THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline Model Hamiltonians and qualitative considerations...

THE STATE UNIVERSITY OF NEW JERSEY

RUTGERS

Outline Model Hamiltonians and qualitative

considerations in the physics of materials. Or what do we want to know? An example from the physics of the Mott transition.

Merging band structure methods with many body theory, where to improve? A) basis set? B) parameter estimates of your model Hamiltonian C) DMFT impurity solver? D) Improvements of DMFT ? An intro to Cellular DMFT [G. Kotliar S. Savrasov G. Palsson and G. Biroli PRL87, 186401 2001]


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Mott transition in the actinide series. B. Johanssen 1974 Smith and Kmetko Phase Diagram 1984.


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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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Robustness of the finite T results

Underlying Landau Free energy which is responsible of all the qualitative features of the phase diagram. Of the frustrated Hubbard model in large d [G. Kotliar EPJB 99]

Around the finite temperature Mott endpoint, the Free energy has a simple Ising like form as in a liquid gas transition [R. Chitra, G. Kotliar E.Lange M. Rozenberg ]

Changing the model (DOS, degeneracy, etc) just changes the coefficients of the Landau theory.


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Robustness of the finite T results and Functional Approach

Different impurity solvers, different values of the Landau coefficients, as long as they preserve the essential (non) analytic properties of the free energy functional.

The functional approach can be generalized to combine DFT and DMFT [R. Chitra G. Kotliar , S. Savrasov and G. Kotliar]

Justification for applying simple models to some aspects of the crossover in Ni(SeS)2And V2O3.


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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.


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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 : One band case (Kotliar Murthy and Rozenberg 2001, cond-matt 0110625)


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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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A (non comprehensive )list of extensions of DMFT

Two impurity method. [A. Georges and G. Kotliar, A. Schiller PRL75, 113 (1995)]

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

Continuous version [periodic cluster] M. Katsenelson and A. Lichtenstein PRB 62, 9283 (2000).

Extended DMFT [H. Kajueter and G. Kotliar

Rutgers Ph.D thesis 2001, Q. Si and J L Smith PRL 77 (1996)3391 ] Coulomb interactions R . Chitra

Cellular DMFT [PRL87, 186401 2001]


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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 cavity construction

†

0 0 0

[ ] ( )[ ( , ')] ( ')o o o oS Go c Go c U 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


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Elements of the Dynamical Mean Field Construction and Cellular DMFT, G. Kotliar S. Savrasov G. Palsson and G. Biroli PRL 2001

Definition of the local degrees of freedom Expression of the Weiss field in terms of the

local variables (I.e. the self consistency condition)

Expression of the lattice self energy in terms of the cluster self energy.


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Cellular DMFT : Basis selection


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Lattice action


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Elimination of the medium variables


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Determination of the effective medium.


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Connection between cluster and lattice self energy.

The estimation of the lattice self energy in terms of the cluster energy has to be done using additional

information Ex. Translation invariance

•C-DMFT is manifestly causal: causal impurity solvers result in causal self energies and Green functions (GK S. Savrasov G. Palsson and G. Biroli PRL 2001)•In simple cases C-DMFT converges faster than other causal cluster schemes.


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Improved estimators

• Improved estimators for the lattice self energy are available (Biroli and Kotliar)


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Real Space Formulation of the DCA approximation of Jarrell et.al.


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Affleck Marston model.


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Convergence test in the Affleck Marston


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Convergence of the self energy


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Recent application to high Tc

A. Perali et.al. cond-mat 2001, two patch model, phenomenological fit of the functional form of the vertex function of C-DMFT to experiments in optimally doped and overdoped cuprates

Flexibility in the choice of basis seems important.


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Extended DMFT electron phonon


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Extended DMFT e.ph. Problem


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E-DMFT classical case, soft spins


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E-DMFT classical case Ising limit


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E-DMFT test in the classical case[Bethe Lattice, S. Pankov 2001]


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Advantage and Difficulties of E-DMFT

The transition is first order at finite temperatures for d< 4

No finite temperature transition for d less than 2 (like spherical approximation)

Improved values of the critical temperature


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Conclusion

For “first principles work” there are several many body tools waiting to be used, once the one electron aspects of the problem are clarified.

E-DMFT or C-DMFT for Ni, and Fe ? Promising problem: Qualitative aspects of

the Mott transition within C-DMFT ?? Cuprates?

Realistic Theories of Correlated Materials

ITP, Santa-Barbara

July 27 – December 13 (2002)

O.K. Andesen, A. Georges,

G. Kotliar, and A. Lichtenstein

http://www.itp.ucsb.edu/activities/future/


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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)


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Recent phase diagram of the frustrated Half filled Hubbard model with semicircular DOS (QMC Joo and Udovenko PRB2001).


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Case study: IPT half filled Hubbard one band

(Uc1)exact = 2.1 (Exact diag, Rozenberg, Kajueter, Kotliar 1995) , (Uc1)IPT =2.4

(Uc2)exact =2.95 (Projective self consistent method, Moeller Si Rozenberg Kotliar PRL 1995 ) (Uc2)IPT =3.3

(TMIT ) exact =.026+_ .004 (QMC Rozenberg Chitra and Kotliar PRL 1999), (TMIT )IPT =.5

(UMIT )exact =2.38 +- .03 (QMC Rozenberg Chitra and Kotliar PRL 1991), (UMIT )IPT =2.5 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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The Mott transition as a bifurcation in effective action

[ , ]G [ , ]0

G

G

2 [ , ]0cG

G G

Zero mode with S=0 and p=0, couples generically

Divergent compressibility (R. Chitra and G.Kotliar


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Realistic implementation of the self consistency condition

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

•H and S, do not commute•Need to do k sum for each frequency •DMFT implementation of Lambin Vigneron tetrahedron integration (Poteryaev et.al 1987)


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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)


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Good method to study the Mott phenomena

Evolution of the electronic structure between the atomic limit and the band limit. Basic solid state problem. Solved by band theory when the atoms have a closed shell. Mott’s problem: Open shell situation.

The “”in between regime” is ubiquitous central them in strongly correlated systems. Some unorthodox examples

Fe, Ni, Pu …………….


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Two Roads for calculations of the electronic structure of correlated materials

Crystal Structure +atomic positions

Correlation functions Total energies etc.

Model Hamiltonian


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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)


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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


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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

w

w s fl f

r l

r rr r

- +Ñ - - - -

- - - +

+ + +-

F - F

å

åò ò

ò òå

1[ ] ( 1)

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

abi

n T G i ew

w+

= å

[ ( ), ( ), ]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


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LDA+DMFT

The light, SP (or SPD) electrons are extended, well described by LDA

The heavy, D (or F) electrons are localized,treat by DMFT.

LDA already contains an average interaction of the heavy electrons, substract this out by shifting the heavy level (double counting term)

The U matrix can be estimated from first principles of viewed as parameters


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Spectral Density Functional : effective action construction (Fukuda, Valiev and Fernando , Chitra and GK).

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


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


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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.


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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 [ [ ]]atW

G G

Weiss field


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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

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


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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 References V. Anisimov, A. Poteryaev, M. Korotin, A.

Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359-7367 (1997).

ALichtensteinandM.KatsenelsonPhys.Rev.B57,6884(1988).

S.SavrasovandG.Kotliar,funcionalformulationforfullselfconsistentimplementation(2001)


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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

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


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



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Case study in f electrons, Mott transition in the actinide series


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Pu: Anomalous thermal expansion (J. Smith LANL)


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Small amounts of Ga stabilize the phase


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Delocalization-Localization across the actinide series

o f electrons in Th Pr U Np are itinerant . From Am on they are localized. Pu is at the boundary.

o Pu has a simple cubic fcc structure,the phase which is easily stabilized over a wide region in the T,p phase diagram.

o The phase is non magnetic.o Many LDA , GGA studies ( Soderlind et. Al 1990,

Kollar et.al 1997, Boettger et.al 1998, Wills et.al. 1999) give an equilibrium volume of the an equilibrium volume of the phasephaseIs 35% lower than experimentIs 35% lower than experiment

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


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Problems with LDA

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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Problems with LDA LSDA predicts magnetic long range order which is

not observed experimentally (Solovyev et.al.) 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 and Wills). This is usually taken as an indication that Pu is a weakly correlated system


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

described with LDA + correction. 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, 4 5f electrons are band-like while one 5f electron is deloclized.

Same situation in LDA + U (Savrasov and Kotliar, Bouchet et. Al. ) Delta Pu has U=4,

Alpha Pu has U =0.


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Problems with the conventional viewpoint of Pu

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.

Only the structural and elastic properties are completely different.


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Pu Specific Heat


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Anomalous ResistivityJ. Smith LANL


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MAGNETIC SUSCEPTIBILITY


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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.

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


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Pu: DMFT total energy vs Volume


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Lda vs Exp Spectra


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


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Earlier Studies of Magnetic Anisotropy

Erickson Daalderop


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Case study Fe and Ni Archetypical itinerant ferromagnets LSDA predicts correct low T moment Band picture holds at low T Main challenge, finite T properties

(Lichtenstein’s talk). Magnetic anisotropy puzzle. LDA predicts the

incorrect easy axis for Nickel . LDA Fermi surface has features which are not

seen in DeHaas Van Alphen ( Lonzarich)


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Iron and Nickel: crossover to a real space picture at high T


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Photoemission Spectra and Spin Autocorrelation: Fe (U=2, J=.9ev,T/Tc=.8) (Lichtenstein, Katsenelson,GK prl 2001)


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Photoemission and T/Tc=.8 Spin Autocorrelation: Ni (U=3, J=.9 ev)


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

0 3( )q

Meff

T Tc

0 3( )q

Meff

T Tc


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Ni and Fe: theory vs exp ( T=.9 Tc)/ ordered moment

Fe 1.5 ( theory) 1.55 (expt) Ni .3 (theory) .35 (expt)

eff high T moment

Fe 3.1 (theory) 3.12 (expt) Ni 1.5 (theory) 1.62 (expt)

Curie Temperature Tc

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


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Fe and Ni Satellite in minority band at 6 ev, 30 % reduction

of bandwidth, exchange splitting reduction .3 ev Spin wave stiffness controls the effects of spatial

flucuations, it is about twice as large in Ni and in Fe

Mean field calculations using measured exchange constants(Kudrnovski Drachl PRB 2001) right Tc for Ni but overestimates Fe , RPA corrections reduce Tc of Ni by 10% and Tc of Fe by 50%.


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Ni moment


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Fe moment\


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Magnetic anisotropy Ni


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Magnetic anisotropy Fe


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Magnetic anisotropy


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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


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


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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 et.al) , CDMFT ( Savrasov and GK )

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)


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Outlook

Extensions of DMFT implemented on model systems, (e.g. Motome and GK ) 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

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


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


THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline Model Hamiltonians and qualitative considerations...

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Transcript of THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline Model Hamiltonians and qualitative considerations...