Transcript The Mott Transition and the Challenge of Strongly Correlated Electron Systems. G. Kotliar

The Mott Transition and the
Challenge of Strongly Correlated
Electron Systems.
G. Kotliar
Physics Department and Center for
Materials Theory
Rutgers
PIPT Showcase Conference UBC Vancouver May 12th 2005
Outline
• Correlated Electron Materials.
• Dynamical Mean Field Theory.
• The Mott transition problem: qualitative insights
from DMFT.
• Towards first principles calculations of the
electronic structure of correlated materials. Pu
Am and the Mott transition across the actinide
series.
The Standard Model of Solids
• Itinerant limit. Band Theory. Wave picture
of the electron in momentum space. .
Pauli susceptibility.
• Localized model. Real space picture of
electrons bound to atoms. Curie
susceptibility at high temperatures, spinorbital ordering at low temperatures.
Correlated Electron Materials
• Are not well described by either the itinerant or
the localized framework .
• Compounds with partially filled f and d shells.
Need new starting point for their description.
Non perturbative problem. New reference frame
for computing their physical properties.
• Have consistently produce spectacular “big”
effects thru the years. High temperature
superconductivity, colossal magneto-resistance,
huge volume collapses……………..
Large Metallic Resistivities
 1
e2 k F ( k F l )
  1Mott
h
(100 cm)1
Transfer of optical spectral weight non local in
frequency Schlesinger et. al. (1994), Vander Marel
(2005) Takagi (2003 ) Neff
depends on T
Breakdown of the standard model
of solids.
• Large metallic resistivities exceeding the Mott
e k (k l )

limit.
Maximum metallic
h
resistivity 200 ohm cm
1
2
F
F
• Breakdown of the rigid band picture. Anomalous
transfer of spectral weight in photoemission and
optics.
• The quantitative tools of the standard model fail.
MODEL HAMILTONIAN AND OBSERVABLES

 (t
i , j  ,
ij
  ij )(c c j  c c )  U  ni ni
Parameters:
†
i
†
j i
i
U/t , T, carrier concentration, frustration :
1
A( k , )   Im[G( k ,  )]   Im[
]
   k  ( , k )
A( )   A( k , )
Local Spectral Function
k
A( )    ( k   )
Limiting case itinerant electrons
k
Limiting case localized electrons
Hubbard bands
A( )   ( B   )   ( A   )
U   A  B
Limit of large lattice coordination
1
d
tij ~
d   ij nearest neighbors
1
 c c j  ~
d
†
i
  tij  ci† c j  ~ d
j ,
Uni  ni  ~O(1)
1
d
1
~ O (1)
d
Metzner Vollhardt, 89
1
G (k , i ) 
i   k  (i )
Muller-Hartmann 89
Mean-Field Classical vs Quantum
Classical case
-
å
J ij Si S j - h å Si
i, j
i
H MF = - heff So
heff
å
j

 (t
i , j  ,
b
ij
  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
- 1
0
G
D ( w)
m0 = áS0 ñH MF ( heff )
heff =
Quantum case
J ij m j + h
G = ­ áco†s (iwn )cos (iwn )ñSMF (D )
G (iwn )[D ] =
å
k
A. Georges, G. Kotliar Phys. Rev. B 45, 6497(1992)
1
1
[D (iwn ) - tk + m]
G (iwn )[D ]
Realistic Descriptions of Materials
and a First Principles Approach to
Strongly Correlated Electron
Systems.
• Incorporate realistic band structure and orbital
degeneracy.
• Incorporate the coupling of the lattice degrees
of freedom to the electronic degrees of freedom.
• Predict properties of matter without empirical
information.
LDA+DMFT V. Anisimov, A. Poteryaev, M.
Korotin, A. Anokhin and G. Kotliar, J. Phys.
Cond. Mat. 35, 7359 (1997).
•
The light, sp (or spd) electrons are extended, well
described by LDA .The heavy, d (or f) electrons are
localized treat by DMFT. Use Khon Sham
Hamiltonian after substracting the average energy
already contained in LDA.
• Add to the substracted Kohn Sham Hamiltonian a
frequency dependent self energy, treat with DMFT.
In this method U is either a parameter or is
estimated from constrained LDA
•
• Describes the excitation spectra of many strongly
correalted solids. .
Spectral Density Functional
• Determine the self energy , the density and the
structure of the solid self consistently. By
extremizing a functional of these quantities.
(Chitra, Kotliar, PRB 2001, Savrasov, Kotliar,
PRB 2005). Coupling of electronic degrees of
freedom to structural degrees of freedom. Full
implementation for Pu. Savrasov and Kotliar
Nature 2001.
• Under development. Functional of G and W, self
consistent determination of the Coulomb
interaction and the Greens functions.
Mott transition in V2O3 under pressure
or chemical substitution on V-site. How does the electron go
from localized to itinerant.
The Mott transition and Universality
Same behavior at high
tempeartures, completely
different at low T
COHERENCE INCOHERENCE CROSSOVER
T/W
Phase diagram of a Hubbard model with partial frustration at integer
filling. M. Rozenberg et.al., Phys. Rev. Lett. 75, 105-108 (1995). .
V2O3:Anomalous transfer of spectral weight
Th. Pruschke and D. L. Cox and M. Jarrell, Europhysics Lett. , 21 (1993), 593
M. Rozenberg G. Kotliar H. Kajueter G Tahomas D. Rapkikne J Honig and P
Metcalf Phys. Rev. Lett. 75, 105 (1995)
Anomalous transfer of optical spectral
weight, NiSeS. [Miyasaka and Takagi
2000]
Anomalous Resistivity and Mott
transition Ni Se2-x Sx
Crossover from Fermi liquid to bad metal to
semiconductor to paramagnetic insulator.
Single-site DMFT and expts
Conclusions.
• Three peak structure, quasiparticles and
Hubbard bands.
• Non local transfer of spectral weight.
• Large metallic resistivities.
• The Mott transition is driven by transfer of
spectral weight from low to high energy as we
approach the localized phase.
• Coherent and incoherence crossover. Real and
momentum space.
• Theory and experiments begin to agree on a
broad picture.
Mott Transition in the Actinide Series
Pu phases: A. Lawson Los Alamos Science 26,
(2000)
LDA underestimates the volume of fcc Pu by 30%.
Within LDA fcc Pu has a negative shear modulus.
LSDA predicts  Pu to be magnetic with a 5 ub moment.
Experimentally it is not.
Treating f electrons as core overestimates the volume by 30 %
Total Energy as a function of volume for PU
(Savrasov, Kotliar, Abrahams, Nature ( 2001)
Non magnetic correlated state of fcc Pu.
Double well structure and 
Pu
Qualitative explanation of negative thermal expansion[ G. Kotliar J.Low
Temp. Physvol.126, 1009 27. (2002)]See also A . Lawson et.al.Phil.
Mag. B 82, 1837 ]
Natural consequence of the conclusions on the model Hamiltonian level. We
had two solutions at the same U, one metallic and one insulating. Relaxing the
volume expands the insulator and contract the metal.
Phonon Spectra
• Electrons are the glue that hold the atoms
together. Vibration spectra (phonons)
probe the electronic structure.
• Phonon spectra reveals instablities, via
soft modes.
• Phonon spectrum of Pu had not been
measured.
Phonon freq (THz) vs q in delta Pu X.
Dai et. al. Science vol 300, 953, 2003
Inelastic X Ray. Phonon energy
10 mev, photon energy 10 Kev.
E = Ei - Ef
Q =ki - kf
DMFT Phonons in fcc -Pu
C11 (GPa)
C44 (GPa)
C12 (GPa)
C'(GPa)
Theory
34.56
33.03
26.81
3.88
Experiment
36.28
33.59
26.73
4.78
( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003)
(experiments from Wong et.al, Science, 22 August 2003)
J. Tobin et. al. PHYSICAL REVIEW B 68,
155109 ,2003
First Principles DMFT Studies of Pu
• Pu strongly correlated element, at the
brink of a Mott instability.
• Realistic implementations of DMFT : total
energy, photoemission spectra and
phonon dispersions of delta Pu.
• Clues to understanding other Pu
anomalies. Qualitative Insights and
quantitative studies. Double well. Alpha
and Delta Pu.
Approach the Mott point from the right Am under
pressureExperimental Equation of State (after Heathman et.al, PRL 2000)
“Soft”
Mott Transition?
“Hard”
Density functional based electronic structure calculations:
 Non magnetic LDA/GGA predicts volume 50% off.
 Magnetic GGA corrects most of error in volume but gives m~6B
(Soderlind et.al., PRB 2000).
 Experimentally, Am has non magnetic f6 ground state with
J=0 (7F0)
Mott transition in open (right) and
closed (left) shell systems.
Realization in Am ??
S
gT
Log[2J+1]
S
Tc
???
Uc
U
J=0
g ~1/(Uc-U)
U
Cluster Extensions of Single Site DMFT
 latt (k , ) 0 ( ) 
1 ( )(cos kx  cos ky )   2 ( )(cos kx.cos ky )  .......
Conclusions Future Directions
• DMFT: Method under development, but it already gives
new insights into materials…….
• Exciting development: cluster extensions. Allows us to
see to check the accuracy of the single site DMFT
corrections, and obtain new physics at lower
temperatures and closer to the Mott transition where the
single site DMFT breaks down.
• Captures new physics beyond single site DMFT , i.e. d
wave superconductivity, and other novel aspects of the
Mott transition in two dimensional systems.
• Allow us to focus on deviations of experiments from
DMFT.
• DMFT and RG developments
Some References
• Reviews: A. Georges G. Kotliar W.
Krauth and M. Rozenberg RMP68 , 13,
(1996).
• Reviews: G. Kotliar S. Savrasov K.
Haule V. Oudovenko O. Parcollet and C.
Marianetti. Submitted to RMP (2005).
• Gabriel Kotliar and Dieter Vollhardt
Physics Today 57,(2004)
Am Equation of State: LDA+DMFT Predictions
(Savrasov Kotliar Haule Murthy 2005)
Self-consistent evaluations of total energies with
LDA+DMFT .
Accounting for full atomic multiplet structure using Slater integrals:
F(0)=4.5 eV, F(2)=8 eV, F(4)=5.4 eV, F(6)=4 eV
New algorithms allow studies of complex structures.
Theoretical P(V) using LDA+DMFT
Predictions for Am I
LDA+DMFT predictions:
 Non magnetic f6 ground state
with J=0 (7F0)
 Equilibrium Volume:
Vtheory/Vexp=0.93
 Bulk Modulus: Btheory=47 GPa
Experimentally B=40-45 GPa
Predictions for Am II
Predictions for Am III
Predictions for Am IV
Photoemission Spectrum from 7F0 Americium
LDA+DMFT Density of States
Matrix Hubbard I Method
F(0)=4.5 eV
F(2)=8.0 eV
F(4)=5.4 eV
F(6)=4.0 eV
Experimental Photoemission Spectrum
(after J. Naegele et.al, PRL 1984)
J. C. Griveau et. al. (2004)
K. Haule , Pu- photoemission with
DMFT using vertex corrected NCA.
Cluster Extensions of DMFT
Pu is not MAGNETIC, alpha and
delta have comparable
susceptibility and specifi heat.
More important, one would like to be able to evaluate from the theory itself when
the approximation is reliable!! And captures new fascinating aspects of the
immediate vecinity of the Mott transition in two dimensional systems…..
Cluster Extensions of Single Site DMFT
 latt (k , ) 0 ( ) 
1 ( )(cos kx  cos ky )   2 ( )(cos kx.cos ky )  .......
Some References
• Reviews: A. Georges G. Kotliar W.
Krauth and M. Rozenberg RMP68 , 13,
(1996).
• Reviews: G. Kotliar S. Savrasov K.
Haule V. Oudovenko O. Parcollet and C.
Marianetti. Submitted to RMP (2005).
• Gabriel Kotliar and Dieter Vollhardt
Physics Today 57,(2004)
Evolution of the Spectral Function with
Temperature
Anomalous transfer of spectral weight connected to the
proximity to the Ising Mott endpoint (Kotliar Lange nd
Rozenberg Phys. Rev. Lett. 84, 5180 (2000)
Total Energy as a function of volume for Pu
W
(ev) vs iw (a.u. 27.2 ev)
(Savrasov, Kotliar, Abrahams, Nature ( 2001)
Non magnetic correlated state of fcc Pu.
Zein Savrasov and Kotliar (2004)
DMFT : What is the dominant atomic configuration
,what is the fate of the atomic 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
• More realistic calculations,
(GGA+U),itineracy, crystal fields G7 G8,

ML=-3.9
Mtot=1.1. S. Y. Savrasov and G.
Kotliar, Phys. Rev. Lett., 84, 3670 (2000)
• This moment is quenched or screened by
spd electrons, and other f electrons. (e.g.
alpha Ce).
 Contrast Am:(5f)6

Anomalous Resistivity
PRL 91,061401 (2003)
• Approach the Mott transition, if the
localized configuration has an OPEN shell
the mass increases as the transition is
approached.
Consistent theory, entropy increases
monotonically as U  Uc .
• Approach the Mott transition, if the
localized configuration has a CLOSED
shell. We have an apparent paradox. To
approach the Mott transitions the bands
have to narrow, but the insulator has not
entropy.. SOLUTION: superconductivity
intervenes.
Mott transition into an open (right) and closed (left) shell
systems. AmAt room pressure a localised 5f6 system;j=5/2.
S = -L = 3: J = 0 apply pressure ?
S
.g T
Log[2J+1]
S
???
Uc
U
S=0
g ~1/(Uc-U)
U
•BACKUPS
Strong Correlation Anomalies cannot be understood within the
standard model of solids, based on a RIGID BAND
PICTURE,e.g.“Metallic “resistivities that rise without sign of
saturation beyond the Mott limit, temperature dependence of the
integrated optical weight up to high frequency
C. Urano et. al. PRL 85, 1052 (2000)
RESTRICTED SUM RULES
H hamiltonian, J electric current , P polarization


0
 ( )d 
Below energy

  


0
( )d

iV

 ne 2
, J ,P
  P, J H 
eff
iV
eff
eff
m

  Peff , J eff  
ApreciableT dependence
found.  n  

k
2
k
k
k
2
M. Rozenberg G. Kotliar and H. Kajueter PRB 54, 8452, (1996).
Ising critical endpoint! In V2O3
P. Limelette et.al. Science 302, 89 (2003)
.
ARPES measurements on NiS2-xSex
Matsuura et. Al Phys. Rev B 58 (1998) 3690. Doniaach and Watanabe Phys. Rev. B 57, 3829 (1998) Mo
al., Phys. Rev.Lett. 90, 186403 (2003).
et
Am under pressure. Lindbaum
et.al. PRB 63,2141010(2001)
Functional formulation. Chitra and Kotliar Phys. Rev. B 62, 12715 (2000)
and Phys. Rev.B (2001)
+
G = - < y ( x ') y † ( x ) >
.
1
f ( x)VC - 1 ( x, x ')f ( x ') +
ò
ò
2
ò if ( x)y
†
( x )y ( x )
< f ( x ')f ( x ) > - < f ( x ') > < f ( x ) > = W
1
1
G[G,W , M , P ]  TrLn[G0 1  M ]  Tr[G ]  TrLn[VC1  P ]  Tr[ P ]W  Ehartree  [G,W ]
2
2
Introduce Notion of Local Greens functions, Wloc, Gloc
G=Gloc+Gnonloc
.
Ex. Ir>=|R, > Gloc=G(R , R ’) R,R’
[G,W ]
 EDMFT [Gloc ,Wloc , Gnonloc  0,Wnonloc  0]
One can also view

Sum of 2PI graphs
as an approximation to an exact Spetral Density Functional
of Gloc and Wloc.
Model Hamiltonians and Observables

 (t
i , j  ,
ij
  ij )(c c j  c ci )  U  ni ni
†
i
U/t
Doping  or chemical potential
Frustration (t’/t)
T temperature
†
j
i
Outlook
 The Strong Correlation Problem:How to deal
with a multiplicity of competing low temperature
phases and infrared trajectories which diverge in
the IR
Strategy: advancing our understanding scale
by scale
Generalized cluster methods to capture longer
range magnetic correlations
New structures in k space?
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?
• LDA+DMFT functional computes total energies
opens the way to the computation of phonon
frequencies in correlated materials (S. Savrasov
and G. Kotliar 2002). Combine linear response
and DMFT.
Epsilon Plutonium.
Phonon entropy drives the
epsilon delta phase transition
• Epsilon is slightly more delocalized than delta,
has SMALLER volume and lies at HIGHER
energy than delta at T=0. 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 following
Drumont and G. Ackland et. al. PRB.65, 184104
(2002); (and neglecting electronic entropy). TC
~ 600 K.
Further Approximations.
o
The light, SP (or SPD) electrons are extended, well described by LDA .The heavy,
d(or f) electrons are localized treat by DMFT.LDA Kohn Sham Hamiltonian already
contains an average interaction of the heavy electrons, subtract this out by shifting the
heavy level (double counting term) .
S ( r, r ', w) = d( r - r ')V ( r ) xc - S Ra beH f
o
Ra
( r )(S a b ( w) - Edc )f
Rb
( r ')
Truncate the W operator act on the H sector only. i.e.
W ( r, r ', w) = S Ra bgdeH f Ra ( r )f Rb ( r )Wa bgd ( w)f Rg ( r ')f Rd ( r ')
• Replace W() by a static U. This quantity can be estimated by a
constrained LDA calculation or by a GW calculation with light
electrons only. e.g.
M.Springer and F.Aryasetiawan,Phys.Rev.B57,4364(1998)
T.Kotani,J.Phys:Condens.Matter12,2413(2000). FAryasetiawan M Imada A Georges G Kotliar S
Biermann and A Lichtenstein cond-matt (2004)
or the U matrix can be adjusted empirically.
• At this point, the approximation can be derived
from a functional G[  Gloc] (Savrasov and
Kotliar 2001)
• FURTHER APPROXIMATION, ignore charge self
consistency, namely set Vxc[  ]  Vxc[  LDA ]
LDA+DMFT V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G.
Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997) See also . A Lichtenstein and
G
M. Katsnelson PRB 57, 6884 (1988).
Reviews:Held, K., I. A. Nekrasov, G. Keller, V. Eyert, N. Bl•umer, A. K.
McMahan, R. T. Scalettar, T. Pruschke, V. I. Anisimov, and D.
Vollhardt, 2003, Psi-k Newsletter #56, 65.
• Lichtenstein, A. I., M. I. Katsnelson, and G. Kotliar, in Electron
Correlations and Materials Properties 2, edited by A. Gonis, N.
Kioussis, and M. Ciftan (Kluwer Academic, Plenum Publishers, New
York), p. 428.
• Georges, A., 2004, Electronic Archive, .lanl.gov, condmat/ 0403123 .
LDA+DMFT Self-Consistency loop
c ka | ­ Ñ 2 + Vxc (r ) | c ka = H LMTO ( k )
Impurity
Solver
G
0
Edc
G
U

S.C.C.
DMFT
r (r) = T
å
iw
G( r, r, iw)e
iw0+
nHH = T
å
iw
+
GHH ( r , r , iw)eiw 0
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
U dkj dil ® U ijkl
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
LDA+DMFT functional
G LDA  DMFT [  (r ) m (r ) G a b VKS(r ) BKS( 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
 Sum of local 2PI graphs with local U
matrix and local G
F DC [G ] = Un(n - 1)
1
2
n= T
å (G
+
i0
ab (iw)e
abiw
)
Anomalous Resistivity
PRL 91,061401 (2003)
The Mott Transiton across the
Actinides Series.
H  H cluster  H cluster exterior  H exterior
H
H cluster  H cluster exterior  H exterior
Simpler "medium" Hamiltonian
Medium of free
electrons :
impurity model.
Solve for the
medium using
Self Consistency
G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87,
186401 (2001)
Other cluster extensions (DCA Jarrell Krishnamurthy, M
Hettler et. al. Phys. Rev. B 58, 7475 (1998)Katsnelson and
Lichtenstein periodized scheme. Causality issues O. Parcollet,
G. Biroli and GK Phys. Rev. B 69, 205108 (2004)
Mott transition in layered organic conductors
cond-mat/0004455, Phys. Rev. Lett. 85, 5420 (2000)
S Lefebvre et al.
k-(ET)2X are across Mott
transition
ET =
Insulating
anion layer
X-1
conducting
ET layer
modeled to triangular lattice
]+1
[(ET)2
t’
modeled to triangular lattice
t’
t
t
Single-site DMFT as a zeroth order
picture ?
Finite T Mott tranisiton in CDMFT
Parcollet Biroli and GK PRL, 92, 226402. (2004))
Evolution of the spectral
function at low frequency.
A(  0, k )vs k
Ek=t(k)+Re( k ,   0)  
g k = Im( k ,   0)
A( k ,   0) 
gk
g k 2  Ek 2
If the k dependence of the self energy is
weak, we expect to see contour lines
corresponding to t(k) = const and a
height increasing as we approach the
Fermi surface.
Evolution of the k resolved Spectral
Function at zero frequency. (QMC
study Parcollet Biroli and GK PRL, 92, 226402. (2004))
A( )
 0, k )vs k
U/D=2
U/D=2.25
Uc=2.35+-.05, Tc/D=1/44. Tmott~.01 W
Momentum Space
Differentiation the high
temperature story
T/W=1/88
Actinies , role of Pu in the
periodic table
CMDFT Studies of the Mott
Transition
• cond-mat/0308577 [PRL, 92, 226402. (2004) ]
• Cluster Dynamical Mean Field analysis of the Mott transition
• : O. Parcollet, G. Biroli, G. Kotliar
• cond-mat/0411696 [abs, ps, pdf, other] :
• Dynamical Breakup of the Fermi Surface in a doped Mott Insulator
M. Civelli (1), M. Capone (2), S. S. Kancharla (3), O. Parcollet (4),
G. Kotliar
• cond-mat/0502565
• Title: Short-Range Correlation Induced Pseudogap in Doped Mott
Insulators
• B. Kyung, S. S. Kancharla, D. Sénéchal, A. -M. S. Tremblay, M.
Civelli, G. Kotliar
Two paths for calculation of
electronic structure of
strongly correlated materials
Crystal structure +Atomic
positions
Model Hamiltonian
Correlation Functions Total
Energies etc.
DMFT ideas can be used in both cases.
Electrons in a Solid:the Standard Model
Band Theory: electrons as waves.
Landau Fermi Liquid Theory.
Rigid bands , optical transitions ,
thermodynamics, transport………
•Quantitative Tools. Density
Functional Theory+Perturbation
Theory.
­ Ñ / 2 + VKS ( r )[r ] y kj = ekj y kj
2
Mean-Field Classical vs Quantum
Quantum case

 (t
i , j  ,
b
ij
  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
D ( w)
1
[iwn + m- S (iwn )] = D (iwn ) G (iwn )[D ]
G (iwn )[D ] =
å
k
1
[[iwn + m- S (iwn )] - tk ]
G = ­ áco†s (iwn )cos (iwn )ñSMF (D )
G (iwn )[D ] =
A. Georges, G. Kotliar Phys. Rev. B 45, 6497(1992)
å
k
1
[D (iwn ) -
1
- tk ]
G (iwn )[D ]
Phase Diag: Ni Se2-x Sx
Mott transition in systems
with close shell.
• Resolution: as the Mott transition is
approached from the metallic side,
eventually superconductivity intervenes to
for a continuous transition to the localized
side.
• DMFT study of a 2 band model for
Buckminster fullerines Capone et. al.
Science 2002.
• Mechanism is relevant to Americium.
Mott transition in systems
with close shell.
• Resolution: as the Mott transition is
approached from the metallic side,
eventually superconductivity intervenes to
for a continuous transition to the localized
side.
• DMFT study of a 2 band model for
Buckminster fullerines Capone et. al.
Science 2002.
• Mechanism is relevant to Americium.
Mott transition in layered organic conductors
cond-mat/0004455
S Lefebvre et al.