The Mott transition in f electron systems, Pu, a dynamical mean field perspective Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers

The Mott transition in f electron systems, Pu, a dynamical mean field perspective

Gabriel Kotliar

Physics Department and

Center for Materials Theory

Rutgers University

ASR2002

Tokai Japan November 12-24 2002

Collaborators: S. Savrasov (NJIT)

THE STATE UNIVERSITY OF NEW JERSEY

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Mott transition in the actinide series (Smith Kmetko phase diagram)


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Small amounts of Ga stabilize the phase (A. Lawson LANL)


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Outline Introduction: some Pu puzzles. Introduction to DMFT. Some qualitative insights from model

Hamiltonian studies. Towards and understanding of elemental Pu. Conclusions


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Plutonium Puzzles

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% Is 35% lower than experimentlower than experiment

o This is the largest discrepancy ever known in DFT based calculations.


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DFT Studies 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% 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

U/W is not so different in alpha and delta


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Pu Specific Heat


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Anomalous Resistivity


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Pu is NOT MAGNETIC


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Plutonium puzzles.

How to think about the alpha and delta phases and compute their physical properties?

Why does delta have a negative thermal expansion?

Why do minute amount of impurities stablize delta?

Where does epsilon fit? Why is it smaller than delta?


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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 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


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Extensions of DMFT. Renormalizing the quartic term in the local

impurity action.

EDMFT. Taking several sites (clusters) as local

entity.

CDMFT Combining DMFT with other methods.

LDA+DMFT, GWU.


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DMFT: effective action point of view. R. Chitra and G. Kotliar Phys. Rev. B 62, 12715 (2000), B63, 115110 (2001)

Identify observable, A. Construct an exact functional of <A>=a, [a] which is stationary at the physical value of a.

Construct approximations to the functional to perform practical calculations.

Example: Density functional theory (Fukuda et. al.),density, LDA, GGA.

Example: model DMFT. Observable: Local Greens function Gii (). Exact functional [Gii ()DMFT Approximation the functional keeping 2PI graphs


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Spectral Density Functionals for Electronic Structure Calculations, Sergej Savrasov and Gabriel Kotliar, cond-mat/0106308 Effective action construction. Introduce local orbitals, R(r-R), 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

http://www.physics.rutgers.edu/~kotliar/papers/spectra.pdf



http://xxx.lanl.gov/format/cond-mat/0106308


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Construct approximate functional which gives the LDA+DMFT equations. V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359-7367 (1997).

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 (Gunnarson and Anisimov, McMahan et.al. Hybertsen et.al) of viewed as parameters

http://www.physics.rutgers.edu/~kotliar/papers/jpcm35_7359.pdf




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Outer loop relax

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 + =

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


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Review

A. Georges G. Kotliar W. Krauth M. Rozenberg. Rev Mod Phys 68,1 (1996)

Kotliar, Savrasov, in Kotliar, Savrasov, in New Theoretical approaches New Theoretical approaches to strongly correlated systemsto strongly correlated systems, Edited by A. , Edited by A. Tsvelik, Kluwer Publishers, (2001).Tsvelik, Kluwer Publishers, (2001).

Held Nekrasov Blumer Anisimov and Vollhardt Held Nekrasov Blumer Anisimov and Vollhardt et.al. Int. Jour. of Mod PhysB15, 2611 (2001).et.al. Int. Jour. of Mod PhysB15, 2611 (2001).

A. Lichtenstein M. Katsnelson and G. Kotliar condmat 0211076(2002)


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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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Cerium


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X.Zhang M. Rozenberg G. Kotliar (PRL 1993) A. Georges, G. Kotliar (1992) Phys. Rev. B 45, 6497

Spectral Evolution at T=0 , half filling full frustration, Hubbard bands in the metal, transfer of spectral weight.


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Qualitative phase diagram in the U, T , plane (two band Kotliar Murthy Rozenberg PRL (2002).

Coexistence regions between localized and delocalized spectral functions.

k diverges at generic Mott endpoints


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Minimum in melting curve and divergence of the compressibility at the Mott endpoint

Vsol

Vliq

mdT V

dP S

é ùDê ú=ê úDë û


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Cerium


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Generalized phase diagram

T

U/WStructure, bands,

orbitals


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Case study in f electrons, Mott transition in the actinide series. B. Johanssen 1974 Smith and Kmetko Phase Diagram 1984.


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Outline Introduction: some Pu puzzles. DMFT , qualitative aspects of the Mott

transition in model Hamiltonians. DMFT as an electronic structure method. DMFT results for delta Pu, and some

qualitative insights. Conclusions


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What is the dominant atomic configuration? Local moment?

Snapshots of the f electron Dominant configuration:(5f)5

Naïve view Lz=-3,-2,-1,0,1 ML=-5 B

S=5/2 Ms=5 B Mtot=0


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GGA+U bands. Savrasov Kotliar ,Phys. Rev. Lett. 84, 3670-3673, (2000)

http://www.physics.rutgers.edu/~kotliar/papers/prl84_3670.pdf




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How is the Magnetic moment quenched.

L=5, S=5/2, J=5/2, Mtot=Ms=B gJ =.7 B

Crystal fields

GGA+U estimate (Savrasov and Kotliar 2000) ML=-3.9 Mtot=1.1

This bit is quenched by Kondo effect of spd electrons [ DMFT treatment]

Experimental consequence: neutrons large magnetic field induced form factor (G. Lander).


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Pu: DMFT total energy vs Volume S. Savrasov, G. Kotliar, and E. Abrahams, Nature 410, 793 (2001),


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Double well structure and Pu Qualitative explanation

of negative thermal expansion


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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.


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Comments on the HF static limit for Pu

Describes only the Hubbard bands. No QP states.

Single well structure in the E vs V curve.

(Savrasov and Kotliar PRL). Same if one uses a Hubbard one impurity solver.


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Lda vs Exp Spectra


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Pu Spectra DMFT(Savrasov) EXP (Arko Joyce Morales Wills Jashley PRB 62, 1773 (2000)

Summary

LDA

LDA+U

DMFT

Spectra Method E vs V


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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?

A functional, that computes total energies opens the way to the computation of phonon frequencies in correlated materials (S. Savrasov and G. Kotliar 2002)


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Effects of structure. GGA+DMFT_Hubbard1 imp.solver

E-E=350 K

GGA gives

E-E=

-6000 K


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Phonon freq (THz) vs q in delta Pu (S. Savrasov)


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Phonon frequency (Thz ) vs q in epsilon Pu.


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Epsilon plutonium. Compute the energy of the most unstable frozen mode.

Transverse mode at ( 0,pi, pi) with polarization (0,1,-1).

Extrapolate the form of the quartic interaction to the whole Brillouin zone.

Carry out a self consistent Born approximation to obtain the restabilize phones. Recompute the entropy difference between delta and epsilon.

Estimate the critical temperatures: 500-700 K , depending on the detials of the extrapolation.


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Phonon entropy drives the epsilon delta phase transition

Epsilon is slightly more metallic than delta, but it has a much larger phonon entropy than delta.

At the phase transition the volume shrinks but the phonon entropy increases.


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Phonons epsilon


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Outline Introduction: some Pu puzzles. DMFT , qualitative aspects of the Mott

transition in model Hamiltonians. DMFT as an electronic structure method. DMFT results for delta Pu, and some

qualitative insights into other phases. Conclusions


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Conclusions DMFT produces non magnetic state, around a

fluctuating (5f)^5 configuraton with correct volume the qualitative features of the photoemission spectra, and a double minima structure in the E vs V curve.

Correlated view of the alpha and delta phases of Pu. Interplay of correlations and electron phonon interactions (delta-epsilon).

Calculations can be refined.


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Conclusions Phonons matter. Role of electronic entropy. In the making, new generation of DMFT

programs, QMC with multiplets, full potential DMFT, frequency dependent U’s, multiplet effects , combination of DMFT with GW

Other materials, Cerium and Yterbium compounds…………


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Acknowledgements: Development of DMFT

Collaborators: V. Anisimov, R. Chitra, V. Dobrosavlevic, D. Fisher, A. Georges, H. Kajueter, W.Krauth, E. Lange, A. Lichtenstein, G. Moeller, Y. Motome, G. Palsson, S. Pankov, M. Rozenberg,S. Murthy , S. Savrasov, Q. Si, V. Udovenko, X.Y. Zhang

Funding: National Science Foundation.

Department of Energy and LANL.

Office of Naval Research.


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Technical details 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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Technical details

Atomic sphere approximation.

Ignore crystal field splittings in the self energies.

Fully relativistic non perturbative treatment of the spin orbit interactions.


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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,ED….)Analytical Methods•Extension to ordered states. Review: A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)]

G0 G

Im puritySo lver

S .C .C .


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Temperature stabilizes a very anharmonic phonon mode


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LSDA+DMFT functional2 *log[ / 2 ( ) ( )]

( ) ( ) ( ) ( ) ( ) ( )

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

2 | ' |

[ ]

R R

n

n KS

KS n nKS i

LDAext xc

DC

R

Tr i V r r

V r r dr B r m 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

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


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E-DMFT references H. Kajueter and G. Kotliar (unpublished and Kajuter’s

Ph.D thesis (1995)). Q. Si and J L Smith PRL 77 (1996)3391 . R. Chitra and G.Kotliar Phys. Rev. Lett

84, 3678-3681 (2000 ) Y. Motome and G. Kotliar. PRB 62, 12800 (2000) R. Chitra and G. Kotliar

Phys. Rev. B 63, 115110 (2001) S. Pankov and G. Kotliar PRB 66, 045117 (2002)


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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

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 o o oc Go c n n U n nb 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

( , ')Do t t+


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Cluster extensions of DMFT

Two impurity method. [A. Georges and G. Kotliar (1995 unpublished ) and RMP 68,13 (1996) , A. Schiller PRL75, 113 (1995)]

M. Jarrell et al Dynamical Cluster Approximation [Phys. Rev. B 7475 1998]

Periodic cluster] M. Katsenelson and A. Lichtenstein PRB 62, 9283 (2000).

G. Kotliar S. Savrasov G. Palsson and G. Biroli Cellular DMFT [PRL87, 186401 2001]


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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)


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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


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DMFT MODELS.


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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 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


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Example: Single site DMFT, functional formulation

Express in terms of Weiss field (G. Kotliar EPJB 99)

[ , ] 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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1

10

1( ) ( )

( )n nn k nk

G i ii t i

w ww m w

-


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


†

0 0

( ) ( , ') ( ') ( , ') o o o o o oc Go c n n U n nb 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

( , ')Do t t+


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1

10

1( ) ( )

( )n nn k nk

G i ii t i

w ww m w

-


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 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


Weiss field


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Case study: IPT half filled Hubbard one band (Uc1)exact = 2.2+_.2 (Exact diag, Rozenberg, Kajueter, Kotliar PRB

1996) , confirmed by Noack and Gebhardt (1999) (Uc1)IPT =2.6

(Uc2)exact =2.97+_.05(Projective self consistent method, Moeller Si Rozenberg Kotliar Fisher PRL 1995 ), (Confirmed by R. Bulla 1999) (Uc2)IPT =3.3

(TMIT ) exact =.026+_ .004 (QMC Rozenberg Chitra and Kotliar PRL 1999), (TMIT )IPT =.045

(UMIT )exact =2.38 +- .03 (QMC Rozenberg Chitra and Kotliar PRL 1999), (UMIT )IPT =2.5 (Confirmed by Bulla 2001)

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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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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Interfacing DMFT in calculations of the electronic structure of correlated materials

Crystal Structure +atomic positions

Correlation functions Total energies etc.

Model Hamiltonian


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E-DMFT+GW effective action

G=

D=


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E-DMFT +GW P. Sun and G. Kotliar Phys. Rev. B 2002


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LDA+DMFT and LDA+U • Static limit of the LDA+DMFT functional , • with atom HF reduces to the LDA+U functional

of Anisimov Andersen and Zaanen.

Crude approximation. Reasonable in ordered Mott insulators. Short time picture of the systems.

• Total energy in DMFT can be approximated by LDA+U with an effective U . Extra screening processes in DMFT produce smaller Ueff.

• ULDA+U < UDMFT

®


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Minimum in melting curve and divergence of the compressibility at the Mott endpoint

( )dT V

dp S

Vsol

Vliq


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Interface DMFT with electronic structure.

Derive model Hamiltonians, solve by DMFT

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

GW or screened HF, heavy electrons by DMFT [E-DMFT frequency dependent interactionsGK and S. Savrasov, P.Sun and GK cond-matt 0205522]

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


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LDA+DMFT-outer loop relax

G0 G

Im puritySolver

S .C .C .

0( ) ( , , ) i

i

r T G r r i e w

w

r w+

= å


DMFT

U

Edc

0( , , )HHi

HH

i

n T G r r i e w

w

w+

= å

ff &


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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


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LDA+DMFT and LDA+U • Static limit of the LDA+DMFT functional , • with atom HF reduces to the LDA+U functional

of Anisimov Andersen and Zaanen.

Crude approximation. Reasonable in ordered Mott insulators. Short time picture of the systems.

• Total energy in DMFT can be approximated by LDA+U with an effective U .

®


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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 G. Kotliar and E. Abrahams

(Nature 2001) MIT in V2O3: K. Held et al (PRL 2001) Magnetism of Fe, Ni: A. Lichtenstein M. Katsenelson and

G. Kotliar et al PRL (2001) transition in Ce: K. Held A. Mc Mahan R. Scalettar (PRL

2000); M. Zolfl T. et al PRL (2000). 3d doped Mott insulator La1-xSrxTiO3 Anisimov et.al 1997,

Nekrasov et.al. 1999, Udovenko et.al 2002) ………………..


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LDA+DMFT References

Anisimov Poteryaev Korotin Anhokin and Kotliar J. Phys. Cond. Mat. 35, 7359 (1997).

Lichtenstein and Katsenelson PRB (1998).

Reviews: Kotliar, Savrasov, in Kotliar, Savrasov, in New Theoretical approaches New Theoretical approaches to strongly correlated systemsto strongly correlated systems, Edited by A. Tsvelik, , Edited by A. Tsvelik, Kluwer Publishers, (2001).Kluwer Publishers, (2001).

Held Nekrasov Blumer Anisimov and Vollhardt et.al. Int. Held Nekrasov Blumer Anisimov and Vollhardt et.al. Int. Jour. of Mod PhysB15, 2611 (2001).Jour. of Mod PhysB15, 2611 (2001).

A. Lichtenstein M. Katsnelson and G. Kotliar (2002)


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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


DMFT

U

E

0( , , )HHi

HH

i

n T G r r i e w

w

w+

= å


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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 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


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Comments on LDA+DMFT• Static limit of the LDA+DMFT functional , with = HF reduces to LDA+U

• Gives the local spectra and the total energy simultaneously, treating QP and H bands on the same footing.

• Luttinger theorem is obeyed.• Functional formulation is essential for

computations of total energies, opens the way to phonon calculations.


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References

LDA+DMFT: 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.SavrasovG.Kotliarfuncionalformulationforfullselfconsistentimplementationofaspectraldensityfunctional.

ApplicationtoPuS.SavrasovG.KotliarandE.Abrahams(Nature2001).


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Debye temperatures


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References

Long range Coulomb interactios, E-DMFT. R. Chitra and G. Kotliar

Combining E-DMFT and GW, GW-U , G. Kotliar and S. Savrasov

Implementation of E-DMFT , GW at the model level. P Sun and G. Kotliar.

Also S. Biermann et. al.


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Wilson and Kadowaki Woods Ratio


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Dependence on structure

Expt: V-V=.54 A

Theory: V-V=.35 A


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Dynamical Mean Field Theory(DMFT)Review: A. Georges G. Kotliar W. Krauth M. Rozenberg. Rev Mod Phys 68,1 (1996)

Local approximation (Treglia and Ducastelle PRB 21,3729), local self energy, as in CPA.

Exact the limit defined by Metzner and Vollhardt prl 62,324(1989) inifinite.

Mean field approach to many body systems, maps lattice model onto a quantum impurity model (e.g. Anderson impurity model )in a self consistent medium for which powerful theoretical methods exist. (A. Georges and G. Kotliar prb45,6479 (1992)). See also M. Jarrell (PRL 1992) .Connect local spectra and lattice total energy.


RUTGERS

Correlation betwee the Minimum of the melting point and the Mott transition endpoint.

Divergence of the compressibility at the Mott transition endpoint.

Rapid variation of the density of the solid as a function of pressure, in the localization delocalization crossover region.

Slow variation of the volume as a function of pressure in the liquid phase

Elastic anomalies, more pronounced with orbital degeneracy.


RUTGERS

Localization delocalization transition and f electrons.

Mott phenomena. Evolution of the electronic structure between the atomic limit and the band limit in an open shell situation.

The “”in between regime” is ubiquitous central them in strongly correlated systems, gives rise to interesting physics. Example Mott transition across the actinide series [ B. Johansson Phil Mag. 30,469 (1974)]

Connection between local spectra and cohesive energy using Anderson impurity models foreshadowed by J. Allen and R. Martin PRL 49, 1106 (1982) in the context of KVC for cerium.


RUTGERS

DMFT and f electrons. These views of the localization delocalization transition are not

orthogonal and were incorporated into a more complete Dynamical Mean Field picture of the Mott transition.

G. Kotliar, EPJ-B, 11, (1999), 27 . A. Georges G. Kotliar W. Krauth M. Rozenberg. Rev Mod Phys 68,1 (1996) . Moeller Q. Si G. Kotliar M. Rozenberg and D. S Fisher, PRL 74 (1995) 2082.

DMFT: Powerful new tool for studying f electrons. Qualitative insights into complex materials. Turn technology developed to solve models into

practical quantitative electronic structure method , to study eg. PU.


RUTGERS

Evolution of the Spectral Function with Temperature near Mott endpoint.

Anomalous transfer of spectral weight connected to the proximity to the Ising Mott endpoint (Kotliar Lange and Rozenberg Phys.Rev.Lett.84,5180(2000)


RUTGERS

DMFT: Effective Action point of view.R. Chitra and G. Kotliar Phys Rev. B.(2000), (2001).


Example, density in DFT theory. (Fukuda et. al.) When a is local, it gives an exact mapping onto a local

problem, defines a Weiss field. The method is useful when practical and accurate

approximations to the exact functional exist. Example: LDA, GGA, in DFT.

It is useful to introduce a Lagrange multiplier conjugate to a, [a,

It gives as a byproduct a additional lattice information.


RUTGERS

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


RUTGERS

Mott transition in layered organic conductors S Lefebvre et al. Ito et.al, Kanoda’s talk Bourbonnais talk

Magnetic Frustration


RUTGERS

Ultrasound study of

Fournier et. al. (2002)


RUTGERS

Comparaison with LDA+U


RUTGERS

DMFT: Effective Action point of view.R. Chitra and G. Kotliar Phys Rev. B.(2000), Phys. Rev. B 63, 115110 (2001)


Example, density in DFT theory. (Fukuda et. al.) When a is local, it gives an exact mapping onto a local

problem, defines a Weiss field. The method is useful when practical and accurate

approximations to the exact functional exist. Example: LDA, GGA, in DFT.


RUTGERS

Example: DMFT for lattice model (e.g. single band Hubbard).

Observable: Local Greens function Gii ().

Exact functional [Gii () DMFT Approximation to the functional.

[ , ] log[ ] ( ) ( ) [ ]DMFT DMFTij ii iin n niG Tr i t Tr i G i Gw w w-G S =- - S - S +Få

[ ] Sum of 2PI graphs with local UDMFT atom ii

i

GF = Få


RUTGERS

Wilson and Kadowaki Woods Ratio


RUTGERS

Comparaison with the Hartree Fock static limit: GGA+U.

E-E=350 K

Volume, total energies are OK much better than LDA, but no double minima

Documents

The Mott transition in f electron systems, Pu, a dynamical mean field perspective Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers