Transcript jerusalem - Department of Physics and Astronomy

Dynamical Mean Field Theory
of the Mott Transition
Gabriel Kotliar
Physics Department and
Center for Materials Theory
Rutgers University
Jerusalem Winter School
January 2002
OUTLINE OF THE COURSE


Motivation . Electronic structure of
correlated materials, limiting cases and
open problems. The standard model of
solids and its failures.
Introduction to the Dynamical Mean Field
Theory (DMFT). Cavity construction.
Statistical Mechanical Analogies. Lattice
Models and Quantum Impurity models.
Functional derivation.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Outline


The limit of large lattice coordination.
Ordered phases. Correlation functions.
Techniques for solving the Dynamical Mean
Field Equations. [ Trieste School June 17-22
2002]
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Outline



The Mott transition. Early ideas. Brinkman Rice.
Hubbard. Slater.
Analysis of the DMFT equations: existence of a Mott
transition.
The Mott transition within DMFT. Overview of some
important results of DMFT studies of the Hubbard
Model. Electronic Structure of Correlated Materials.
Canonical Phase diagram of a fully frustrated
Hubbard model. Universal and non universal
aspects of the physics of strongly correlated
materials.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Outline




Analysis of the DMFT equations. Existence
of a Mott transition. Analysis from large U
and small U.
The destruction of the metallic phase.
Landau analysis. Uc1 . Uc2.
The Mott transition endpoint.
A new look at experiments.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Outline
The electronic structure of real materials.
Examples of problems where DMFT gives
new insights, and quantitative
understanding: itinerant ferromagnetism,
Fe, Ni. Volume collapse transitions, actinide
physics. Doping driven Mott transition
titanites.

THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Outline

New directions, beyond single site DMFT.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Realistic
Theories of
Correlated
Materials
ITP, Santa-Barbara workshop
July 29 – December 16 (2002)
O.K. Andesen, A. Georges,
G. Kotliar, and A. Lichtenstein
Contact: [email protected]
Conference: November 25-29, (2002)
The promise of Strongly Correlated
Materials



Copper Oxides. High Temperature
Superconductivity.
Uranium and Cerium Based Compounds.
Heavy Fermion Systems.
(LaSr)MnO3 Colossal Magnetoresistence.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
The Promise of Strongly Correlated
Materials.

High Temperature Superconductivity in doped filled
Bucky Balls (B. Battlog et.al Science)
Thermoelectric response in CeFe4 P12 (H. Sato et
al. cond-mat 0010017).
Large Ultrafast Optical Nonlinearities Sr2CuO3 (T
Ogasawara et.al cond-mat 000286)
 Theory will play an important role in optimizing their
physical properties.

THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
How to think about the electron
in a solid?
ne 2

m
Drude
Sommerfeld

Bloch,
Periodic potential
Bands, k in Brillouin zone
e 2 k F (k F l )
h
Maximum metallic
resistivity 200 mohm cm
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Standard Model
High densities, electron as a wave, band theory, kspace
Landau: Interactions Renormalize Away
One particle excitations: quasi-particle bands
Density Functional Theory in Kohn Sham
Formulation, successful computational tool for total
energy, and starting point
For perturbative calculation of spectra, Si Au, Li, Na
……………………
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Standard Model : Metals
Predicts low temperature dependence of thermodynamics
and transport
RH
const
T
ST
2
CV ~ T

Hall Coefficient
const
Resistivity
Thermopower
Specific Heat
Susceptibility
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Quantitative Tools
:
Density Functional Theory with approximations
suggested by the Kohn Sham formulation,
(LDA GGA) is a successful computational tool
for the total energy, and a good starting point
for perturbative calculation of spectra, GW,
transport.……………………
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
aB
Mott : correlations localize the
electron

Array of hydrogen atoms is insulating if
a>>aB
e_

e_
e_
e_
Superexchange
Think in real space , atoms
High T : local moments
Low T: spin orbital order
RUTGERS
THE STATE UNIVERSITY OF NEW JERSEY

1
T
Mott : Correlations localize the
electron
Low densities, electron behaves as a particle,use atomic
physics, 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….)
Rich structure of Magnetic and Orbital Ordering at low T
Quantitative calculations of Hubbard bands and exchange
constants, LDA+ U, Hartree Fock. Atomic Physics.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Localization vs Delocalization
Strong Correlation Problem
•A large number of compounds with electrons which are
not close to the well understood limits (localized or
itinerant).
•These systems display anomalous behavior (departure
from the standard model of solids).
•Neither LDA or LDA+U or Hartree Fock works well
•Dynamical Mean Field Theory: Simplest approach to
the electronic structure, which interpolates correctly
between atoms and bands
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Mott transition in layered organic conductors
al. cond-mat/0004455
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
S Lefebvre et
Failure of the Standard Model:
Miyasaka and
NiSe2-xSx
Takagi (2000)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Failure of the standard model :
Anomalous Resistivity:LiV2O4
Takagi et.al. PRL 2000
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS


0
 ( )d  Neff
Failure of the Standard
Model: Anomalous Spectral
Weight Transfer
Optical Conductivity of FeSi for
T=,20,20,250 200 and 250 K from
Schlesinger et.al (1993)


0
 ( )d
Neff depends on T
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Strong Correlation Problem





Large number of compounds (d,f,p….). Departure
from the standard model.
Hamiltonian is known. Identify the relevant
degrees of freedom at a given scale.
Treat the itinerant and localized aspect of the
electron
The Mott transition, head on confrontation with this
issue
Dynamical Mean Field Theory simplest approach
interpolating between that bands and atoms
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Hubbard model

 (t
i , j  ,
ij
 m ij )(c c j  c ci )  U  ni ni
†
i
†
j
i
U/t
Doping d or chemical potential
Frustration (t’/t)
T temperature
Mott transition as a function of doping, pressure
RUTGERS
temperature etc.
THE STATE UNIVERSITY OF NEW JERSEY
Limit of large lattice
coordination
tij ~
1
d   ij nearest neighbors
d
1
 c c j  ~
d
Uni  ni  ~O(1)
†
i
  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
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Mean-Field : Classical
-
å
i, j
heff =
å
J ij Si S j - h å Si
i
H MF = - heff So
J ij m j + h
j
m0 = áS0 ñH MF ( heff )
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
DMFT Impurity cavity construction: [A.
R[
Georges, G. Kotliar, PRB, (1992)]


i , j  ,
(tij  m ij )(ci† c j  c †j ci )  U  ni  ni 
i
b
S [Go] =
b
òò
0
G0-
1
b
co†s (t )[Go(t , t ')]cos (t ') + U ò no­ no
0
0
Weiss field
(iwn ) = iwn + m - D (iwn )
GL (iwn ) = ­ áco† (iwn )co (iwn )ñS (G0 )
D (iwn ) = - G -
é
)
D( z ) = ê
ê
ê
ë
å
k
1
(iwn ) + R[G (i wn )]
1
z - tk
ù
ú
ú
ú
û
)
R[ D( z )] = z
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Comments on DMFT




Exact in both atomic and band limits
Weiss field is a function
Multiple energy scales in a correlated
electron problem, non linear coupling
between them.
Frezes spatial fluctuations but treats
quantum fluctuations exactly, local view of
the quantum many body problem.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Example: semicircular DOS
)
1
R[ D ( z )] = t 2 z +
z
2
†
o
D (iwn ) = t ác (iwn )co (iwn )ñS (G0 )
b
S [Go] =
b
òò
0
b
co†s (t )[Go(t , t ')]cos (t ') + U ò no­ no¯
0
0
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
DMFT Impurity cavity construction: [A.
Georges, G. Kotliar, PRB, (1992)]


i , j  ,
(tij  m ij )(ci† c j  c †j ci )  U  ni  ni 
i
b
S [Go] =
- 1
0
G
b
b
òò
0
co†s (t )[Go(t , t ')]cos (t ') + U ò no­ no¯
0
Weiss field
0
(iwn ) = iwn + m - D (iwn )
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
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
+ S (iwn )
1
Solving the DMFT equations
D (iwn ) G
0
Impurity
Solver
G0
Impurity
Solver
G

S.C.C.
G

S.C.C.
•Wide variety of
computational tools (QMC,
NRG,ED….)
•Analytical Methods
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
A. Georges, G. Kotliar (1992)
Phys. Rev. B 45, 6497
Mean-Field : Classical vs Quantum
Quantum case
Classical case
-
å
J ij Si S j - h å Si
i, j

 (t
i , j  ,
ij
 m ij )(ci† c j  c †j ci )  U  ni  ni 
i
i
b
H MF = - heff So
b
b
†
ò ò cos (t )[
0
0
¶
+ m- D (t - t ')]cos (t ') + ò no­ no¯
¶t
0
heff
D ( w)
m0 = áS0 ñH MF ( heff )
heff =
å
J ij m j + h
GL = ­ áco† (iwn )co (iwn )ñH MF (D )
G (iwn ) =
å
k
j
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
1
[D (iwn ) +
1
- ek ]
G (iwn )[D ]
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 ]
i

Express in terms of Weiss field (semicircularDOS)
  Fimp

D (i ) 2
F [ D ]  T  
 Fimp [D ]
2
t
†
†

L
[
f
,
f
]

f
( i ) D ( i ) f ( i )
loc


†
 ,
 Log[  df dfe
]
The Mott transition as bifurcation point in functionals
oG[G] or F[D], (G. Kotliar EPJB 99)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
DMFT for lattice hamiltonians
k independent  k dependent G, Local Approximation
Treglia et. al 1980
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
How to compute  ?
View locally the lattice problem as a (multiorbital)
Anderson impurity model
The local site is now embedded in a medium
characterized by
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
How to determine the medium

Use the impurity model to compute  and
the impurity local Greens function. Require
that impurity local Greens function equal to
the lattice local Greens function.
Weiss field
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
ucible vertex G is momentum independent
Response functions
 is q dependent but irreducible vertex G is momentum independent
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Evaluation of the Free energy.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Solving the DMFT equations
G
0
Impurity
Solver
G0
Impurity
Solver
G

S.C.C.
G

S.C.C.
•Wide variety of
computational tools (QMC,
NRG,ED….)
•Analytical Methods
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Review of DMFT, technical tools
for solving DMFT eqs..,
applications, references……

A. Georges, G. Kotliar, W. Krauth and M.
Rozenberg Rev. Mod. Phys. 68,13 (1996)]
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
DMFT: Methods of Solution
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Mott transition: Early ideas.
Half filling.
Evolution of the one electron spectra [physical
quantity measured in photoemission and BIS]
as a function of control parameters. ( U/t,
pressure, temperature )
 Hubbard, begin in paramagnetic insulator.
As U/t is reduced Hubbard bands merge.
Gap closure. Mathematical description, closure
of equations of motion, starting from atoms
(I.e. large U). Incoherent motion, no fermi
surface.

THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Mott transition: early ideas.
Brinkman and Rice. Gutzwiller.
Begin in paramagnetic metallic state, as U/t
approaches a critical value the effective
mass diverges. Luttinger fermi surface.

1
 Z  (Uc  U )
*
m
Mathematical description, variational wave
function, slave bosons, quantum coherence
and double occupancy.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Slave bosons: mean field
+fluctuations



Fluctuations of the slave bosons around the
saddle point gives rise to Hubbard bands.
Starting from the insulating side, in a
paramagnetic state, the gap closes at the
same U, where Z vanishes.
No satisfactory treatement of finite
temperature properties.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Mott vs Slater
Mott: insulators in the absence of magnetic
long range order.
e.g. Vanadium Oxide Nickel Oxide. Mott
transition in the paramagnetic state .
• Slater: insulating behavior as a
consequence of antiferromagnetic long
range order. Double the unit cell to convert
a Mott insulator into a band insulator.

THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
A time-honored example:
Mott transition in V2O3 under pressure
or chemical substitution on V-site
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Local view of the spectral
function
Partition function of the Anderson impurity
model : gas of kinks [Anderson and Yuval]
Metallic state, proliferation of
kinks.
D(i)  i si gn()
D(i)  i
Insulating state. Kinks are
confined.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Local view of the spectral
function.




Consistent treatement of quasiparticles and
collective modes.
Kinky paths, with may spin fluctuations: low
energy resonance [Abrikosov Suhl Resonance]
Confined kinks, straight paths, Hubbard bands.
[control the insulator partition function]
Strongly correlated metal has both.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Spectral Evolution at T=0
half filling full frustration
X.Zhang M. Rozenberg G. Kotliar (PRL 1993)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Destruction of the metal
1
 Z  (Uc 2  U )
*
m
The gap is well formed at Uc2,
when the metal is destroyed.
Hubbard bands are well formed in
the metal.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Parallel development: Fujimori et.al
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
gap  (U  Uc1)
Destruction of the insulator



Continue the insulating solution below Uc2.
Coexistence of two solutions between Uc1
and Uc2
Mott Hubbard gap vanishes linearly at Uc1.
gap  (U  Uc1)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Recent calculation of the phase diagram of
the frustrated Half filled Hubbard model with
semicircular DOS (QMC Joo and Udovenko
PRB2001).
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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).
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Schematic DMFT phase diagram
Hubbard model (partial
frustration)
Rozenberg et.al. PRL (1995)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Kuwamoto Honig and Appell
PRB (1980)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Phase Diag: Ni Se2-x Sx
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Insights from DMFT
 Low temperatures several
competing phases . Their relative
stability depends on chemistry
and crystal structure
High temperature behavior
around Mott endpoint, more
universal regime, captured by
simple models treated within
DMFT
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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…
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 2000)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
.
ARPES measurements on NiS2-xSex
Matsuura et. Al Phys. Rev B 58 (1998) 3690
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Insights from DMFT: think in
term of spectral functions
(branch cuts) instead of well
defined QP (poles )
Resistivity near the metal insulator endpoint ( Rozenberg
et. Al 1995) exceeds the Mott limit
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Anomalous Resistivity and Mott
transition Ni Se2-x Sx
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Anomalous resisitivity near
Mott transition.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Anomalous transfer of spectral
weight in v2O3
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Anomalous transfer of spectral
weight in v2O3
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Anomalous transfer of spectral
weight in v2O3
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Anomalous transfer of
spectral weight in heavy
fermions [Rozenberg etal]
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Insights from DMFT

Mott transition as a bifurcation of an
effective action

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. [Non local in
frequency] Real and momentum space.
–
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Anomalous Resistivity:LiV2O4
Takagi et.al. PRL 2000
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Mott transition in layered organic conductors
al. cond-mat/0004455
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
S Lefebvre et
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS

Mott
Standard Model
Odd # electrons -> metal
Even # electrons -> insulator
Theoretical foundation: Sommerfeld, Bloch and
Landau
Computational tools DFT in LDA
Transport Properties, Boltzman equation , low
temperature dependence
e2 k F (k F l ) of transport coefficients

h
  Mott
Typical Mott values of the resistivity 200 mOhmcm
Residual instabilites SDW, CDW, SC
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Failure of the “Standard Model”:
Cuprates
Anomalous Resistivity
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
DMFT
Formulation as an electronic structure
method (Chitra and Kotliar)
Density vs Local Spectral Function
Extensions to treat strong spatial
inhomogeneities. Anderson Localization
(Dobrosavlevic and Kotliar),Surfaces
(Nolting),Stripes (Fleck Lichtenstein and
Oles)
Practical Implementation (Anisimov and
Kotliar, Savrasov, Katsenelson and
Lichtenstein)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
DMFT
Spin Orbital Ordered States
Longer range interactions Coulomb,
interactions, Random Exchange (Sachdev
and Ye, Parcollet and Georges, Kajueter and
Kotliar, Si and Smith, Chitra and Kotliar,)
Short range magnetic correlations.
Cluster Schemes. (Ingersent and Schiller,
Georges and Kotliar, cluster expansion in real
space, momentum space cluster DCA Jarrell
et.al., C-DMFT Kotliar et. al ).
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Strongly Correlated Electrons
Competing Interaction
Low T, Several Phases Close in Energy
Complex Phase Diagrams
Extreme Sensitivity to Changes in External
Parameters
Need for Quantitative Methods
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Landau Functional
G. Kotliar EPJB (1999)
  Fimp
D (i ) 2
FLG [D]  T  
 Fimp [D]
2
t
 Lloc [ f † , f ] 
f † ( i ) D ( i ) f ( i )
†
 , 
 Log[  df dfe
]

Lloc [ f , f ]   [ f † [
†
0
d
 e f ] f  Uf † f  f † f  ]d
d
Mettalic Order Parameter: D(i )
 Fimp
D(i )
 T    f† (i ) f (i ) D  2TG (i )[D]  2
D(i )
t
Spin Model Analogy:
h2
 FLG [h]  
 Log[[ch]2  h]
2J
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
LDA functional
GLDA[  ( r )]
Conjugate field, VKS(r)
GLDA[  (r ), VKS (r )]
­ Tr log[iwn + Ñ 2 / 2 ­ VKS ] ­
ò
Vext (r )r ( r ) dr +
1
2
ò
ò
VKS (r )r (r )dr
r (r )r ( r ')
drdr '+ ExcLDA [r ]
| r- r'|
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Minimize LDA functional
VKS (r ) = Vext (r ) +
r (r ) =
å
kj
f {e kj )y kj
*
LDA
xc
dE [r ]
r (r )
dr '+
| r- r'|
dr (r )
ò
(r )y kj(r ) =
å e
iwn 0+
wn
tr
[iwn + Ñ 2 / 2 ­ VKS ]
Kohn Sham eigenvalues,
auxiliary quantities.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Ising character of the transfer of
spectral weight
Ising –like dependence of the photo-emission intensity
and the optical spectral weight near the Mott transition
endpoint
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Spectral Evolution at T=0
half filling full frustration
X.Zhang M. Rozenberg G. Kotliar (PRL 1993)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Parallel development: Fujimori et.al
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS