Transcript Correlated Electrons: A Dynamical Mean Field (DMFT

Dynamical Mean Field Approach to
strongly Correlated Electrons
Gabriel Kotliar
Rutgers University
Theoretical and Experimental Magnetism Meeting
3-4 August Cosener house , Abingdon, Oxfordhisre UK
Support :National Science Foundation. Department of
Energy (BES).
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).
The Mott transition across in actinides
Cuprate Superconductors: doping the Mott
insulator.
DMFT Cavity Construction. A. Georges and G. Kotliar PRB 45, 6479 (1992). Happy
marriage of atomic and band physics.
1
G(k , i ) 
i   k  (i )
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).
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)..
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 DMFT+electronic 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.
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.
T
Configurational cordinate, doping, T, U, structure
Photoemission and Localization
Trends in Actinides
alpa->delta volume collapse transition
F0=4,F2=6.1
F0=4.5,F2=7.15
Curie-Weiss
F0=4.5,F2=8.11
Curium has large magnetic moment and orders antif
Pu does is non magnetic.
Tc
The “DMFTvalence” in the
late actinides
Minimum in melting curve and divergence of the
compressibility at the Mott endpoint
Vsol
Vliq
dT
V
(
)
dp
S
DMFT Phonons in fcc d-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)
Resistivity of Am under pressure. J. C. Griveau
Rebizant Lander and Kotliar PRL 94, 097002 (2005).
Photomission Spectra of Am under pressure. Sunca. Onset of
mixed valence. Savrasov Haule Kotliar (2005)
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.
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 U’s and CDMFT as the
mean field scheme.
RVB phase diagram of the Cuprate
Superconductors. Superexchange.
Flux-S+iD spin liquid. [Affleck
and Marston , G Kotliar]
G. Kotliar and J. Liu Phys.Rev. B
38,5412 (1988)
Related approach using wave functions:T. 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)
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
CDMFT+ED, 4+ 8 sites t’=0) .
cond-mat/0508205 Anomalous superconductivity in doped
Mott insulator:Order 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).
Superconducting DOS
d  .06
d =.08
d
.1
d = .16
Superconductivity is destroyed by transfer of
spectral weight.. Similar to slave bosons d wave
RVB. M. Capone et. al
Doping Driven Mott transiton at low temperature, in 2d
(U=16 t=1, t’=-.3 ) Hubbard model
Spectral Function A(k,ω→0)= -1/π G(k, ω →0) vs k
K.M. Shen et.al. 2004
Antinodal Region
Senechal et.al
PRL94 (2005)
Nodal Region
2X2 CDMFT
Civelli et.al. PRL 95 (2005)
Nodal Antinodal Dichotomy and pseudogap. T.
Stanescu and GK cond-matt 0508302
Optics and RESTRICTED SUM RULES
H hamiltonian, J electric current , P polarization


0
 ( )d 
Below energy


iV


0
Heff , J eff , Peff
 2 k
 nk 2
k
k
  P, J  
 ( )d 
 ne2
m

  Peff , J eff  
iV
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
Optics and RESTRICTED SUM RULES


0
 ( )   s( )d  T  n(T )   T  s(T )
n
<T>n is defined for
T> Tc, while <T>s
exists only for
T<Tc . <T>n is a strong
function of temperature in the
normal state. Carbone et. al
(2006) .
Hubbard versus t-J model
Kinetic energy in Hubbard model:
•Moving of holes
•Excitations between Hubbard bands
Hubbard model
U
Drude
t2/U t
Experiments
Excitations into upper
Hubbard band
Kinetic energy in t-J model
•Only moving of holes
Drude
J-t
intraband
interband
transitions
t-J model
no-U
~1eV
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
Phys Rev. B 72, 092504 (2005)
Exchange energy decreases and gives
largest contribution to condensation energy
Haule and Kotliar (2006)
Coarsed grained or “local
“ susceptibility around
()
Scalapino White PRB 58, 8222 (1988)
 Si .S j >=
k

3


d

b( )  "(k ,  ) cos q.Ri  Rj
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.
Mean-Field : Classical vs Quantum
Classical case
-
å
J ij Si S j - hå Si
i, j
Quantum case
 (t

i , j  ,
i
HMF = - heff So
b
ij
 d ij )(ci† c j  c†j ci )  U  ni  ni 
i
b
b
¶
†
ò ò cos (t )[ ¶ t + m- D (t - t ')]cos (t ') + U ò no­ no¯
0 0
0
heff
D (w)
m0 = áS0 ñHMF ( heff )
heff =
å
J ij m j + h
j
Phys. Rev. B 45, 6497
G = ­ áco†s (iwn )cos (iwn )ñSMF (D )
G (iwn ) =
å
k
1
[D (iwn ) -
1
- ek ]
G (iwn )[D ]
A. Georges, G. Kotliar (1992)
Anomalous Self Energy. (from Capone et.al.) Notice the
remarkable increase with decreasing doping! True
superconducting pairing!! U=8t
Significant Difference with Migdal-Eliashberg.
<l.s> in the late actinides [DMFT results:
K. Haule and J. Shim ]
a-U
XAS and EELS
J. Tobin et.al. PRB 72,085109
(2005)
Double well structure and d Pu
Qualitative explanation of negative thermal expansion[Lawson, 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)=Fphonons+
Finvar
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.
“Invar model “ for Pu-Ga. Lawson et. al.Phil. Mag.
(2006) Data fits if the excited state has zero stiffness.
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)
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
LDA+CDMFT or GW+CDMFT 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.
.
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).
M. Capone and GK cond-mat 0511334 . Competition fo
superconductivity and antiferromagnetism.
Temperature dependence of the spectral
weight of CDMFT in normal state.
Carbone, see also ortholani for CDMFT.
Finite temperature view of the phase
diagram t-J model.
K. Haule and GK (2006)
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.
UPS of alpha-U
GGA
- He I (hv=21.21eV), He II (hv=40.81eV)
- f-electron features is enhanced in He II spectra.
Opeil et al. PRB(2006)
-LDA+DMFT reproduces peaks
near -1eV, 0.3eV, and EF
-The peak near -3eV
corresponds to U 6d states.
n_f=2.94
Cluster Extensions of Single Site DMFT
latt (k , ) 0 ( ) 
1 ( )(cos kx  cos ky )  2 ( )(cos kx.cos ky )  .......
Many Techniques for
solving the impurity
model: QMC, (FyeHirsch), NCA,
ED(Krauth –Caffarel),
IPT, …………For a
review see Kotliar et.
Al to appear in RMP
(2006)
n_5/2=2.41
n_7/2=0.53
How is the Mott insulator
approached from the
superconducting state ?
Work in collaboration with M. Capone M Civelli O Parcollet
• In BCS theory the order parameter is tied to
the superconducting gap. This is seen at
U=4t, but not at large U.
• How is superconductivity destroyed as one
approaches half filling ?
Superconducting State t’=0
•
•
•
•
•
•
•
•
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!
• 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.
•Can we connect the
superconducting state with the
“underlying “normal” state “ ?
What does the underlying “normal”
state look like ?
Follow the “normal state” with doping. Civelli et.al. PRL 95,
106402 (2005)
Spectral Function A(k,ω→0)= -1/π G(k, ω →0) vs k U=16 t,
t’=-.3
A(  0, k )vs k
K.M. Shen et.al. 2004
Ek=t(k)+Re(k ,   0)  
 k = Im(k ,   0)
k
A(k ,   0)  2
 k  Ek 2
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
Nodal Antinodal Dichotomy and pseudogap. T.
Stanescu and GK cond-matt 0508302
Optics and RESTRICTED SUM RULES
H hamiltonian, J electric current , P polarization


0
 ( )d 
Below energy


iV


0
Heff , J eff , Peff
 2 k
 nk 2
k
k
  P, J  
 ( )d 
 ne2
m

  Peff , J eff  
iV
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
Larger frustration: t’=.9t U=16t
n=.69 .92 .96
M. Civelli M. CaponeO. Parcollet and GK
PRL (20050
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.
•
. Spectral weight integrated up to 1 eV of the three BSCCO
films. a) underdoped, Tc=70 K; b) ∼ optimally doped, Tc=80 K; c)
overdoped, Tc=63 K; the full
symbols 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:
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 LDA+CDMFT or
GW+CDMFT 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.
.
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.
• 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.
<b> coherence
order parameter. k,  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 s+id (
G. Kotliar (1988) Affleck and Marston (1988) . Spectrum of
excitation have point zerosUpon doping they become a d –
wave superconductor. (Kotliar and Liu 1988). .
The simplest model of high Tc’s
t-J, PW Anderson
Hubbard-Stratonovich ->(to keep some out-of-cluster quantum fluctuations)
BK Functional, Exact
cluster in k space
cluster in real space
Evolution of the spectral function
at low frequency.
A(  0, k )vs k
Ek=t(k)+Re(k ,   0)  
 k = Im(k ,   0)
k
A(k ,   0)  2
 k  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.
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).
Mean-Field : Classical vs Quantum
Classical case
-
å
Quantum case
J ij Si S j - hå Si
i, j

i , j  ,
i
HMF = - heff So
Easy!!!
h
áS ñ=eff
th[b heff ]
 (t
b
ij
 d ij )(ci† c j  c†j ci )  U  ni  ni 
i
b
b
¶
†
m- D (t - t ')]cos (t ') + U ò no­ no¯
ò ò cos (t )[ ¶ t + Hard!!!
0 0
0
QMC: J. Hirsch R. Fye (1986)
NCA : T. Pruschke and N. Grewe (1989)
PT : Yoshida and Yamada (1970)
NRG: Wilson (1980)
D (w)
0
m0 = áS0 ñHMF (heff ) IPT: Georges Kotliar (1992). .
†
G
(
i
w
)
=
­
á
c
os
n
os (iwn )cos (iwn )ñSMF (D )
QMC: M. Jarrell, (1992),
heff =
å
j
Jij m j +
NCA T.Pruschke D. Cox and M. Jarrell
(1993),
1
G
(
i
w
)
=
h ED:Caffarel Krauth nand Rozenberg
(1994) 1
å
Projective method: G Moellerk (1995).
[D (1999)
(iwn ) - t (k ) + m]
NRG: R. Bulla et. al. PRL 83, 136
G
(
i
w
)
[
D
]
n
,……………………………………...
• Pruschke et. al Adv. Phys. (1995)
• Georges et. al RMP
A.(1996)
Georges, G. Kotliar (1992)
DMFT Qualitative Phase diagram of a
frustrated Hubbard model at integer filling
T/W
Georges et.al.
RMP (1996)
Kotliar
Vollhardt
Physics Today
(2004)
Single site DMFT and kappa organics. Qualitative phase
diagram Coherence incoherence crosover.
Finite T Mott tranisiton in CDMFT O. Parcollet
G. Biroli and GK PRL, 92, 226402. (2004))
CDMFT results Kyung et.al. (2006)
•
•
•

•
•
CDMFT
:
methodological
comments
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
-(k,)+= /b2 -(+b2 t) (cos kx + cos ky)/b2 +l
b--------> b(k),  ----- (), l  l (k )
Better description of the incoherent state, more general
functional form of the self energy to finite T and higher
frequency.
S latt (k , w) = S 11 + S 23 (cos kx + cos ky )
+ S 24 cos kx cos ky
Further extensions by periodizing cumulants rather than self energies. Stanescu and
GK (2005)
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.
• Normal State at low temperatures.
Dependence on periodization scheme.
Energetics and phase separation. Right
U=16t Left U=8t
Evolution of the spectral function
at low frequency.
A(  0, k )vs k
Ek=t(k)+Re(k ,   0)  
 k = Im(k ,   0)
k
A(k ,   0)  2
 k  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.
Temperature Depencence of
Integrated spectral weight
Phase diagram
t’=0
E Energy difference between the normal
and superconducing state of the t-J model.
K. Haule (2006)
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 .
LS vs jj coupling in Am and Cm
Temperature dependence of the spectral
weight of CDMFT in normal state.
Carbone, see also Toschi et.al for CDMFT.
UPS of alpha-U
GGA
- He I (hv=21.21eV), He II (hv=40.81eV)
- f-electron features is enhanced in He II spectra.
Opeil et al. PRB(2006)
-LDA+DMFT reproduces peaks
near -1eV, 0.3eV, and EF
-The peak near -3eV
corresponds to U 6d states.
n_f=2.94
<l.s> in the late actinides [DMFT results:
K. Haule and J. Shim ]
a-U
Why is Epsilon Pu (which is smaller than delta
Pu) stabilized at higher temperatures ??Compute
phonons in bcc structure.
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.
Double well structure and d Pu
Qualitative explanation of negative thermal expansion[Lawson, 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)=Fphonons+
Finvar
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.
“Invar model “ for Pu-Ga. Lawson et. al.Phil. Mag.
(2006) Data fits if the excited state has zero stiffness.
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.
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)
Am equation of state. LDA+DMFT.New acceleration
technique for solving DMFT equations S. Savrasov K.
Haule G. Kotliar cond-mat. 0507552 (2005)
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.
DMFT Phonons in fcc d-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)
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.
Double well structure and d Pu
Qualitative explanation of negative thermal expansion[Lawson, 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)=Fphonons+
Finvar
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.
“Invar model “ for Pu-Ga. Lawson et. al.Phil. Mag.
(2006) Data fits if the excited state has zero stiffness.
a-U
Why is Epsilon Pu (which is smaller than delta
Pu) stabilized at higher temperatures ??Compute
phonons in bcc structure.
What can we learn from “small”
Cluster-DMFT?
Phase diagram
t’=0
The simplest model of high Tc’s
t-J, PW Anderson
Hubbard-Stratonovich ->(to keep some out-of-cluster quantum fluctuations)
BK Functional, Exact
cluster in k space
cluster in real space
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 U’s 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
-(k,)+= /b2 -(+b2 t) (cos kx + cos ky)/b2 +l
b--------> b(k),  ----- (), 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 ]
Exact Baym Kadanoff functional ofwo variables. G,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
S latt (k , w) = S 11 + S 23 (cos kx + cos ky )
+ S 24 cos kx cos ky
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
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 L’huillier (1987).
• Mean field is too uniform on the Fermi surface,
in contradiction with ARPES.[Penetration depth,
Wen and Lee ][Raman spectra, sacutto’s talk,
Photoemission ]
• Description of the incoherent finite temperature
regime.
Development of DMFT in its plaquette version may solve
some of these problems.!!