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LDA+DMFT

To implement step 3 we use

Notice the Weiss field,E and self energies use

only heavy block, while H is full.


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Sir Nevill Mott

gabriel kotliar physics department and center for materials theory rutgers university

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