applications of dmft to actinide materials gabriel kotliar physics department and center for...

Applications of DMFT to actinide materials

Gabriel Kotliar

Physics Department and

Center for Materials Theory

Rutgers University

THE STATE UNIVERSITY OF NEW JERSEY

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Outline Introduction. Pu puzzles.[ LosAlamos Science

(2000)] DMFT , qualitative aspects of the Mott transition from

model Hamiltonians. [A. Georges, G. Kotliar, W. Krauth

and M. Rozenberg Rev. Mod. Phys. 68,13 (1996 )] LDA+DMFT results for delta Pu, and epsilon Pu.

Some qualitative insights. [X. Dai, S. Y. Savrasov, G.

Kotliar, A. Miglori, H. Ledbetter,E. Abrahams preprint] Approaching the Mott transition from the localized

side, a first look at Am.


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


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Physics of Pu


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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

Alternative approach Wills et. al. (5f)4 core+ 1f(5f)in conduction band.


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Shear anisotropy fcc Pu (GPa)

C’=(C11-C12)/2 = 4.78

C44= 33.59

C44/C’ ~ 8 Largest shear anisotropy in any element!

LDA Calculations (Bouchet) C’= -48


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Pu is NOT MAGNETIC


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Pu Specific Heat


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Anomalous Resistivity


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

Concepts : three peak structure and transfer of spectral weigth. Evolution at T=0 half filling full frustration


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Evolution of the Spectral Function with Temperature

(Kotliar Lange and Rozenberg Phys. Rev. Lett. 84, 5180 (2000)


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Generalized phase diagram

T

U/WStructure, bands,

orbitals


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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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Minimum of the melting point near Pu.

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


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Cerium


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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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Energy vs Volume [GGA+U]


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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]


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Pu: DMFT total energy vs Volume (Savrasov Kotliar and Abrahams 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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W (ev) vs (a.u. 27.2 ev) N.Zein G. Kotliar and S. Savrasov

iw

Spectra –E (V)

LDA

LDA+U

DMFT

Spectra Method E vs V


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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

Spectra –E (V)

LDA

LDA+U

DMFT

Spectra Method E vs V


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Phases of Pu


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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?

LDA+DMFT functional 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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Phonon freq (THz) vs q in delta Pu (X. Dai et. al. Preprint 2003)


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Phonons in delta Pu

Computed using extension of the linear response technique (developed for LDA+ LMTO by Savrasov ) to LDA+DMFT using a Hubbard 1 impurity solver.

Unusual, near degeneracy of transverse and longitudinal sound velocity along (0,0,1). Not measured yet.


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Shear anisotropy. Expt. vs Theory

C’=(C11-C12)/2 = 4.78 GPa C’=3.9 GPa

C44= 33.59 GPa C44=33.0 GPa

C44/C’ ~ 7 Largest shear anisotropy in any element!

C44/C’ ~ 8.4


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Elastic constants theory (LDA+DMFT with a Hubbard1 solver, Dai et. al. and experiments,( Letbetter and Moment )

C11 (GPa) C44 (GPa) C12 (GPa) C'(GPa) Theory 34.56 33.03 26.81 3.88 Expt 36.28 33.59 26.73 4.78

The Cauchy relation (c44=c12)(is not badly violated, angular forces are not

playing a big role ).


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Phases of Pu


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Phonon frequency (Thz ) vs q in epsilon Pu.


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Frozen phonon calculation.


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Temperature stabilizes a very anharmonic mode in epsilon Pu


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Phonon entropy drives the epsilon delta phase transition

Epsilon is slightly more delocalized than delta, has SMALLER volume and lies at HIGHER energy than delta at T=0. But it has a much larger phonon entropy than delta.

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

Estimates of the phase transition following Drumont and Ackland et. al. PRB.65, 184104 (2002); (and neglecting electronic entropy). TC ~ 600 K.


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Negative thermal expansion of Pu revisited. The distortion described by C'

is very soft, nearly liquid, . C' measures the rigidity against the volume conserving tetragonal deformation. fcc to a bcc along a Bain path. Previous LDA+ U and [Bouchet et. al. ] and our DMFT study show that the total energy difference between phase and phases is quite small ~ 1000K. Pu can sample the bcc structure, which has lower volume by thermal fluctuation along Bain path.

d

d

e


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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).


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Conclusions Outstanding 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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Open vs Closed Shell

g

gU/W

U/W


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Americium metal: a different kind of Mott transition.

Approach the Mott transition, if the localized configuration has an OPEN shell the mass increases as the transition is approached.

Consistent theory, entropy increases monotonically as U Uc .

Approach the Mott transition, if the localized configuration has a CLOSED shell. We have an apparent paradoxto approach the Mott trnasitions the bands have to narrow, but the insulator has not entropy.. SOLUTION: superconductivity intervenes.


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Mott transition in systems with close shell.

Resolution: as the Mott transition is approached from the metallic side, eventually superconductivity intervenes to for a continuous transition to the localized side.

DMFT study of a 2 band model for Buckminster fullerines Capone et. al. Science 2002.

Mechanism is relevant to Americium.


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Phase diagram (Lindbaum et. al. PRB 2003)


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Theoretical LDA+U caculations [S. Murthy and GK]


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Am Photoemission (Negele)


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Am LDA+U Fat Bands


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LDA+U computations on Am

What’s good with LDA+U for Americium?

With U ~4 equilibrium volume and photoemission are roughly OK.

o What’s wrong with LDA+U for Americium?

Being HF-like, it lacks the configuration interaction which is responsible for stabilizing the J=0 configuration. To be done with DMFT!


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Many Body Physics and Electronic Structure both are needed to describe Actinides.

The actinide series provides realizations of two different Mott transitions ( Pu , Am ).

Qualitative understanding of the differences between alpha and delta.

Very unusual phonon dynamics is responsible for the delta to epsilon transition.


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


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

®


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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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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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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.


RUTGERS

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


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GWU

1) Form basis set. (e.g. Phi, Phi_dot, Chi ) 2) Form Hamiltonian, Overlap and Coulomb

interaction matrices. 3) Solve the extended DMFT equations using the

results of 2) as input, compute G, Sigma, Pi. 4) Go to step 2, improve H. 5) Go to step 2 to improve basis set. 6) Evaluate total energy.


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

®


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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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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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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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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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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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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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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 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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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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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+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

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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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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Energy difference between epsilon and delta


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Energy vs Volume

Success story : Density Functional Linear Success story : Density Functional Linear ResponseResponse

Tremendous progress in ab initio modelling of lattice dynamics& electron-phonon interactions has been achieved(Review: Baroni et.al, Rev. Mod. Phys, 73, 515, 2001)

(Savrasov, PRB 1996)

Results for NiO: PhononsResults for NiO: Phonons

Solid circles – theory, open circles – exp. (Roy et.al, 1976)

DMFT Savrasov and GK PRL 2003


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DMFT for Mott insulators


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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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E-DMFT+GW effective action

G=

D=


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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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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)


RUTGERS

Technical details

Atomic sphere approximation.

Ignore crystal field splittings in the self energies.

Fully relativistic non perturbative treatment of the spin orbit interactions.


RUTGERS

LDA+U bands. (Savrasov GK ,PRL 2000). Similar work Bouchet et. al. 2000


RUTGERS

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

Spectral Evolution at T=0 half filling full frustration

applications of dmft to actinide materials gabriel kotliar physics department and center for...

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