Theoretical and Experimental Magnetism Meeting

1
Theoretical and Experimental Magnetism Meeting
Dynamical Mean Field Approach to strongly
Correlated Electrons

• Gabriel Kotliar
• Rutgers University

3-4 August Cosener house , Abingdon, Oxfordhisre
UK
Support National Science Foundation.
Department of Energy (BES).
2
Outline

• Motivation. Introduction to DMFT ideas.
• Application to the late actinides.
• Application to Cuprate Supeconductors.

Collaborators M. Civelli K. Haule (Rutgers )
Ji-Hoon Shim (Rutgers) S. Savrasov (UCDavis
) A.M. Tremblay B. Kyung V.
Kancharla (Sherbrook) M. Capone (Rome) O
Parcollet(Saclay).
3
The Mott transition across in actinides
4
Cuprate Superconductors doping the Mott
insulator.
5
DMFT Cavity Construction. A. Georges and G.
Kotliar PRB 45, 6479 (1992). Happy marriage of
atomic and band physics.
Extremize a functional of the local spectra.
Local self energy.
Reviews A. Georges G. Kotliar W. Krauth and M.
Rozenberg RMP68 , 13, 1996 Gabriel Kotliar and
Dieter Vollhardt Physics Today 57,(2004). G.
Kotliar S. Savrasov K. Haule V. Oudovenko O.
Parcollet and C. Marianetti (to appear in RMP).
6
Mott transition in one band model. Review
Georges et.al. RMP 96
T/W
Phase diagram of a Hubbard model with partial
frustration at integer filling. Rozenberg et.
al. PRL 1995 Evolution of the Local Spectra as a
function of U,and T. Mott transition driven by
transfer of spectral weight Zhang Rozenberg
Kotliar PRL (1993)..
7
DMFT electronic structure method
Basic idea of DMFT reduce the quantum many body
problem to a one site or a cluster of sites
problem, in a medium of non interacting electrons
obeying a self-consistency condition. (A.
Georges et al., RMP 68, 13 (1996)). DMFT in the
language of functionals DMFT sums up all local
diagrams in BK functional
Basic idea of DMFTelectronic structure method
(LDA or GW) For less correlated bands (s,p)
use LDA or GW For correlated bands (f or d) with
DMFT add all local diagrams. Gives total energy
and spectra
Technical Implementation is Involved. Different
Impurity Solvers. ED-NCA- Expansions in t and U
, etc Different forms of Self consistency
conditions for the bath in the clusters case.
Different levels of complexity in the
description of the electronic structure, simple
models to all electron calculations. Review G.
Kotliar, S. Savrasov K. Haule, V. Oudovenko O
Parcollet, C. Marianetti . Review of Modern
Physics 2006.
8
Mean Field Approach

• Follow different states as a function of
  parameters.
• Second step compare free energies.
• Work in progress. Solving the DMFT equations
  are non trivial.

Configurational cordinate, doping, T, U, structure
9
Photoemission and Localization Trends in Actinides
alpa-gtdelta volume collapse transition
F04,F26.1
F04.5,F27.15
F04.5,F28.11
Curium has large magnetic moment and orders
antif Pu does is non magnetic.
10
The DMFT-valence in the late actinides
11
Minimum in melting curve and divergence of the
compressibility at the Mott endpoint
12
DMFT Phonons in fcc d-Pu
( Dai, Savrasov, Kotliar,Ledbetter, Migliori,
Abrahams, Science, 9 May 2003)
(experiments from Wong et.al, Science, 22 August
2003)
13
Resistivity of Am under pressure. J. C. Griveau
Rebizant Lander and Kotliar PRL 94, 097002 (2005).
14
Photomission Spectra of Am under pressure.
Sunca. Onset of mixed valence. Savrasov Haule
Kotliar (2005)
15
Theoretical Approach P.WAnderson,1987

• Connection of the cuprate anomalies to the
  proximity to a doped Mott insulator without
  magnetic long range order.Spin Liquid
• Study low energy one band models, Hubbard and
  t-J.

Needed. a good mean field theory of the problem.
RVB physics requires a plaquette as a reference
frame.
16
.
CDMFT study of cuprates

• A functional 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. al 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 ) and
  vertex corrected NCA as a solvers to study
  larger Us and CDMFT as the mean field scheme.
  

17
RVB phase diagram of the Cuprate Superconductors.
Superexchange.

• Flux-SiD spin liquid. Affleck and Marston , G
  Kotliar

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)
18
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) .
19
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).
20
Superconducting DOS
Superconductivity is destroyed by transfer of
spectral weight.. Similar to slave bosons d
wave RVB. M. Capone et. al
21
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)
22
Nodal Antinodal Dichotomy and pseudogap. T.
Stanescu and GK cond-matt 0508302
23
Optics and RESTRICTED SUM RULES
Below energy
Low energy sum rule can have T and doping
dependence . For nearest neighbor it gives the
kinetic energy. Use it to extract changes in KE
in superconducing state
24
Optics and RESTRICTED SUM RULES
ltTgtn is defined for Tgt Tc, while ltTgts
exists only for TltTc . ltTgtn is a strong function
of temperature in the normal state. Carbone et.
al (2006) .
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Hubbard versus t-J model

• Kinetic energy in Hubbard model
• Moving of holes
• Excitations between Hubbard bands

Hubbard model
U
Drude
t2/U t
Excitations into upper Hubbard band

• Kinetic energy in t-J model
• Only moving of holes

Drude
t-J model
J-t
no-U
27
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)
28
Haule and Kotliar (2006) Coarsed grained or
local susceptibility around (pp)
Scalapino White PRB 58, 8222 (1988)
29
Conclusion

• DMFT versatile tool for advancing our
  understanding, and predicting properties of
  strongly correlated materials.
• Theoretical spectroscopy in the making.
• Substantial work is needed to refine the tool.
• Great opportunity for experimental-theoretical
  interactions.
• Refine the questions and our understanding by
  focusing on differences between the DMFT results
  and the experiments.

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Mean-Field Classical vs Quantum
Classical case
Quantum case
A. Georges, G. Kotliar (1992)
Phys. Rev. B 45, 6497
33
Anomalous Self Energy. (from Capone et.al.)
Notice the remarkable increase with decreasing
doping! True superconducting pairing!! U8t
Significant Difference with Migdal-Eliashberg.
34
ltl.sgt in the late actinides DMFT results K.
Haule and J. Shim
35
a-U
36
J. Tobin et.al. PRB 72,085109 (2005)
XAS and EELS
37
Double well structure and d Pu

• Qualitative explanation of negative thermal
  expansionLawson, A. C., Roberts J. A., Martinez,
  B., and Richardson, J. W., Jr. Phil. Mag. B, 82,
  1837,(2002). G. Kotliar J.Low Temp. Physvol.126,
  1009 27. (2002)

F(T,V)FphononsFinvar
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.
38
Invar model for Pu-Ga. Lawson et. al.Phil.
Mag. (2006) Data fits if the excited state has
zero stiffness.
39
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)

40
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.

.
41
Correlations Magnetism and Structure across the
actinide series a Dynamical Mean Field Theory
Perspective
G.Kotliar Physics Department and Center for
Materials Theory Rutgers University. .
Collaborators K. Haule (Rutgers ) Ji-Hoon
Shim (Rutgers) S. Savrasov (UCDavis ) A.M.
Tremblay B. Kyung (Sherbrook) M. Capone (Rome) O
Parcollet(Saclay).
Support DOE- BES DOE-NNSA .
Expts. M. Fluss J. C Griveaux G Lander A.
Lawson A. Migliori J.Singleton J.Smith J Thompson
J. Tobin

• Plutonium Futures Asilomar July 9-13 (2006).

42
M. Capone and GK cond-mat 0511334 . Competition
fo superconductivity and antiferromagnetism.
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45
Temperature dependence of the spectral weight of
CDMFT in normal state. Carbone, see also
ortholani for CDMFT.
46
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Finite temperature view of the phase diagram t-J
model.K. Haule and GK (2006)
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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.

50
UPS of alpha-U
GGA
- He I (hv21.21eV), He II (hv40.81eV) -
f-electron features is enhanced in He II
spectra. Opeil et al. PRB(2006)
51

• LDADMFT reproduces peaks near -1eV, 0.3eV, and
  EF
• The peak near -3eV corresponds to U 6d states.

n_f2.94
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Cluster Extensions of Single Site DMFT
Many Techniques for solving the impurity model
QMC, (Fye-Hirsch), NCA, ED(Krauth Caffarel),
IPT, For a review see Kotliar et. Al to
appear in RMP (2006)
55
n_5/22.41 n_7/20.53
56
How is the Mott insulatorapproached from the
superconducting state ?
Work in collaboration with M. Capone M Civelli O
Parcollet
57

• In BCS theory the order parameter is tied to the
  superconducting gap. This is seen at U4t, but
  not at large U.
• How is superconductivity destroyed as one
• approaches half filling ?

58
Superconducting State t0

• Does it superconduct ?
• Yes. Unless there is a competing phase.
• Is there a superconducting dome ?
• Yes. Provided U /W is above the Mott transition .
• Does the superconductivity scale with J ?
• Yes. Provided U /W is above the Mott transition .
• Is superconductivity BCS like?
• Yes for small U/W. No for large U, it is RVB
  like!

59

• The superconductivity scales
• with J, as in the RVB approach.
• Qualitative difference between large and small U.
  The superconductivity goes to zero at half
  filling ONLY above the Mott transition.

60
Can we connect the superconducting state with the
underlying normal state ? What does the
underlying normal state look like ?
61
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
62
Nodal Antinodal Dichotomy and pseudogap. T.
Stanescu and GK cond-matt 0508302
63
Optics and RESTRICTED SUM RULES
Below energy
Low energy sum rule can have T and doping
dependence . For nearest neighbor it gives the
kinetic energy. Use it to extract changes in KE
in superconducing state
64
Larger frustration t.9t U16tn.69 .92 .96
M. Civelli M. CaponeO. Parcollet and GK PRL
(20050
65
Add equation for the difference between the
methods.

• Can compute kinetic energy from both the integral
  of sigma and the expectation value of the kinetic
  energy.
• Treats normal and superconducting state on the
  same footing.

66
. 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
67
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 LDACDMFT or GWCDMFT could be then
  be used to account semiquantitatively for the
  large body of experimental data by studying
  more realistic models of the material.
• B) Or we do not, in which case other degrees of
  freedom, or inhomgeneities or long wavelength non
  Gaussian modes are essential as many authors
  have surmised.
• Too early to tell, talk presented some evidence
  for A.

.
68
Issues

• What aspects of the unusual properties of the
  cuprates follow from the fact that they are doped
  Mott insulators using a DMFT which treats exactly
  and in an umbiased way all the degrees of freedom
  within a plaquette ?
• Solution of the model at a given energy scale,
• Physics at a given energy
• Recent Conceptual Advance DMFT (in its single
  site a cluster versions) allow us to address
  these problems.
• A) Follow various metastable states as a function
  of doping.
• B) Focus on the physics on a given scale at at
  time. What is the right reference frame for high
  Tc.

69

• 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). .
70
The simplest model of high Tcs
t-J, PW Anderson
Hubbard-Stratonovich -gt(to keep some
out-of-cluster quantum fluctuations)
BK Functional, Exact
71
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.
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Dynamical Mean Field Theory. Cavity Construction.
A. Georges and G. Kotliar PRB 45, 6479 (1992).
Reviews A. Georges W. Krauth G.Kotliar and M.
Rozenberg RMP (1996)G. Kotliar and D. Vollhardt
Physics Today (2004).
74
Mean-Field Classical vs Quantum
Classical case
Quantum case
Hard!!!
Easy!!!
QMC J. Hirsch R. Fye (1986) NCA T. Pruschke
and N. Grewe (1989) PT Yoshida and Yamada
(1970) NRG Wilson (1980)
A. Georges, G. Kotliar (1992)
75
DMFT Qualitative Phase diagram of a frustrated
Hubbard model at integer filling

T/W
Georges et.al. RMP (1996) Kotliar Vollhardt
Physics Today (2004)
76
Single site DMFT and kappa organics. Qualitative
phase diagram Coherence incoherence crosover.
77
Finite T Mott tranisiton in CDMFT O. Parcollet G.
Biroli and GK PRL, 92, 226402. (2004))
CDMFT results Kyung et.al. (2006)
78
.

• Functional 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.
• Can study different states on the same footing
  allowing for the full frequency dependence of all
  the degrees of freedom contained in the
  plaquette.
• 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 )
• Better description of the incoherent state, more
  general functional form of the self energy to
  finite T and higher frequency.

CDMFT methodological comments
Further extensions by periodizing cumulants
rather than self energies. Stanescu and GK (2005)
79
Early SB DMFT.

• There are two regimes, one overdoped one
  underdoped.
• Tc has a dome-like shape.
• High Tc superconductivity is driven by
  superexchange.
• Normal state at low doping has a pseudogap a low
  doping with a d wave symmetry.

80

• Normal State at low temperatures.

81
Dependence on periodization scheme.
82
Energetics and phase separation. Right U16t Left
U8t
83
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.
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85
Temperature Depencence of Integrated spectral
weight
Phase diagram
86
E Energy difference between the normal and
superconducing state of the t-J model. K. Haule
(2006)
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Conclusion

• DMFT studies of electrons and lattice
  displacements.
• Valence changes and transfers of spectral
  weight. Consistent picture of Pu-Am-Cm.
• Alpha and delta Pu, screened (5f)5
  configuration. Differ in the degree of screening.
• Different views Pu non magnetic (5f)6, Pu
  magnetic
• Magnetism and defects.
• Important role of phonon entropy in phase
  transformations .

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LS vs jj coupling in Am and Cm
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92
Temperature dependence of the spectral weight of
CDMFT in normal state. Carbone, see also Toschi
et.al for CDMFT.
93
UPS of alpha-U
GGA
- He I (hv21.21eV), He II (hv40.81eV) -
f-electron features is enhanced in He II
spectra. Opeil et al. PRB(2006)
94

• LDADMFT reproduces peaks near -1eV, 0.3eV, and
  EF
• The peak near -3eV corresponds to U 6d states.

n_f2.94
95
ltl.sgt in the late actinides DMFT results K.
Haule and J. Shim
96
a-U
97
Why is Epsilon Pu (which is smaller than delta
Pu) stabilized at higher temperatures ??Compute
phonons in bcc structure.
98
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 T0. 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.

99
Double well structure and d Pu

• Qualitative explanation of negative thermal
  expansionLawson, A. C., Roberts J. A., Martinez,
  B., and Richardson, J. W., Jr. Phil. Mag. B, 82,
  1837,(2002). G. Kotliar J.Low Temp. Physvol.126,
  1009 27. (2002)

F(T,V)FphononsFinvar
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.
100
Invar model for Pu-Ga. Lawson et. al.Phil.
Mag. (2006) Data fits if the excited state has
zero stiffness.
101
Approach

• 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..
• Work in progress. The framework and the resulting
  equations are very non trivial to solve.

102
Approach the Mott point from the right Am under
pressure
Experimental Equation of State
(after Heathman et.al, PRL 2000)
Mott Transition?
Soft
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 m6mB
• (Soderlind et.al., PRB 2000).
• Experimentally, Am has non magnetic f6 ground
  state with J0 (7F0)

103
Am equation of state. LDADMFT.New acceleration
technique for solving DMFT equations S. Savrasov
K. Haule G. Kotliar cond-mat. 0507552 (2005)
104
Photoemission spectra using Hubbard I solver
Lichtenstein and Katsnelson, PRB 57, 6884,(1998
), Svane cond-mat 0508311 and Sunca . Savrasov
Haule and Kotliar cond-mat 0507552 Hubbard
bands width is determined by multiplet
splittings.
105
DMFT Phonons in fcc d-Pu
( Dai, Savrasov, Kotliar,Ledbetter, Migliori,
Abrahams, Science, 9 May 2003)
(experiments from Wong et.al, Science, 22 August
2003)
106
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 T0. 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.

107
Double well structure and d Pu

• Qualitative explanation of negative thermal
  expansionLawson, A. C., Roberts J. A., Martinez,
  B., and Richardson, J. W., Jr. Phil. Mag. B, 82,
  1837,(2002). G. Kotliar J.Low Temp. Physvol.126,
  1009 27. (2002)

F(T,V)FphononsFinvar
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.
108
Invar model for Pu-Ga. Lawson et. al.Phil.
Mag. (2006) Data fits if the excited state has
zero stiffness.
109
a-U
110
Why is Epsilon Pu (which is smaller than delta
Pu) stabilized at higher temperatures ??Compute
phonons in bcc structure.
111
What can we learn from small Cluster-DMFT?
Phase diagram
112
The simplest model of high Tcs
t-J, PW Anderson
Hubbard-Stratonovich -gt(to keep some
out-of-cluster quantum fluctuations)
BK Functional, Exact
113
.
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 ) and
  vertex corrected NCA as a solvers to study
  larger Us 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

114
Exact Baym Kadanoff functional ofwo variables.
GS,G. Restric to the degrees of freedom that
live on a plaquette and its supercell extension..
Maps the many body problem onto a self consistent
impurity model
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 in Press.
Tremblay Kyung Senechal cond-matt 0511334
115
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).
• Mean field is too uniform on the Fermi surface,
  in contradiction with ARPES.Penetration depth,
  Wen and Lee Raman spectra, sacuttos talk,
  Photoemission
• Description of the incoherent finite
  temperature regime.

Development of DMFT in its plaquette
version may solve some of these
problems.!!