Download Electronic Structure Near the Mott transition

Electronic Structure Near
the Mott transition
Gabriel Kotliar
Physics Department and
Center for Materials Theory
Rutgers University
Outline

Introduction to the strong correlation
problem and to the Mott transition
Some dynamical mean field ideas

Applications to the Mott transition problem: some
insights from studies of model Hamiltonians.

Towards an electronic structure method: applications to
materials.

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RUTGERS
The electron in a solid: wave picture
Momentum Space , bands, k in Brillouin zone is good
quantum number.

e 2 k F (k F l )
h
Maximum metallic
resistivity 200 mohm cm
Landau Fermi liquid theory interactions renormalize
away at low energy, simple band picture in effective
field holds.
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RUTGERS
Standard Model of Solids

Qualitative predictions: low temperature dependence of
thermodynamics and transport
 ~ const
CV ~ T
RH ~ const
Optical response, transitions between bands.
Qualitative predictions. Filled bands-Insulators, Unfilled
bands metals. Odd number of electrons metallicity.
Quantitative tools: DFT, LDA, GGA, total energies,good
starting point for spectra, GW,and transport
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RUTGERS
The electron in a solid: particle picture.

NiO, MnO, …Array of atoms is insulating if a>>aB. Mott:
correlations localize the electron
e_
e_
e_
e_
•Superexchange

•Think in real space , solid collection of atoms
•High T : local moments, Low T spin-orbital order
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1
T
Mott : Correlations localize the electron
Low densities, electron behaves as a particle,use
atomic physics, work in real space.
•One particle excitations: Hubbard Atoms: sharp excitation
lines corresponding to adding or removing electrons. In
solids they broaden by their incoherent motion, Hubbard
bands (eg. bandsNiO, CoO MnO….)
•H H H+ H H H
motion of H+ forms the lower
Hubbard band
•H H H H- H H
motion of H_ forms the upper
Hubbard band
• Quantitative calculations of Hubbard bands and
exchange constants, LDA+ U, Hartree Fock. Atomic
RUTGERS
Physics.
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Localization vs Delocalization
Strong Correlation Problem
• A large number of compounds with electrons in partially filled
shells, are not close to the well understood limits (localized
or itinerant). Non perturbative problem.
•These systems display anomalous behavior
(departure from the standard model of solids).
•Neither LDA or LDA+U or Hartree Fock work well.
•Dynamical Mean Field Theory: Simplest approach
to electronic structure, which interpolates correctly
between atoms and bands. Treats QP bands and
Hubbard bands.
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RUTGERS
Correlated Materials do big things




Huge resistivity changes V2O3.
Copper Oxides. .(La2-x Bax) CuO4 High
Temperature Superconductivity.150 K in the
Ca2Ba2Cu3HgO8 .
Uranium and Cerium Based Compounds.
Heavy Fermion Systems,CeCu6,m*/m=1000
(La1-xSrx)MnO3 Colossal Magnetoresistance.
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Strongly Correlated Materials.



Large thermoelectric response in CeFe4 P12
(H. Sato et al. cond-mat 0010017). Ando
et.al. NaCo2-xCuxO4 Phys. Rev. B 60,
10580 (1999).
Huge volume collapses, Ce, Pu……
Large and ultrafast optical nonlinearities
Sr2CuO3 (T Ogasawara et.a Phys. Rev.
Lett. 85, 2204 (2000) )
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RUTGERS
The Mott transition
Electronically driven MIT.
 Forces to face directly the localization
delocalization problem.
 Relevant to many systems, eg V2O3
 Techniques applicable to a very broad
range or problems.

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Mott transition in V2O3 under pressure
or chemical substitution on V-site
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Universal and non universal
features. Top to bottom approach
to correlated materials.


Some aspects at high temperatures,
depend weakly on the material (and on the
model).
Low temperature phase diagram, is very
sensitive to details, in experiment (and in
the theory).
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Mott transition in layered organic conductors
al. cond-mat/0004455, Phys. Rev. Lett. 85, 5420 (2000)
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S Lefebvre et
Failure of the Standard Model:
Miyasaka and
NiSe2-xSx
Takagi (2000)
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RUTGERS
Phase Diagrams :V2O3, Ni Se2-x Sx Mc Whan
et. Al 1971,. Czek et. al. J. Mag. Mag. Mat. 3, 58 (1976),
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Outline





Introduction to the strong correlation problem
and to the Mott transition.
DMFT ideas
Applications to the Mott transition problem:
some insights from studies of models.
Towards an electronic structure method:
applications to materials: NiO, Pu, Fe, Ni,
LaSrTiO3, ……….
Outlook
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Hubbard model

 (t
i , j  ,
ij
 m ij )(c c j  c ci )  U  ni ni
†
i
†
j
i
U/t
Doping  or chemical potential
Frustration (t’/t)
T temperature
Mott transition as a function of doping, pressure
RUTGERS
temperature etc.
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Limit of large lattice
coordination
tij ~
1
 c c j  ~
d
1
d   ij nearest neighbors
d
†
i
Uni  ni  ~O(1)
  tij  ci† c j  ~ d
j ,
1
d
1
~ O (1)
d
Metzner Vollhardt, 89
1
G (k , i ) 
i   k  (i )
Muller-Hartmann 89
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Mean-Field : Classical vs Quantum
Classical case
-
å
J ij Si S j - h å Si
i, j
i
H MF = - heff So
Quantum case

 (t
ij
i , j  ,
b
 m 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 ñH MF ( heff )
heff =
å
J ij m j + h
G = ­ áco†s (iwn )cos (iwn )ñSMF (D )
G (iwn ) =
å
k
j
Phys. Rev. B 45, 6497
1
[D (iwn ) -
THE STATE UNIVERSITY OF NEW JERSEY
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1
- ek ]
G (iwn )[D ]
A. Georges, G. Kotliar (1992)
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, clusters……..
Review: A. Georges, G. Kotliar, W. Krauth and M.
Rozenberg Rev. Mod. Phys. 68,13 (1996)]

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tools
DMFT: Effective Action point of view.
R. Chitra and G. Kotliar Phys Rev. B.
(2000).




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.
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Example: DMFT for lattice model (e.g.
single band Hubbard).Muller Hartman 89,
Chitra and Kotliar 99.



Observable: Local Greens function Gii ().
Exact functional G [Gii () ].
DMFT Approximation to the functional.
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Extensions of DMFT.
Renormalizing the quartic term in the local
impurity action.
EDMFT.
 Taking several sites (clusters) as local
entity.
CDMFT
 Combining DMFT with other methods.
LDA+DMFT, GW+EDMFT or “GWU”…..

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Outline





Introduction to the strong correlation
problem.
Essentials of DMFT
Applications to the Mott transition problem:
some insights from studies of models.
Towards an electronic structure method:
applications to materials
Outlook
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RUTGERS
Schematic DMFT phase diagram Hubbard model
(partial frustration). Evidence for QP peak in V2O3
from optics.
M. Rozenberg G. Kotliar H. Kajueter G Thomas D. Rapkine J Honig and P
Metcalf Phys. Rev. Lett. 75, 105 (1995)
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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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Spectral Evolution at T=0 half filling full
frustration. Three peak structure.
X.Zhang M. Rozenberg G. Kotliar (PRL 1993)
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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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Insights from DMFT


Three peak structure of the density of
states.
In the strongly correlated metallic regime
the Hubbard bands are well formed.
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Insights from DMFT
The Mott transition is driven
by transfer of spectral weight
from low to high energy as we
approach the localized phase
Control parameters: doping,
temperature,pressure…
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
What about experiments?
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Parallel development: Fujimori et.al
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Mott transition in V2O3 under pressure
or chemical substitution on V-site
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Anomalous transfer of optical
spectral weight V2O3
:M Rozenberg G. Kotliar and H. Kajuter Phys. Rev. B 54, 8452 (1996).
M. Rozenberg G. Kotliar H. Kajueter G Tahomas D. Rapkikne J Honig and P Metcalf
Phys. Rev. Lett. 75, 105 (1995)
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Anomalous transfer of spectral
weight in v2O3
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Anomalous Spectral Weight Transfer:
Optics
H hamiltonian, J electric current , P polarization


 ne2
0  ( )d  iV   P, J ]  m
Below energy


0
 ( )d 

iV
H eff , J eff , Peff

  Peff , J eff  
ApreciableT dependence
found.
Schlesinger et.al (FeSi) PRL 71 ,1748 , (1993) B Bucher et.al. Ce2Bi4Pt3PRL
72, 522 (1994), Rozenberg et.al. PRB 54, 8452, (1996).
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ARPES measurements on NiS2-xSex
.Matsuura et. al Phys. Rev B 58 (1998) 3690. Doniach and Watanabe Phys. Rev. B 57,
3829 (1998)
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RUTGERS
Anomalous transfer of optical spectral
weight, NiSeS. [Miyasaka and Takagi
2000]
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Anomalous Resistivity and Mott
transition Ni Se2-x Sx
Insights from DMFT: think in term of spectral
functions (branch cuts) instead of well defined
QP (poles )
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Recent exps. Moo et. al. (2003)
Theory Held et. al.
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•Transport in 2d organics. Limlet et. al.
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Strong correlation anomalies



Metals with resistivities which exceed the
Mott Ioffe Reggel limit.
Transfer of spectral weight which is non
local in frequency.
Dramatic failure of DFT based
approximations in predicting physical
properties.
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Conclusions: generic aspects



Three peak structure, quasiparticles and
Hubbard bands.
Non local transfer of spectral weight.
Large resistivities.
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Insights from DMFT
.


Important role of the incoherent part of the
spectral function at finite temperature
Physics is governed by the transfer of
spectral weight from the coherent to the
incoherent part of the spectra. Real and
momentum space pictures are needed as
synthesized in DMFT.
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Outline





Introduction to the strong correlation problem.
Essentials of DMFT
Applications to the Mott transition problem: some
insights from studies of models.
Towards an electronic structure method:
applications to materials: Pu, Fe, Ni, Ce, LaSrTiO3,
NiO,MnO,CrO2,K3C60,2d and quasi-1d organics,
magnetic semiconductors,SrRuO4,V2O3………….
Outlook
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RUTGERS
Interface DMFT with
electronic structure.



Derive model Hamiltonians, solve by DMFT
(or cluster extensions).
Full many body aproach, treat light electrons by
GW or screened HF, heavy electrons by DMFT .
Treat correlated electrons with DMFT and light
electrons with DFT (LDA, GGA +DMFT)
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RUTGERS
DMFT: Effective Action point of view.
R. Chitra and G. Kotliar Phys Rev. B.
(2000).




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.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Spectral Density Functional : effective
action construction (Chitra and GK).




Introduce local orbitals, aR(r-R)orbitals, and local
GF
G(R,R)(i ) =  dr ' dr  (r ) *G(r , r ')(i ) a (r ')
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)]
Approximate functional using DMFT insights.
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Very Partial list of application of
realistic DMFT to materials








QP bands in ruthenides: A. Liebsch et al (PRL 2000)
N phase of Pu: Savrasov GK and Abrahams (Nature
2001)
Dai Savrasov GK Migliori Letbetter and Abrahams (Science
2003)
MIT in V2O3: K. Held et al (PRL 2001)
Magnetism of Fe, Ni: A. Lichtenstein et al PRL (2001)
J-G transition in Ce: K. Held et al (PRL 2000); M. Zolfl
et al PRL (2000).
3d doped Mott insulator La1-xSrxTiO3 (Anisimov et.al
1997, Nekrasov et.al. 1999, Udovenko et.al 2003)
Paramagnetic Mott insulators. NiO MnO, Savrasov and
RUTGERS
GK( PRL 2002)……………………
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Case study in f electrons, Mott transition in
the actinide series. B. Johanssen 1974
Smith and Kmetko Phase Diagram 1984.
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RUTGERS
Physics of Pu
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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  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  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
 Alternative approach Wills et. al. (5f)4 core+
1f(5f)in conduction band.

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RUTGERS
Shear anisotropy fcc Pu (GPa)

C’=(C11-C12)/2

C44= 33.59


= 4.78
C44/C’ ~ 8 Largest shear anisotropy in any
element!
LDA Calculations (Bouchet et. al.) C’= -48
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Pu: DMFT total energy vs Volume
(Savrasov Kotliar and Abrahams 2001)
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Phonon freq (THz) vs q in delta Pu X.
Dai et. al. Science vol 300, 953, 2003
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Functional approach allows
computation of linear
response.(S. Savrasov and GK
2002)
Apply to NiO, canonical Mott insulator.
U=8 ev, J=.9ev
Simple Impurity solver Hubbard 1.
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RUTGERS
Results for NiO: Phonons (Savrasov and Kotliar
PRL 2002)
Solid circles – theory, open circles – exp. (Roy et.al, 1976)
DMFT
Phases of Pu
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Dai et. al.
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Epsilon Plutonium.
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Outline





Introduction to the strong correlation
problem.
Essentials of DMFT
Applications to the Mott transition problem:
some insights from studies of models.
Towards an electronic structure method:
applications to materials:
Outlook
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RUTGERS
Outlook


Local approach to strongly correlated
electrons.
Many extensions, make the approach
suitable for getting insights and quantitative
results in correlated materials.
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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 to the electronic structure of
correlated materials.
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Outlook



Systematic improvements, short range
correlations, cluster methods, improved
mean fields.
Improved interfaces with electronic
structure.
Exploration of complex strongly correlated
materials. Correlation effects on surfaces,
large molecules, systems out of equilibrium,
illumination, finite currents, aeging.
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Acknowledgements: Development of DMFT
Collaborators: V. Anisimov,G. Biroli, R. Chitra, V.
Dobrosavlevic, X. Dai, D. Fisher, A. Georges, H.
Kajueter, K. Haujle, W.Krauth, E. Lange, A. Lichtenstein,
G. Moeller, Y. Motome, O. Parcollet , 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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Expts’ Wong et. al.
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E-DMFT references






H. Kajueter and G. Kotliar (unpublished and
Kajuter’s Ph.D thesis (1995)).
Q. Si and J L Smith PRL 77 (1996)3391 .
R. Chitra and G.Kotliar Phys. Rev. Lett 84, 36783681 (2000 )
Y. Motome and G. Kotliar. PRB 62, 12800 (2000)
R. Chitra and G. Kotliar Phys. Rev. B 63, 115110
(2001)
S. Pankov and G. Kotliar PRB 66, 045117 (2002)
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DMFT Impurity cavity construction


i , j  ,
b
i
  Vij ni n j
i , j 
b
òò
0
D0-
(tij  m 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 )
Cluster extensions of DMFT




Two impurity method. [A. Georges and G. Kotliar
(1995 unpublished ) and RMP 68,13 (1996) , A.
Schiller PRL75, 113 (1995)]
M. Jarrell et al Dynamical Cluster Approximation
[Phys. Rev. B 7475 1998]
Periodic cluster] M. Katsenelson and A.
Lichtenstein PRB 62, 9283 (2000).
G. Kotliar S. Savrasov G. Palsson and G. Biroli
Cellular DMFT [PRL87, 186401 2001]
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C-DMFT
G0 ¾ ¾
® G0ac b
tij ¾ ¾
® tija b
S ¾¾
®S
C:DMFT The lattice self energy is inferred
from the cluster self energy.
Alternative approaches DCA (Jarrell et.al.)
Periodic clusters (Lichtenstein and Katsnelson)
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c
ab
C-DMFT: test in one dimension.
(Bolech, Kancharla GK cond-mat 2002)
Gap vs U, Exact solution
Lieb and Wu,
Ovshinikov
Nc=2 CDMFT
vs Nc=1
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
DMFT plus other methods.




DMFT+ LDA , 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 and G.Kotliar, funcional formulation for full self
consistent implementation. Savasov Kotliar and Abrahams .
Application to delta Pu Nature (2001)
Combining EDMFT with GW. Ping Sun and Phys. Rev. B 66,
085120 (2002). G. Kotliar and S. Savrasov in New
Theoretical Approaches to Strongly Correlated Systems,
A. M. Tsvelik Ed. 2001 Kluwer Academic Publishers. 259301 . cond-mat/0208241
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
ARPES measurements on NiS2-xSex
.Matsuura et. Al Phys. Rev B 58 (1998) 3690. Doniaach and Watanabe Phys. Rev. B 57,
3829 (1998)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
LDA+DMFT Self-Consistency loop
c ka | ­ Ñ 2 + Vxc (r ) | c ka = H LMTO ( k )
Impurity
Solver
G
0
E
G
U

S.C.C.
DMFT
r (r) = T
å
G( r, r, iw)e
iw0+
nHH = T
å
iw
iw
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
+
GHH ( r , r , iw)eiw 0
QP in V2O3 was recently
found Mo et.al
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Ni and Fe: theory vs exp



m( T=.9 Tc)/ mB
ordered moment
Fe 1.5 ( theory)
Ni .3
(theory)
meff / mB
1.55 (expt)
.35 (expt)
high T moment
Fe 3.1 (theory) 3.12 (expt)
Ni 1.5 (theory) 1.62 (expt)
Curie Temperature Tc


Fe 1900
Ni 700
( theory)
(theory)
1043(expt)
631 (expt)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
LDA+DMFT Self-Consistency loop
c ka | ­ Ñ 2 + Vxc (r ) | c ka = H LMTO ( k )
Impurity
Solver
G
0
E
G
U

S.C.C.
DMFT
r (r) = T
å
G( r, r, iw)e
iw0+
nHH = T
å
iw
iw
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
+
GHH ( r , r , iw)eiw 0
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
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
+
i0
ab (iw)e
)
Anomalous Spectral Weight Transfer:
Optics
H hamiltonian, J electric current , P polarization


 ne2
0  ( )d  iV   P, J ]  m
Below energy


0
 ( )d 

iV
H eff , J eff , Peff

  Peff , J eff  
AppreciableT dependence
found.
Schlesinger et.al (FeSi) PRL 71 ,1748 , (1993) B Bucher et.al. Ce2Bi4Pt3PRL
72, 522 (1994), Rozenberg et.al. PRB 54, 8452, (1996).
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Comments on LDA+DMFT
•
•
•
•
Static limit of the LDA+DMFT functional ,
with F= FHF 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.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
LSDA+DMFT functional
G LDA  DMFT [ r (r ) m (r ) G a b VKS(r ) BKS(r ) ab]
­ Tr log[iwn + Ñ 2 / 2 ­ VKS ­ c *a R (r )S a bc b R (r )] ­
ò
VKS (r )r (r )dr -
ò
å
Vext (r )r (r )dr +
òB
1
2
ò
(r )m(r )dr KS
å
TrS (iwn )G (iwn ) +
iwn
r (r )r (r ')
drdr '+ ExcLDA [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
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
+
i0
ab (iw)e
)
DMFT: Effective Action point of view.
R. Chitra and G. Kotliar Phys Rev. B.
(2000).






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.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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)]

THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
tools
LDA+DMFT Self-Consistency loop
c ka | ­ Ñ 2 + Vxc (r ) | c ka = H LMTO ( k )
G
0
Impurity
Solver
Edc
G
U

S.C.C.
DMFT
r (r) = T
å
G( r, r, iw)e
iw0+
nHH = T
å
iw
iw
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
+
GHH ( r , r , iw)eiw 0