Transcript Issues in Strongly Correlated Electron Physics: A DMFT

Elemental Plutonium: a strongly correlated metal
Gabriel Kotliar
Physics Department and
Center for Materials Theory
Rutgers University
Collaborators: S. Savrasov
(NJIT) X. Dai( Rutgers )
Physics of Pu
The Problem:
This?
Or this?
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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 .
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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
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Mott transition in the actinide series
(Smith Kmetko phase diagram)
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Small amounts of Ga stabilize the d
phase (A. Lawson LANL)
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Shear anisotropy.

C’=(C11-C12)/2

C44= 33.59


4.78
19.70
C44/C’ ~ 8 Largest shear anisotropy in any
element!
LDA Calculations (Bouchet) C’= -48
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Plutonium Puzzles
o
o
o
o
o
DFT in the LDA or GGA is a well established tool
for the calculation of ground state properties.
Many studies (Freeman, Koelling 1972)APW
methods
ASA and FP-LMTO Soderlind et. Al 1990, Kollar
et.al 1997, Boettger et.al 1998, Wills et.al. 1999)
give
an equilibrium volume of the d phase Is 35%
lower than experiment
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 d 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
a phase of Pu, when full potential LMTO
(Soderlind Eriksson and Wills) is used. This is
usually taken as an indication that a Pu is a
weakly correlated system
 Alterantive approach Wills et. al. (5f)4 core+
1f(5f)in conduction band.

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Pu Specific Heat
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Anomalous Resistivity
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Pu is NOT MAGNETIC
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Specific heat and susceptibility.
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Problems with the conventional
viewpoint of a 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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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
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What do we want from
materials theory?
New concepts , qualitative ideas
 Understanding, explanation of existent
experiments, and predictions of new ones.
 Quantitative capabilities with predictive
power.

Notoriously difficult to achieve in strongly
correlated materials.
We have solved “the hydrogen atom problem” of
strongly correlated electron systems.
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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)
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Generalized phase diagram
T
Structure,
bands,
orbitals
U/W
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Qualitative phase diagram in the U, T , m plane (two
band Kotliar Murthy Rozenberg PRL (2002).


Coexistence regions between localized and
delocalized spectral functions.
k diverges at generic Mott endpoints
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Mott transition in layered organic conductors
et al. Ito et.al, Kanoda’s talk Bourbonnais talk
Magnetic
Frustration
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S Lefebvre
Ultrasound study of
Fournier et. al. (2002)
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Minimum in melting curve and divergence of
the compressibility at the Mott endpoint
Vsol
Vliq
dT
V
(
)
dp
S
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Minimum of the melting point




Divergence of the compressibility at the Mott
transition endpoint.
Rapid variation of the density of the solid as
a function of pressure, in the localization
delocalization crossover region.
Slow variation of the volume as a function of
pressure in the liquid phase
Elastic anomalies, more pronounced with
orbital degeneracy.
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Minimum in melting curve and divergence of
the compressibility at the Mott endpoint
Vsol
Vliq
dT
V
(
)
dp
S
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Cerium
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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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Solving the DMFT equations
Impurity
G
G
0
Solver


G0
Impurity
Solver
G

S.C.C.
S.C.C.
•Wide
variety
of
computational
(QMC,ED….)Analytical Methods
•Extension to ordered states.
Review: A. Georges, G. Kotliar, W. Krauth and M.
Rozenberg Rev. Mod. Phys. 68,13 (1996)]

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tools
Realistic DMFT loop
iw ® iwOk
é H LL
ê
êH HL
ë
tk ® H LMTO (k ) - E
H LH ù
ú= H LMTO
H HH ú
û
iG0- 1 = iwnO + e - D
é0
0 ù
ú
D=ê
ê0 D HH ú
ë
û
é0
0 ù
ú
S=ê
ê0 S HH ú
ë
û
S HH (iwn )[G0 ] = G0- 1 (iwn ) + [áca† (iwn )cb (iwn )ñS (G0 )
é
G0- 1 (iwn ) = êê
êë
å
k
ù -1
1
ú
+ S HH (iwn )
iwnOk - H LMTO (k ) - E - S (iwn ) ú
ú
ûHH
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LDA+DMFT-outer loop relax
c ka | - Ñ 2 + Vxc (r ) | c ka = H LMTO ( k )
Impurity
Solver
G
0
f f&
Edc
G

U
S.C.C.
DMFT
r (r) = T
å
G ( r , r , i w) e
iw 0+
nHH = T
å
iw
iw
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+
GHH ( r , r , iw)eiw 0
f f&
Outer loop relax
c ka | - Ñ 2 + Vxc (r ) | c ka = H LMTO ( k )
Edc
G
G,
Impurity
Imp. Solver:
Solver
Hartree-Fock
0
SCC
DMFT
LDA+U
r (r) = T
U
å
G ( r , r , i w) e
iw 0+
nHH = T
å
iw
iw
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+
GHH ( r , r , iw)eiw 0
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
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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 mB
S=5/2 Ms=5 mB
Mtot=0
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LDA+U bands. (Savrasov GK ,
PRL 2000).
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Magnetic moment


L=5, S=5/2, J=5/2, Mtot=Ms=mB gJ =.7 mB
Crystal fields G7 +G8

GGA+U estimate (Savrasov and Kotliar 2000)
ML=-3.9 Mtot=1.1

This bit is quenched by Kondo effect of spd
electrons [ DMFT treatment]
Experimental consequence: neutrons large
magnetic field induced form factor (G. Lander).

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Pu: DMFT total energy vs Volume
(Savrasov Kotliar and Abrahams 2001)
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Double well structure and d Pu
Qualitative explanation
of negative thermal expansion
Sensitivity to impurities which easily raise the
energy of the a -like minimum.
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Dynamical Mean Field View of Pu
(Savrasov Kotliar and Abrahams, Nature 2001)


Delta and Alpha Pu are both strongly
correlated, the DMFT mean field free energy
has a double well structure, for the same
value of U. One where the f electron is a bit
more localized (delta) than in the other
(alpha).
Is the natural consequence of the model
Hamiltonian phase diagram once electronic
structure is about to vary.
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Comments on the HF static
limit


Describes only the Hubbard bands.
No QP states.
Single well structure in the E vs V curve.
(Savrasov and Kotliar PRL)

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DOS, st./[eV*cell]
Lda vs Exp Spectra
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Spectral Evolution at T=0
half filling full frustration
X.Zhang M. Rozenberg G. Kotliar (PRL 1993)
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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
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Summary
Spectra
Method
LDA
LDA+U
DMFT
E vs V
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)
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Energy vs Volume
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Energy vs Volume
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Success story : Density Functional Linear Response
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: Phonons
Solid circles – theory, open circles – exp. (Roy et.al, 1976)
DMFT Savrasov and GK PRL 2003
DMFT for Mott insulators
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Phonon freq (THz) vs q in
delta Pu (Dai et. al. )
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Shear anisotropy. Expt. vs Theory

C’=(C11-C12)/2 = 4.78 GPa

C44= 33.59 GPa


C’=3.37GPa
C44=19.7 GPa
C44/C’ ~ 8 Largest shear anisotropy in any
element!
C44/C’ ~ 6
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Phonon frequency (Thz ) vs q
in epsilon Pu.
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Temperature stabilizes a very
anharmonic phonon mode
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Phonons epsilon
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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.

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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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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.
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Conclusions


Outsanding question: electronic entropy, lattice
dynamics.
In the making, new generation of DMFT
programs, QMC with multiplets, full potential
DMFT, frequency dependent U’s, multiplet
effects , combination of DMFT with GW
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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
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DMFT MODELS.
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Mean-Field : Classical vs Quantum
Classical case
-
å
J ij Si S j - hå Si
i, j
i
HMF = - heff So
Quantum case

 (t
ij
i , j  ,
b
+ md ij )(ci† c j + c†j ci ) + U  ni  ni 
i
b
b
¶
†
c
(
t
)[
ò ò os ¶ t + m- D (t - t ')]cos (t ') + U ò no- no¯
0 0
0
heff
D (w)
m0 = áS0 ñHMF ( heff )
heff =
å
J ij m j + h
GL = - áco† (iwn )co (iwn )ñHMF (D )
G (iwn ) =
k
j
Phys. Rev. B 45, 6497
å
1
[D (iwn ) -
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1
- ek ]
G (iwn )[D ]
A. Georges, G. Kotliar (1992)
Example: Single site DMFT, functional formulation
G[S , G] = - Tr log[iwn - tij - S ] - TrS (iwn )G(iwn ) + F [G]
F DMFT =
å
F atom [Gii ]
Local self energy (Muller Hartman 89)
i
Express in terms of Weiss field
(G. Kotliar EPJB 99)
(i ) 2
F []  T 
+ Fimp []
2
t
†
†

L
[
f
,
f
]

f
( i )  ( i ) f ( i )
loc


†
 ,
  Fimp  Log[  df dfe
]

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DMFT Impurity cavity construction


i , j  ,
b
i
+  Vij ni n j
i , j 
b
òò
0
D0-
(tij + md ij )(ci† c j + c†j ci ) + U  ni ni
co†s (t )Go(t , t ')cos (t ') + no- no¯U d(t , t ')+ Do(t , t
')
no- no¯
0
1
é
(iwn ) = ê
ê
ê
ë
- 1
å
k
ù
1
ú
Vk - P (iwn ) ú
ú
û
+ P (iwn )
S (iwn )[G0 ] = G0- 1 (iwn ) + [áca† (iwn )cb (iwn )ñS (G0 ) ]- 1
P (iwn )[G0 ] = D0- 1 (iwn ) + [án0 (iwn )n0(iwn )ñS () ]é
- 1
G0 (iwn ) = ê
ê
ê
ë
1
- 1
å
k
ù
1
ú
iwn - tk + m- S (iwn ) ú
ú
û
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+ S (iwn )
DMFT Review: A. Georges, G. Kotliar, W. Krauth
and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)]


i , j  ,
(tij + md ij )(ci† c j + c†j ci ) + U  ni ni
b
S [Go] =
i
b
òò
0
b
co†s (t )[Go(t , t ')]cos (t ') + U ò no- no¯
0
0
G0- 1 (iwn ) = iwn + m- D (iwn )
Weiss field
GL (iwn ) = - áco† (iwn )co (iwn )ñS (G0 )
S (iwn )[G0 ] = G0- 1 (iwn ) + [áca† (iwn )cb (iwn )ñS (G0 ) ]é
- 1
G0 (iwn ) = ê
ê
ê
ë
- 1
å
k
ù
1
ú
iwn - tk + m- S (iwn ) ú
ú
û
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+ S (iwn )
1
Case study: IPT half filled Hubbard one band

(Uc1)exact = 2.2+_.2 (Exact diag, Rozenberg, Kajueter, Kotliar
PRB 1996) , confirmed by Noack and Gebhardt (1999)
(Uc1)IPT =2.6

(Uc2)exact =2.97+_.05(Projective self consistent method, Moeller
Si Rozenberg Kotliar Fisher PRL 1995 ), (Confirmed by R.
Bulla 1999) (Uc2)IPT =3.3

(TMIT ) exact =.026+_ .004 (QMC Rozenberg Chitra and Kotliar
PRL 1999), (TMIT )IPT =.045

(UMIT )exact =2.38 +- .03 (QMC Rozenberg Chitra and Kotliar
PRL 1999), (UMIT )IPT =2.5 (Confirmed by Bulla 2001)
For realistic studies errors due to other sources (for example
the value of U, are at least of the same order of magnitude).

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Spectral Density Functional

The exact functional can be built in perturbation
theory in the interaction (well defined diagrammatic
rules )The functional can also be constructed from
the atomic limit, but no explicit expression exists.

DFT is useful because good approximations to the
exact density functional GDFT[r(r)] exist, e.g. LDA,
GGA
A useful approximation to the exact functional can
be constructed, the DMFT +LDA functional.

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Interfacing DMFT in calculations of the
electronic structure of correlated materials
Crystal Structure
+atomic positions
Model
Hamiltonian
Correlation functions
Total energies etc.
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LDA+DMFT functional
G LDA + DMFT [ r (r ) G a b VKS(r ) ab]
- Tr log[iwn + Ñ 2 / 2 - VKS - c *a R ( r )S a b c b R ( r )] -
ò
VKS ( r )r ( r ) dr -
ò
å
Vext ( r )r ( r ) dr +
å
TrS (iwn )G (iwn ) +
iwn
1
2
ò
r ( r )r ( r ')
LDA
drdr '+ E xc
[r ] +
| r- r '|
F [G ] - F DC
R
F Sum of local 2PI graphs with local U
matrix and local G
F DC [G ] = Un(n - 1)
1
2
n= T
å (G
abiw
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+
i0
ab (iw)e
)
LDA+DMFT and LDA+U
Static limit of the LDA+DMFT functional ,
• with Fatom® FHF 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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LDA+DMFT and LDA+U
Static limit of the LDA+DMFT functional ,
• with Fatom® FHF 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 .
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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 New Theoretical approaches
to strongly correlated systems, Edited by A. Tsvelik,
Kluwer Publishers, (2001).
Held Nekrasov Blumer Anisimov and Vollhardt et.al. Int.
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
F= FHF reduces to LDA+U
Gives the local spectra and the total energy
simultaneously, treating QP and H bands on
the same footing.
Luttinger theorem is obeyed.
Functional formulation is essential for
computations of total energies, opens the way
to phonon calculations.
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References





LDA+DMFT:
V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and
G. Kotliar, J. Phys. Cond. Mat. 35, 7359-7367 (1997).
A Lichtenstein and M. Katsenelson Phys. Rev. B 57,
6884 (1988).
S. Savrasov G.Kotliar funcional formulation for full
self consistent implementation of a spectral density
functional.
Application to Pu S. Savrasov G. Kotliar and E.
Abrahams (Nature 2001).
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DMFT: Effective Action point of view.
R. Chitra and G. Kotliar Phys Rev. B.
(2000), (2001).
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Identify observable, A. Construct an exact functional of
<A>=a, G [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 l conjugate
to a, G [a, l ].
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
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Introduce local orbitals, caR(r-R), and local GF
G(R,R)(i ) =
dr ' dr c R (r ) *G (r , r ')(i ) ca R (r ')
 
The exact free energy can be expressed as a
functional of the local Greens function and of the
density by introducing sources for r(r) and G and
performing a Legendre transformation,
G[r(r),G(R,R)(i)]
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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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References
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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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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)
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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
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Multiorbital situation and several atoms per
unit cell considerably increase the size of the
space H (of heavy electrons).
QMC scales as [N(N-1)/2]^3 N dimension of
H
Fast interpolation schemes (Slave Boson at
low frequency, Roth method at high
frequency, + 1st mode coupling correction),
match at intermediate frequencies. (Savrasov
et.al 2001)
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Technical details

Atomic sphere approximation.
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Ignore crystal field splittings in the self energies.

Fully relativistic non perturbative treatment of the
spin orbit interactions.
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