dynamical mean field theory or metallic plutonium

Dynamical Mean Field Theory or Metallic Plutonium

Gabriel Kotliar

Physics Department and

Center for Materials Theory

Rutgers University

IWOSMA

Berkeley October 2002

Collaborators: S. Savrasov (NJIT) and Xi Dai (Rutgers)

THE STATE UNIVERSITY OF NEW JERSEY

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Mott PhenomenaEvolution 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)]

Revisit the problem using a new insights and new techniques from the solution of the Mott transition problem within dynamical mean field theory in the model Hamiltonian context.

Use the ideas and concepts that resulted from this development to give physical qualitative insights into real materials.

Turn the technology developed to solve simple models into a practical quantitative electronic structure method .


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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.

Identificaton of Kondo resonance n Ce , PRB 28, 5347 (1983).


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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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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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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


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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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Specific heat and susceptibility.


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Problems with the conventional viewpoint of Pu

U/W is not so different in alpha and delta 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.


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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).


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

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

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.


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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


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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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X.Zhang M. Rozenberg G. Kotliar (PRL 1993)

Spectral Evolution at T=0 half filling full frustration


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Phase Diagrams :V2O3, Ni Se2-x Sx Mc Whan et.

Al 1971,. Czek et. al. J. Mag. Mag. Mat. 3, 58 (1976),


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

Magnetic Frustration


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Cerium


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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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Ultrasound study of

Fournier et. al. (2002)


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Minimum of the melting point

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.


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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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Cerium


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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 approximate functional

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


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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

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 &


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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.

• 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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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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Spectral Density Functional : 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


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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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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). A Lichtenstein and M. Katsenelson Phys. Rev. B

57, 6884 (1988). S. Savrasov G.Kotliar funcional formulation

for full self consistent implementation of a spectral density functional.

Application to Pu S. Savrasov G. Kotliar and E. Abrahams (Nature 2001).


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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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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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LDA+U bands. (Savrasov GK ,PRL 2000).


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Magnetic moment

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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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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Pu: DMFT total energy vs Volume (Savrasov 00)


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Double well structure and Pu Qualitative explanation

of negative thermal expansion

Sensitivity to impurities which easily raise the energy of the -like minimum.


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


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Dependence on structure


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Comments on the HF static limit

Describes only the Hubbard bands. No QP states.

Single well structure in the E vs V curve.

(Savrasov and Kotliar PRL)


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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)


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Comparaison with LDA+U

Summary

LDA

LDA+U

DMFT

Spectra Method E vs V


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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.

Calculations can and should be refined and extended.


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Conclusions Outsanding question: electronic entropy, lattice

dynamics. 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


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DMFT EXPERIMENTS


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Pu: Anomalous thermal expansion ( Smith and Boring )


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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

† †

, ,

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

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

-

-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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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 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


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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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Combining LDA and DMFT The light, SP electrons well described by LDA. The heavier D

electrons treat by DMFT. LDA already contains an average interaction of the heavy

electrons, subtract this out by shifting the heavy level (double counting term, Edc , review Anismov Aersetiwan and Lichtenstein )

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


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DMFT MODELS RESULTS


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QMC calculationof n vs (Kotliar Murthy Rozenberg PRL 2002, 2 band, U=3.0)

diverges at generic Mott endpoints


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Compressibilty divergence


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Cerium


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E-DMFT+GW effective action

G=

D=


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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 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


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


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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 Reviews: A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996). Prushke T. Jarrell M. and Freericks J. Adv. Phys. 44,187 (1995)

G0 G

Im puritySo lver

S .C .C .


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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 cluster Greens functions ) as the basic quantities of the theory.

Express the free energy as a functional of these local quantities and the density.

Provide useful approximations to the functional.


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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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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 [ [ ]]atWG G

Weiss field


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Double counting term (Lichtenstein et.al)

Problem : What is the LDA+U functional, a functional of?

What is nab ?


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Plutonium


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PU: (“cubic ALPHA” AND DELTA

dynamical mean field theory or metallic plutonium

Documents

phase of pu

pu puzzles

susceptibility of alpha

resistivity of alpha

mott transition problem

dmft results

example mott transition

d phase