Dynamical Mean Field Theory, Mott transition and Electronic Structure of Actinides Gabriel Kotliar...

Dynamical Mean Field Theory , Mott transition and Electronic Structure of Actinides

Gabriel Kotliar

Physics Department and

Center for Materials Theory

Rutgers University

SCES 2001

Ann Arbor August 6th-10th 2001

THE STATE UNIVERSITY OF NEW JERSEY

RUTGERS

Outline Introduction to Pu Background: DMFT study of the Mott

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

qualitative insights into the “Mott transition across the actinide series”


RUTGERS

Mott transition in the actinide series (Smith Kmetko phase diagram, Johanssen 1974)


RUTGERS

Phase transition with Large Volume changes! Small amounts of Ga stabilize the phase (A. Lawson LANL)


RUTGERS

The bonding problem

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. Full potential and ASA methods 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.


RUTGERS

Conventional viewpoint Alpha Pu is a simple metal, it can be

described with LDA + pert. corrections. In contrast delta Pu is strongly correlated.

Constrained LDA approach (Erickson, Wills, Balatzki, Becker). In Alpha Pu, all the 5f electrons are treated as band like, while in Delta Pu, one 5f electrons are band-like while four 5f electron is localized.

LDA + U (Savrasov andGK Phys. Rev. Lett. 2000 , Bouchet et.al 2000) predicts correct volume of Delta Pu with U=4,Alpha Pu has U =0.


RUTGERS

Problems with the conventional viewpoint of Pu

U/W is not so different in alpha and delta LDA+U, LDA, constrained LDA are not good starting

points to describe the transport and thermodynamics, Pu is a light heavy fermion.

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.


RUTGERS

Pu Specific Heat


RUTGERS

Anomalous Resistivity


RUTGERS

Outline Introduction to Pu

Background: DMFT study of the Mott transition in a toy model

DMFT as an electronic structure method. DMFT results for delta Pu, and some

qualitative insights into the “Mott transition across the actinide series”


RUTGERS

Theoretical approach to the Mott transition problem

Mean field approach to quantum many body systems, constructing equivalent impurity models embedded in a bath to be determined self consistently.

Use and compare exact and approximate numerical techniques (QMC, RG, ED) as well as semianalytical approaches (interpolative schemes) to solve the self consistent impurity model.

Formulation the DMFT equations as saddle points of a functional of the spectral function . Deeper understanding of the validity of the DMFT results.


RUTGERS

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


RUTGERS

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

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


RUTGERS

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

Magnetic Frustration


RUTGERS

Insights from DMFTThe Mott transition is driven by transfer of spectral weight from low to high energy as we approach the localized phase. Fixed density.Control parameters: doping, temperature,pressure…The laws that govern the transfer of spectral weight can be formulated around special points in the phase diagram, where bifurcations take place


RUTGERS

Mott endpoint Transfer of spectral weight at fixed density.

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


RUTGERS

Anomalous transfer of optical spectral weight in NiSeS(Miyasaka and Takagi 2000),Photoemission Matsuura et. Al. 1998


RUTGERS


transition in a toy model

DMFT as an electronic structure method.

DMFT results for delta Pu, and some qualitative insights into the “Mott transition across the actinide series”


RUTGERS

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


RUTGERS

Spectral Density Functional : effective action construction (Chitra and GK 2000).

DFT, consider the exact free energy as a functional of an external potential. Express the free energy as a functional of the density by Legendre transformation. DFT(r)]

Introduce local orbitals, R(r-R)orbitals, and local 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


RUTGERS

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. Motivated by LDA+U


RUTGERS

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

( ) ( ) ( ) ( )

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

2 | ' |

[ ]

R R

n

n KS

KS n n

i

LDAext xc

DC

R

Tr i V r r

V r r dr Tr i G i

r rV r r dr drdr E

r r

G

a b ba

w

w c c

r w w

r rr r

- +Ñ - - S -

- S +

+ + +-

F - F

åò

ò òå

Baym Kadanoff functional of an ATOM . Sum of local 2PI graphs with local U matrix and local G

1[ ] ( 1)

2DC G Un nF = - ( )0( ) iab

abi

n T G i ew

w+

= å

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


RUTGERS

LDA+DMFT: Introduction of a Weiss field, mapping onto impurity models

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


RUTGERS

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

• Removes inconsistencies of this approach,• Only in the orbitally ordered Hartree Fock limit,

the Greens function of the heavy electrons is fully coherent

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

• Luttinger theorem is obeyed.


RUTGERS


transition in a toy model DMFT as an electronic structure method.

DMFT results for delta Pu, and some qualitative insights into the “Mott transition across the actinide series”


RUTGERS

Pu: DMFT total energy vs Volume(Savrasov et.al.2001)


RUTGERS

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 (see also Majumdar and Krishnmurthy 1995).

This result resolves one of the basic paradoxes in the physics of Pu.


RUTGERS

Lda vs Exp Spectra


RUTGERS

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


RUTGERS

PU: (“cubic ALPHA” AND DELTA


RUTGERS

Double well structure and Pu Qualitative explanation

of negative thermal expansion

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


RUTGERS

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

( )dT V

dp S

Vsol

Vliq


RUTGERS

Superconductivity in Am

Atomic state J=0 How to go from a metal to a closed shell

insulator by increasing U. Entropy has to increase, as U increases, but

the insulator has zero entropy! Something has to happen

DMFT study of the problem (Capone Fabrizio and Tossatti) Superonductivity intervenes!


RUTGERS

Conclusion

The character of the localization delocalization in simple( Hubbard) models within DMFT is now fully understood, nice qualitative insights.

This has lead to extensions to more realistic models, and a beginning of a first principles approach interpolating between atoms and band, encouraging results for simple elements, (Savrasov, Kotliar, Abrahams Nature 2001 Pu), Lichtenstein Katsenelson Kotliar (PRL 2001 Fe and Ni).

Outlook: compounds, C-DMFT ………..


RUTGERS

References

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

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).


RUTGERS

Acknowledgements

Useful discussions with A. Lichtenstein, J. Thompson and R. Schrieffer

NSF-DMR , DOE (Basic Energy Sciences)


RUTGERS

1

10

1( ) ( )

( )n nn k nk

G i ii t i

w ww m w

-

-é ùê ú= +Sê ú- + - Sê úë ûå

DMFT 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 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 ( )( )[ ] ( ) [ ( ) ( ) ]n n n n S Gi G G i c i c ia bw w w w- -S = + á ñ

Weiss field


RUTGERS

Outlook Systematic improvements, short range correlations. Take a cluster of sites, include the effect of the rest

in a G0 (renormalization of the quadratic part of the effective action). What to take for G0:

DCA (M. Jarrell Krishnamurthy et.al) , CDMFT ( GK Savrasov Palsson and Biroli)

include the effects of the electrons to renormalize the quartic part of the action (spin spin , charge charge correlations) E. DMFT (Kajueter and GK, Si et.al)


RUTGERS

Outlook

Extensions of DMFT implemented on model systems, carry over to more realistic framework. Better determination of Tcs…………

First principles approach: determination of the Hubbard parameters, and the double counting corrections long range coulomb interactions E-DMFT

Improvement in the treatement of multiplet effects in the impurity solvers, phonon entropies, ……

…


RUTGERS

Wilson and Kadowaki Woods Ratio


RUTGERS

Vanadium Oxide


RUTGERS

. ARPES measurements on NiS2-xSex

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


RUTGERS

T_MIT=.013 Rozenberg et.al 2001


RUTGERS

Realistic DMFT loop

( )k LMTOt H k E® -LMTO

LL LH

HL HH

H HH

H H

é ùê ú=ê úë û

ki i Ow w®

10 niG i Ow e- = + - D

0 0

0 HH

é ùê úS =ê úSë û

0 0

0 HH

é ùê úD =ê úDë û

0

1 †0 0 ( )( )[ ] ( ) [ ( ) ( )HH n n n n S Gi G G i c i c ia bw w w w-S = + á ñ

110

1( ) ( )

( ) ( ) HH

LMTO HH

n nn k nk

G i ii O H k E i

w ww w

--é ùê ú= +Sê ú- - - Sê úë ûå


RUTGERS

V2O3


RUTGERS

Theoretical Foundations: functionals

G. Kotliar and R. Chitra PRB 1999,2000G. Kotliar and S. Savrasov 2001

LDA[ Fukuda et.al, Aliev and Fernando] LDA+U LDA+DMFT


RUTGERS

LDA functional

2log[ / 2 ] ( ) ( )

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

2 | ' |

n KS KS

LDAext xc

Tr i V V r r dr

r rV r r dr drdr E

r r

w r

r rr r

- +Ñ - -

+ +-

ò

ò ò

[ ( )]LDA r

[ ( ), ( )]LDA KSr V r

Conjugate field, VKS(r)


RUTGERS

Minimize LDA functional

[ ]( )( ) ( ) '

| ' | ( )

LDAxc

KS ext

ErV r V r dr

r r r

d rrdr

= + +-ò

0*2

( ) { )[ / 2 ]

( ) ( ) n

n

ikj kj kj

n KSkj

r f tri V

r r ew

w

r e yw

y +=

+Ñ -=å å


RUTGERS

LDA+U functional

2 *log[ / 2 . ( ) ( )]

( ) ( ) ( ) ( )

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

2 | ' |

[ ]

aR bR

n

KS abn KS

R

KS KS

i

LDAext xc

DC

R

Tr i V B r r

V r r dr B r m r dr Tr n

r rV r r dr drdr E

r r

G E

w

w s fl f

r l

r rr r

- +Ñ - - - -

- - - +

+ + +-

F -

å

åò ò

ò òå

[ ( ), ( ), ]LDA U abr m r n

, KS KS ab [ ( ), ( ), V ( ), ( ), ]LDA U a br m r n r B r

1

2 ab abcd cdn U n


RUTGERS

Double counting term (Lichtenstein et.al)

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

What is nab ?


RUTGERS

Functional Approach

† †,

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)


RUTGERS

Functional Approach The functional approach offers a direct

connection to the atomic energies. One is free to add terms which vanish quadratically at the saddle point.

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 .


RUTGERS

Solving the impurity 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

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


RUTGERS

Recent QMC phase diagram of the frustrated Half filled Hubbard model with semicircular DOS ( Joo and Udovenko 2001).


RUTGERS

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).


RUTGERS

NiSeS


RUTGERS

Ising character of Mott endpoint

Singular part of the Weiss field is proportional toMax{ (p-pc) (T- Tc)}1/in mean field and 5 in 3d

couples to all physical quantities which then exhibit a kink at the Mott endpoint. Resistivity, double occupancy,photoemission intensity, integrated optical spectral weight, etc.

Divergence of the specific heat.


RUTGERS

Mott transition endpoint

Rapid variation has been observed in optical measurements in vanadium oxide and nises mixtures

Experimental questions: width of the critical region. Ising exponents or classical exponents, validity of mean field theory

Building of coherence in other strongly correlated electron systems.

Unify concepts from different theoretical approaches, condensation of d and onset of coherence .


RUTGERS

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


RUTGERS

Cerium


RUTGERS

Pu: Anomalous thermal expansion (J. Smith LANL)


RUTGERS

MAGNETIC


RUTGERS

Specific heat and susceptibility.


RUTGERS

Remarks on the literature

The qualitative features found by the Rutgers ENS groups were challenged in a series of publications: Logan and Nozieres (1987) 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 non perturbative aspects of the Mott metal to insulator transition such as the singular behavior of the self energy


RUTGERS

Two Roles for DMFT in calculations of the electronic structure of correlated materials

Crystal Structure +atomic positions

Correlation functions Total energies etc.

Model Hamiltonian


RUTGERS

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 elemental plutonium [ B. Johanssen 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 a simple model Hamiltonian (one band Hubbard, semicircular density of states).

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

Turn the technology developed to solve the toy model into a practical electronic structure method.


RUTGERS

More on DFT

LSDA predicts magnetic long range (Solovyev et.al.0

Experimentally Pu is not magnetic. If one treats the f electrons as part of the core LDA

overestimates the volume by 30% LDA predicts correctly the volume of the phase of

Pu, when full potential LMTO (Soderlind Eriksson and Wills) is used. This is taken as an indication that Pu is a weakly correlated system


RUTGERS

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


RUTGERS

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


RUTGERS

Anomalous Resistivity and Mott transition Ni Se2-x Sx

Miyasaka and Tagaki (2000)


RUTGERS

LDA+DMFT Self-Consistency loop

G0 G

Im puritySo lver

S .C .C .

0( ) ( , , ) i

i

r T G r r i e w

w

r w+

= å

2| ( ) | ( )k xc k LMTOV H ka ac r c- Ñ + =

DMFT

U

E

0( , , )HHi

HH

i

n T G r r i e w

w

w+

= å

Date post:	21-Dec-2015
Category:	Documents
View:	214 times
Download:	1 times

Dynamical Mean Field Theory, Mott transition and Electronic Structure of Actinides Gabriel Kotliar...

Documents