Elemental Plutonium: a strongly correlated metal Gabriel Kotliar Physics Department and Center for...
-
date post
15-Jan-2016 -
Category
Documents
-
view
221 -
download
0
Transcript of Elemental Plutonium: a strongly correlated metal Gabriel Kotliar Physics Department and Center for...
Elemental Plutonium: a strongly correlated metal
Gabriel Kotliar
Physics Department and
Center for Materials Theory
Rutgers University
Collaborators: S. Savrasov (NJIT) X. Dai( Rutgers )
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Physics of Pu
The Problem:This? Or this?
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
For me the problem is :THIS. The Mott Phenomena
Evolution of the electronic structure between the atomic limit and the band limit in an open shell situation.
The “”in between regime” is ubiquitous central them in strongly correlated systems, gives rise to interesting physics. Example Mott transition across the actinide series [ B. Johansson Phil Mag. 30,469 (1974)]
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 .
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Outline Introduction: some Pu puzzles. Results: Minimum of the melting curve, Delta Pu: Most probable valence, size of the
local moment Equilibrium Volume. Photoemission Spectral. Stabilization of Epsilon Pu: Conclusions
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Mott transition in the actinide series (Smith Kmetko phase diagram)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Small amounts of Ga stabilize the phase (A. Lawson LANL)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Shear anisotropy.
C’=(C11-C12)/2 4.78
C44= 33.59 19.70
C44/C’ ~ 8 Largest shear anisotropy in any element!
LDA Calculations (Bouchet) C’= -48
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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
Alterantive approach Wills et. al. (5f)4 core+ 1f(5f)in conduction band.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Pu Specific Heat
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Anomalous Resistivity
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Pu is NOT MAGNETIC
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Specific heat and susceptibility.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Outline Introduction: some Pu puzzles. DMFT , qualitative aspects of the Mott
transition from model Hamiltonians DMFT as an electronic structure method. DMFT results for delta Pu, and some
qualitative insights. Conclusions
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
What do we want from materials theory?
New concepts , qualitative ideas Understanding, explanation of existent
experiments, and predictions of new ones. Quantitative capabilities with predictivepower.
Notoriously difficult to achieve in strongly correlated materials.
We have solved “the hydrogen atom problem” of strongly correlated electron systems.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Evolution of the Spectral Function with Temperature
Anomalous transfer of spectral weight connected to the proximity to the Ising Mott endpoint (Kotliar Lange and Rozenberg Phys. Rev. Lett. 84, 5180 (2000)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Generalized phase diagram
T
U/WStructure, bands,
orbitals
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Mott transition in layered organic conductors S Lefebvre et al. Ito et.al, Kanoda’s talk Bourbonnais talk
Magnetic Frustration
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Ultrasound study of
Fournier et. al. (2002)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Minimum in melting curve and divergence of the compressibility at the Mott endpoint
( )dT V
dp S
Vsol
Vliq
THE STATE UNIVERSITY OF NEW JERSEY
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
Elastic anomalies, more pronounced with orbital degeneracy.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Minimum in melting curve and divergence of the compressibility at the Mott endpoint
( )dT V
dp S
Vsol
Vliq
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Cerium
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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 .
THE STATE UNIVERSITY OF NEW JERSEY
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
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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 &
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Outline Introduction: some Pu puzzles. DMFT , qualitative aspects of the Mott
transition in model Hamiltonians. DMFT as an electronic structure method. Realistic DMFT and Plutonium Conclusions
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
LDA+U bands. (Savrasov GK ,PRL 2000).
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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).
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Pu: DMFT total energy vs Volume (Savrasov Kotliar and Abrahams 2001)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Double well structure and Pu Qualitative explanation
of negative thermal expansion
Sensitivity to impurities which easily raise the energy of the -like minimum.
THE STATE UNIVERSITY OF NEW JERSEY
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.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Lda vs Exp Spectra
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
X.Zhang M. Rozenberg G. Kotliar (PRL 1993)
Spectral Evolution at T=0 half filling full frustration
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Pu Spectra DMFT(Savrasov) EXP (Arko Joyce Morales Wills Jashley PRB 62, 1773 (2000)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Comparaison with LDA+U
Summary
LDA
LDA+U
DMFT
Spectra Method E vs V
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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?
Having a functional, that computes total energies opens the way to the computation of phonon frequencies in correlated materials (S. Savrasov and G. Kotliar 2002)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Energy vs Volume
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
DMFT for Mott insulators
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Phonon freq (THz) vs q in delta Pu (Dai et. al. )
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Shear anisotropy. Expt. vs Theory
C’=(C11-C12)/2 = 4.78 GPa C’=3.37GPa
C44= 33.59 GPa C44=19.7 GPa
C44/C’ ~ 8 Largest shear anisotropy in any element!
C44/C’ ~ 6
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Phonon frequency (Thz ) vs q in epsilon Pu.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Temperature stabilizes a very anharmonic phonon mode
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Phonons epsilon
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Phonon entropy drives the epsilon delta phase transition
Epsilon is slightly more metallic than delta, but it has a much larger phonon entropy than delta.
At the phase transition the volume shrinks but the phonon entropy increases.
Estimates of the phase transition neglecting the
Electronic entropy: TC 600 K.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Conclusions DMFT produces non magnetic state, around a
fluctuating (5f)^5 configuraton with correct volume the qualitative features of the photoemission spectra, and a double minima structure in the E vs V curve.
Correlated view of the alpha and delta phases of Pu. Interplay of correlations and electron phonon interactions (delta-epsilon).
Calculations can be refined.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Acknowledgements: Development of DMFT
Collaborators: V. Anisimov, R. Chitra, V. Dobrosavlevic, X. Dai, D. Fisher, A. Georges, H. Kajueter, W.Krauth, E. Lange, A. Lichtenstein, G. Moeller, Y. Motome, G. Palsson, M. Rozenberg, S. Savrasov, Q. Si, V. Udovenko, I. Yang, X.Y. Zhang
Support: NSF DMR 0096462
Support: Instrumentation. NSF DMR-0116068
Work on Fe and Ni: ONR4-2650
Work on Pu: DOE DE-FG02-99ER45761 and LANL subcontract No. 03737-001-02
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
DMFT MODELS.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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)
THE STATE UNIVERSITY OF NEW JERSEY
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 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
† †
, ,
( )( )ij ij i j j i i ii j i
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+
THE STATE UNIVERSITY OF NEW JERSEY
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 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
THE STATE UNIVERSITY OF NEW JERSEY
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).
THE STATE UNIVERSITY OF NEW JERSEY
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.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Interfacing DMFT in calculations of the electronic structure of correlated materials
Crystal Structure +atomic positions
Correlation functions Total energies etc.
Model Hamiltonian
THE STATE UNIVERSITY OF NEW JERSEY
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
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
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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
®
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
E-DMFT +GW P. Sun and G. Kotliar Phys. Rev. B 2002
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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 .
®
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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).
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Energy difference between epsilon and delta
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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).
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
E-DMFT+GW effective action
G=
D=
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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).
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Technical details
Atomic sphere approximation.
Ignore crystal field splittings in the self energies.
Fully relativistic non perturbative treatment of the spin orbit interactions.