Superconductivity near the Mott transition a Cluster Dynamical Mean Field Theory (CDMFT) perspective

1
Superconductivity near the Mott transition a
Cluster Dynamical Mean Field Theory (CDMFT)
perspective

• Gabriel Kotliar
• Rutgers

Coherence and incoherence in stronly correlated
systems. July 3-7 Rome Italy
Collaborators G. Biroli M . Capone M Civelli
K. Haule O. Parcollet T.D. Stanescu V.
Kancharla A.M.Tremblay B. Kyung D. Senechal A.
Georges
2
References Collaborators

• M. Capone and GK PRB 74, 54513(2006)
• M. Civelli M. Capone A. Georges K. Haule O
  Parcollet T. Stanescu and GK cond-mat 0704.1486
• M. Civelli et. al. PRL 95 106402(2005)
• B. Kyung S. Kancharla D. Senechal A. Ms Tremblay
  M. CIvellli and GK PRB 73 165114(20060.
• K. Haule and GK ( preprint)

Closely related work M. Capone M. Fabrizio C.
Castellani and E. Tosatti Phys. Rev. Lett 93,
047001(2004) . Science 296 2364 (2002).
3
Cuprates
Damascelli, Shen, Hussain, RMP 75, 473 (2003)
4
Kappa Organics
F. Kagawa, K. Miyagawa, K. Kanoda PRB 69
(2004) Nature 436 (2005)
Phase diagram of (XCuN(CN)2Cl) S. Lefebvre
et al. PRL 85, 5420 (2000), P. Limelette, et al.
PRL 91 (2003)
5
Perspective
U/t
Doping Driven Mott Transition
Pressure Driven Mott transtion
d
t/t
6
t-J Hamiltonian RVB P.W. Anderson (1987)
Slave Boson Formulation Baskaran Zhou Anderson
(1987) Ruckenstein Hirschfeld and Appell (1987)
bi bi fsi fsi 1
Other RVB states with d wave symmetry. Flux
phase or sid G. Kotliar (1988) Affleck and
Marston (1988) . Spectrum of excitation have
point zeros like a a d wave superconductor.
7
Superexchange Mechanism proximity to the Mott
transition renormalizes down kinetic energy, but
not the superexchange.

Slave Boson Mean Field Theory Phase Diagram.
Formation of Singlets
Coherent Quasiparticles
Re
8
Problems with the approach.

• Stability of the MFT. Ex. Neel order. Slave
  boson MFT with Neel order predicts AF AND SC.
  Inui et.al. 1988 Giamarchi and Lhuillier
  (1987).
• Gauge fluctuations destablize the mean field
  Ubbens and Lee
• Temperature dependence of the penetration depth
  Wen and Lee , Ioffe and Millis .
  TheoryrTx-T x2 , Exp rT x-T
• Z x . Mean field is too uniform on the Fermi
  surface, in contradiction with ARPES.
• No proper description the incoherent regime and
  the coherent-incoherent and the incoherent
  regime.

9
Dynamical Mean Field Theory

• Map lattice model into quantum impurity problem
  in a self consistent medium.
• The quantum impurity problem is used to generate
  local quantities, i.e. a local self energy.
• From local quantities one reconstruct k
  dependent spectral functions, susceptibilities,
  etc.
• Single site, DMFT k independent self energy
  (cumulant).
• Cluster extensions, incorporate additional k
  dpendence.
• Follow different mean field states, AF, normal,
  supeconductor, etc as a function of parameters.

10
CLUSTER EXTENSIONS umbiased reduction of the
many body problem to a plaquette in a medium.
Reviews Georges et.al. RMP(1996). Th. Maier, M.
Jarrell, Th.Pruschke, M.H. Hettler RMP (2005)
G. Kotliar S. Savrasov K. Haule O. Parcollet
V. Udovenko and C. Marianetti RMP (2006) .
Employ different impurity solvers. ED (Civelli
Capone) CTQMC (Haule)NCA (Haule)
11
Single site DMFT Qualitative Phase diagram of a
frustrated Hubbard model at integer filling

T/W
Synthesis Brinkman Rice Hubbard
Castellani, C., C. DiCastro, D. Feinberg, and J.
Ranninger, 1979, Phys. Rev. Lett. 43, 1957.
12

• Good description of the evolution of the
• spectra and transport, not too close to the
  Mott transition, at relatively high temperatures.
  For example V2O3 ( Rozenberg et. al. 1996)
  K-organics (Limelette et.al. 2002).
• At lower temperatures, closer to the Mott
  transition, cluster description is necessary.
• Study at low temperatures the doping driven Mott
  transition.

13
The approach validates many crucial features of
the RVB theory.
14
Tunnelling DOS (NCA-tJ) Gap (distance between
coherence peaks) increases with decreasing
doping.
15
Order Parameter and Superconducting Gap do not
scale for large U ! ED study in the SC state
Capone and GK PRB (2006) Kancharla et. al.
cond-mat 0508205.
16
CDMFT on a plaquette gives rise to a Dynamical
RVB pictures which retains all the good features
of the previous slave boson mft treatment
17

• The quasiparticle residue, decreases with doping
  but the effective mass (Fermi velocity) remains
  finite. M. Grilli and GK PRL (1990)
• The gap in the tunneling density of states
  increases with decreasing doping.
• The ph asymmetry grows with the approach to the
  Mott insulator.
• Superconducting order parameter does not scale
  with the gap.

18
But with substantial two differences!!! which
have important consequencesa) nodal antinodal
dichotomyb) vD decreses with decreasing doping
in superconductor. Two-gap picture
19
Nodal Antinodal Dichotomy and pseudogap. T.
Stanescu and GK PRB (2006)
20
Nodal Antinodal Dichotomy Civelli et. al. (2007)
21
Follow the normal state with doping. Civelli
et.al. PRL 95, 106402 (2005) Spectral Function
A(k,??0) -1/p G(k, ? ?0) vs k U16 t, t-.3
K.M. Shen et.al. 2004
If the k dependence of the self energy is weak,
we expect to see contour lines corresponding to
Ek const and a height increasing as we approach
the Fermi surface.
2X2 CDMFT
22
Doping Driven Mott transiton at low temperature,
in 2d (U16 t1, t-.3 ) Hubbard model
Spectral Function A(k,??0) -1/p G(k, ? ?0) vs k
K.M. Shen et.al. 2004
Antinodal Region
2X2 CDMFT
Nodal Region
Civelli et.al. PRL 95 (2005)
23
Scaling of the velocity in the superconductor
with doping.
M. Civelli et. al. cond-mat K. Haule and GKgt
24
Consequences for linear term coefficient of the
penetration depth. . K. Haule and GK
25
Experimentstwo superconducing gaps, with
opposite dependence on doping ? Antinodal gap
increases towards the Mott insulator while vD
decreases?

• Coherence and single-particle excitations in the
  high-temperature superconductors. Guy Deutscher
  ,Nature 397, 410-412 (1999) Andreev reflection.
• M. Opel et. al. PRB 61, 9752 (2000) Venturini, F.
  et al., Doping dependence of the electronic Raman
  spectra in Phys. Chem. Solids, 63, 2345 (2001).
  Raman scattering.

26
LeTacon et. al. Two Energy Scales and two
Quasiparticle Dynamics in the Superconducting .
Nature Physics 2, 537 (2006) Raman scattering. .
K. Tanaka, et. al Distinct Fermi-Momentum
Dependent Energy Gaps in Deeply Underdoped Bi2212
. arXivcond-mat/0612048 . ARPES M. C. Boyer
et. al. arXiv0705.1731 . Imaging the Two Gaps
of the High-TC Superconductor Pb-Bi2Sr2CuO6x
Tunnelling. arXiv0705.0111 Spectroscopic
distinction between the normal state pseudogap
and the superconducting gap of cuprate high T_c
superconductors Li Yu, et. al. . C- Axis
Optical Spectrsocopy.
27

• Metodological advantages. We can follow well
  defined phases as a function of parameters ,
  doping temperature.
• Well defined (meta) stable states, in contrast to
  the old slave boson MFT approach.
• CDMFT treats properly the incoherent state, with
  short ranged magnetic correlations.

28
AF and superconductivity M. Capone and GK PRB
74,054513
AFM blue dashed line with circles and dSC red
solid line with squares order parameters as a
function of doping for four values of the
repulsion U/ t4,8,12, and 16. The dSC order
parameter is multiplied by a factor of 10 for
graphical purposes.
29

• Can we continue the superconducting state towards
  the Mott insulating state ?

For U gt 8t YES. For U lt 8t NO,
magnetism really gets in the way.
30
Evolution of the q integrated staggered spin
susceptilibty K. Haule and GK (2006)
31
Conclusions CDMFT studies of superconductivity
near a Mott insulator.

• Captures the essential RVB physics of the
  interplay of the Mott transition and
  superconductivity. Kinetic energy supression.
  Retains the good aspects of the slave boson MFT.
• Solves many problems of the earlier slave boson .
  e.g.doping dependence of T linear term in the
  penetration depth
• Allows the continuation of spin liquid states as
  metastable states. Functional of local spectral
  functions.
• Nodal Antinodal dichotomy, emerges naturally.
• Work in progress. No full solution of the CDMFT
  eqs.and its lattice interpretation, (on the
  same level of single site DMFT), is available
  yet.

32

• Happy Birthday Carlo!!!!

33
(No Transcript)
34
Temperature dependence of the arcs. doping.09
(underdoped) Plaquette DMFT. K. Haule and GK
35
Lines of Zeros and Spectral Shapes. Stanescu and
GK cond-matt 0508302
36
Interpretation in terms of lines of zeros and
lines of poles of G T.D. Stanescu and G.K
cond-matt 0508302
37
(No Transcript)
38
Finite temperature view of the phase diagram
optimal doping in the t-J model.K. Haule and
GK (2006)
39
(No Transcript)
40
(No Transcript)
41
On the accuracy of CDMFT
42
Two Site Cellular DMFT (G.. Kotliar et.al. PRL
(2001)) in the 1D Hubbard model M.Capone
M.Civelli V. Kancharla C.Castellani and GK PRB
69,195105 (2004)T. D Stanescu and GK PRB (2006)
U/t4.
24
43
On the interpretation of CDMFT
44
Doping Driven Mott transiton at low temperature,
in 2d (U16 t1, t-.3 ) Hubbard model
Spectral Function A(k,??0) -1/p G(k, ? ?0) vs k
K.M. Shen et.al. 2004
Antinodal Region
2X2 CDMFT
Senechal et.al PRL94 (2005)
Nodal Region
Civelli et.al. PRL 95 (2005)
45
(No Transcript)
46
RVB phase diagram of the Cuprate Superconductors.
Superexchange.

• The approach to the Mott insulator renormalizes
  the kinetic energy Trvb increases.
• Approach the Mott insulator , Z, charge
  stiffness , TBETcoh goes to zero. M finite.
• Superconducting dome. Pseudogap evolves
  continously into the superconducting state.

G. Kotliar and J. Liu Phys.Rev. B 38,5412 (1988)
Related approach using wave functionsT. M. Rice
group. Zhang et. al. Supercond Scie Tech 1, 36
(1998, Gross Joynt and Rice (1986) M. Randeria
N. Trivedi , A. Paramenkanti PRL 87, 217002
(2001)
47
Doping Driven Mott transiton at low temperature,
in 2d (U16 t1, t-.3 ) Hubbard model
Spectral Function A(k,??0) -1/p G(k, ? ?0) vs k
K.M. Shen et.al. 2004
Antinodal Region
2X2 CDMFT
Senechal et.al PRL94 (2005)
Nodal Region
Civelli et.al. PRL 95 (2005)
48
(No Transcript)
49
Pseudoparticle picture
50
How is the Mott insulatorapproached from the
superconducting state ?
Work in collaboration with M. Capone M Civelli O
Parcollet
51
Nodal Antinodal Dichotomy and pseudogap. T.
Stanescu and GK cond-matt 0508302
52
Superconducting DOS
Superconductivity is destroyed by transfer of
spectral weight. M. Capone et. al. Similar to
slave bosons d wave RVB.
53
Superconductivity in the Hubbard model role of
the Mott transition and influence of the
super-exchange. ( work with M. Capone et.al V.
Kancharla.et.al CDMFTED, 4 8 sites t0) .
54
cond-mat/0508205 Anomalous superconductivity in
doped Mott insulatorOrder Parameter and
Superconducting Gap . They scale together for
small U, but not for large U. S. Kancharla M.
Civelli M. Capone B. Kyung D. Senechal G. Kotliar
andA.Tremblay. Cond mat 0508205 M. Capone
(2006).
55
M. Capone and GK cond-mat 0511334 . Competition
fo superconductivity and antiferromagnetism.
56
Superconducting DOS
Superconductivity is destroyed by transfer of
spectral weight.. Similar to slave bosons d
wave RVB. M. Capone et. al
57
Anomalous Self Energy. (from Capone et.al.)
Notice the remarkable increase with decreasing
doping! True superconducting pairing!! U8t
Significant Difference with Migdal-Eliashberg.
58
Mott Phenomeman and High Temperature
Superconductivity Began Study of minimal model
of a doped Mott insulator within plaquette
Cellular DMFT

• Rich Structure of the normal state and the
  interplay of the ordered phases.
• Work needed to reach the same level of
  understanding of the single site DMFT solution.
• A) Either that we will understand some
  qualitative aspects found in the experiment. In
  which case the next step LDACDMFT or GWCDMFT
  could be then be used make realistic modelling
  of the various spectroscopies.
• B) Or we do not, in which case other degrees of
  freedom, or inhomogeneities or long wavelength
  non Gaussian modes are essential as many authors
  have surmised.
• Too early to tell, talk presented some evidence
  for A.

.
59
(No Transcript)
60
(No Transcript)
61
(No Transcript)
62
(No Transcript)
63
(No Transcript)
64
Outline

• Introduction. Mott physics and high temperature
  superconductivity. Early Ideas slave boson mean
  field theory. Successes and Difficulties.
• Dynamical Mean Field Theory approach and its
  cluster extensions.
• Results for optical conductivity.
• Anomalous superconductivity and normal state.
• Future directions.

65
(No Transcript)
66
(No Transcript)
67
Temperature dependence of the spectral weight of
CDMFT in normal state. Carbone et al, see also
ortholani for CDMFT.
68
Larger frustration t.9t U16tn.69 .92 .96
M. Civelli M. CaponeO. Parcollet and GK PRL
(20050
69
. Spectral weight integrated up to 1 eV of the
three BSCCO films. a) under-doped, Tc70 K b)
optimally doped, Tc80 K c) overdoped, Tc63 K
the fullsymbols are above Tc (integration from
0), the open symbols below Tc, (integrationfrom
0, including th weight of the superfuid).
H.J.A. Molegraaf et al., Science 295, 2239
(2002). A.F. Santander-Syro et al., Europhys.
Lett. 62, 568 (2003). Cond-mat 0111539. G.
Deutscher et. A. Santander-Syro and N. Bontemps.
PRB 72, 092504(2005) . Recent review
70

• P.W. Anderson. Connection between high Tc and
  Mott physics. Science 235, 1196 (1987)
• Connection between the anomalous normal state of
  a doped Mott insulator and high Tc. t-J limit.
• Slave boson approach. ltbgt
  coherence order parameter. k, D singlet formation
  order parameters.Baskaran Zhou Anderson ,
  (1987)Ruckenstein Hirshfeld and Appell (1987)
  .Uniform Solutions. S-wave superconductors.
  Uniform RVB states.

Other RVB states with d wave symmetry. Flux
phase or sid ( G. Kotliar (1988) Affleck and
Marston (1988) . Spectrum of excitation have
point zerosUpon doping they become a d wave
superconductor. (Kotliar and Liu 1988). .
71
The simplest model of high Tcs
t-J, PW Anderson
Hubbard-Stratonovich -gt(to keep some
out-of-cluster quantum fluctuations)
BK Functional, Exact
72
Evolution of the spectral function at low
frequency.
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.
73
(No Transcript)
74
DMFT Qualitative Phase diagram of a frustrated
Hubbard model at integer filling

T/W
Georges et.al. RMP (1996) Kotliar Vollhardt
Physics Today (2004)
75
Single site DMFT and kappa organics. Qualitative
phase diagram Coherence incoherence crosover.
76
Dependence on periodization scheme.
77
Energetics and phase separation. Right U16t Left
U8t
78
Temperature Depencence of Integrated spectral
weight
Phase diagram
79
(No Transcript)
80
Pseudoparticle picture
81
(No Transcript)
82
(No Transcript)
83
(No Transcript)
84
Optical Conductivity near optimal doping. DCA
EDNCA study, K. Haule and GK
85
Behavior of the optical mass and the plasma
frequency.
86
Magnetic Susceptibility
87
References and Collaborators

• References
• M. Capone et. al. in preparation
• M. Capone and G. Kotliar cond-mat
  cond-mat/0603227
• Kristjan Haule, Gabriel Kotliar cond-mat/0605149
• M. Capone and G.K cond-mat/0603227
• Kristjan Haule, Gabriel Kotliar cond-mat/0601478
• Tudor D. Stanescu and Gabriel Kotliar
  cond-mat/0508302
• S. S. Kancharla, M. Civelli, M. Capone, B. Kyung,
  D. Senechal, G. Kotliar, A.-M.S. Tremblay
  cond-mat/0508205
• M. Civelli M. Capone S. S. Kancharla O.
  Parcollet and G. Kotliar Phys. Rev. Lett. 95,
  106402 (2005)

88
P. W. Anderson, Science 235, 1196 (1987)
89
RVB phase diagram of the Cuprate Superconductors.
G. Kotliar and J. Liu Phys.Rev. B 38,5412 (1988)
Related approach using wave functionsT. M. Rice
group. Zhang et. al. Supercond Scie Tech 1, 36
(1998, Gross Joynt and Rice (1986) M. Randeria
N. Trivedi , A. Paramenkanti PRL 87, 217002
(2001)
90
RVB Approach Anderson (1987)

• Understand the physics resulting from the
  proximity to a Mott insulator in the context of
  the simplest models. Leave out disorder,
  electronic structure,phonons
• Follow different states as a function of
  parameters. Second step compare free energies
  which will depend more on the detailed
  modelling..
• Solve the plaquette mean field equations!!!! Work
  in progress.

91
(No Transcript)
92
(No Transcript)
93
(No Transcript)
94
(No Transcript)
95
Kinetic energy change in t-J K Haule and GK
Kinetic energy increases
cluster-DMFT, cond-mat/0601478
Kinetic energy decreases
Kinetic energy increases
cond-mat/0503073
Exchange energy decreases and gives largest
contribution to condensation energy
Phys Rev. B 72, 092504 (2005)
96
(No Transcript)
97
.
CDMFT study of cuprates

• AFunctional of the cluster Greens function.
  Allows the investigation of the normal state
  underlying the superconducting state, by forcing
  a symmetric Weiss function, we can follow the
  normal state near the Mott transition.
• Earlier studies use QMC (Katsnelson and
  Lichtenstein, (1998) M Hettler et. T. Maier
  et. al. (2000) . ) used QMC as an impurity
  solver and DCA as cluster scheme. (Limits U to
  less than 8t )
• Use exact diag ( Krauth Caffarel 1995 ) as a
  solver to reach larger Us
• and smaller Temperature and CDMFT as the
  mean field scheme.
• Recently (K. Haule and GK ) the region near the
  superconducting normal state transition
  temperature near optimal doping was studied
  using NCA DCA-CDMFT .
• DYNAMICAL GENERALIZATION OF SLAVE BOSON ANZATS
• w-S(k,w)m w/b2 -(Db2 t) (cos kx cos ky)/b2
  l
• b--------gt b(k), D -----? D(w), l -----?
  l (k )
• Extends the functional form of self energy to
  finite T and higher frequency.
• Larger clusters can be studied with VCPT CPT
  Senechal and Tremblay, Arrigoni, Hanke

98
RVB phase diagram of the Cuprate Superconductors.
Superexchange.

• The approach to the Mott insulator renormalizes
  the kinetic energy Trvb increases.
• The proximity to the Mott insulator reduce the
  charge stiffness , TBE goes to zero.
• Superconducting dome. Pseudogap evolves
  continuously into the superconducting state.

G. Kotliar and J. Liu Phys.Rev. B 38,5412 (1988)
Related approach using wave functionsT. M. Rice
group. Zhang et. al. Supercond Scie Tech 1, 36
(1998, Gross Joynt and Rice (1986) M. Randeria
N. Trivedi , A. Paramenkanti PRL 87, 217002
(2001)
99
Copper oxide superconducors CuO2
100
Kappa organics
H. Kino H. Fukuyama, J. Phys. Soc. Jpn 65 2158
(1996), R.H. McKenzie, Comments Condens Mat
Phys. 18, 309 (1998)
Y. Shimizu, et al. Phys. Rev. Lett. 91,
107001(2003)
t/t 0.6 - 1.1
101
Photoemission spectra near the antinodal
direction in a Bi2212 underdoped sample.
Campuzano et.al
EDC along different parts of the zone, from Zhou
et.al.
102
Origin of the ph asymmetry
103
(No Transcript)
104
Problems with the approach.

• Stability of the MFT. Ex. Neel order. Slave
  boson MFT with Neel order predicts AF AND SC.
  Inui et.al. 1988 Giamarchi and Lhuillier
  (1987).

105
Copper oxide superconducors CuO2
106
Kappa organics
H. Kino H. Fukuyama, J. Phys. Soc. Jpn 65 2158
(1996), R.H. McKenzie, Comments Condens Mat
Phys. 18, 309 (1998)
Y. Shimizu, et al. Phys. Rev. Lett. 91,
107001(2003)
t/t 0.6 - 1.1
107
Model Hamiltonians
t
t
m
U
t
108
Photoemission spectra near the antinodal
direction in a Bi2212 underdoped sample.
Campuzano et.al
109
RVB phase diagram of the Cuprate Superconductors.
Superexchange.

• Proximity to Mott insulator renormalizes the
  kinetic energy Trvb increases.
• Proximity to the Mott insulator reduce the
  charge stiffness, and QPcoherence scale . T BE
  goes to zero.
• Superconducting dome.
• Pseudogap with d wave symmetry.

G. Kotliar and J. Liu Phys.Rev. B 38,5412 (1988)
Related approach using wave functionsT. M. Rice
group. Zhang et. al. Supercond Scie Tech 1, 36
(1998, Gross Joynt and Rice (1986) M. Randeria
N. Trivedi , A. Paramenkanti PRL 87, 217002
(2001)
110
Approach

• Understand the physics resulting from the
  proximity in the context of the simplest models.
• Leave out disorder, electronic
  structure,phonons,inhomogeneous structures.
• Follow different states as a function of
  parameters.
• Second step compare free energies which will
  depend more on the detailed modelling
• Local (plaquette ) Mott physics. Leave out long
  wavelength collective modes.
• Look at experiments.
• Work in progress. The framework and the resulting
  CDMFT equations are very non trivial to solve.

111
Lower Temperature, AF and SCM. Capone and GK,
SC
AF
SC
AF
AFSC
d
d
112
Mott Phenomeman and High Temperature
Superconductivity Began Study of minimal model
of a doped Mott insulator within plaquette
Cellular DMFT

• Rich Structure of the normal state and the
  interplay of the ordered phases.
• Work needed to reach the same level of
  understanding of the single site DMFT solution.
• A) Either that we will understand some
  qualitative aspects found in the experiment. In
  which case the next step LDACDMFT or GWCDMFT
  could be then be used make realistic modelling
  of the various spectroscopies.
• B) Or we do not, in which case other degrees of
  freedom, or inhomogeneities or long wavelength
  non Gaussian modes are essential as many authors
  have surmised.
• Too early to tell, talk presented some evidence
  for A.

.
113
Dynamical Mean Field Theory. Cavity Construction.
A. Georges and G. Kotliar PRB 45, 6479 (1992).
A(w)
10
114
Finite T, DMFT and the Energy Landscape of
Correlated Materials
T
115
DMFT Qualitative Phase diagram of a frustrated
Hubbard model at integer filling

T/W
Synthesis Brinkman Rice Hubbard Castellani
et.al. Kotliar Ruckenstein Fujimori
116
Single site DMFT and kappa organics. Qualitative
phase diagram Coherence incoherence crosover.
117
Finite T Mott tranisiton in CDMFT O. Parcollet G.
Biroli and GK PRL, 92, 226402. (2004))
CDMFT results Kyung et.al. (2006)
118
Evolution of the spectral function at low
frequency.
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.
119
Evolution of the k resolved Spectral Function at
zero frequency. (Parcollet Biroli and GK PRL,
92, 226402. (2004)) )
U/D2.25
U/D2
Uc2.35-.05, Tc/D1/44. Tmott.01 W
120
Doping Driven Mott transiton at low temperature,
in 2d (U16 t1, t-.3 ) Hubbard model
Spectral Function A(k,??0) -1/p G(k, ? ?0) vs k
K.M. Shen et.al. 2004
Antinodal Region
2X2 CDMFT
Nodal Region
Civelli et.al. PRL 95 (2005)
121
Larger frustration t.9t U16tn.69 .92 .96
M. Civelli M. CaponeO. Parcollet and GK PRL
(20050
122
Larger frustration t.9t U16tn.69 .92 .96
M. Civelli M. CaponeO. Parcollet and GK PRL
(20050