Transcript Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped

Dynamical RVB: Cluster Dynamical
Mean Field Studies of Doped Mott
Insulators.
Gabriel Kotliar
and Center for Materials Theory &
CPHT Ecole Polytechnique Palaiseau & SPHT CEA Saclay, France
Itzykson Meeting 2006, Strongly Correlated Electrons
Cea Saclay June 21-23 2006
Support : Chaire Blaise Pascal Fondation de l’Ecole Normale.
Outline
• Introduction. Mott physics and high temperature
superconductivity. Early Ideas: slave boson mean
field theory. Successes and Difficulties.
• The Dynamical Mean Field Theory approach and
cluster extensions.
• CDMFT results. Normal state.
• Competition of superconductivity and
antiferromagnetism.
• Comparing superconductivity and the normal
state.
• Outlook.
References-Collaborators
• A. Georges (Ecole Polytechnique)
• G. Biroli (CEA-Saclay)-S. Savrasov
(UCDavis) O. Parcollet (CEA-Saclay).
• M Civelli (ILL-Grenoble)T. Stanescu and
M.Capone (U. Rome).
• B. Kyung, D. Senechal A. M. Tremblay
(Sherbrooke)
• K. Haule (Rutgers).
Cuprate Experimental Phase diagram
Damascelli, Shen, Hussain, RMP 75, 473 (2003)
Kappa Organics
F. Kagawa, K. Miyagawa, + K. Kanoda
PRB 69 (2004) +Nature 436 (2005)
Phase diagram of (X=Cu[N(CN)2]Cl)
S. Lefebvre et al. PRL 85, 5420 (2000), P. Limelette, et al. PRL 91 (2003)
Cuprates Open Problems
• What is the mechanism for high temperature
superconductivity. Why is it realized in the
copper oxides?
• What are the relevant energy degrees of
freedom to describe the physics of these
materials at a given energy scale?
• Proper reference frame for understanding the
correlated solid, e.g. are there other
competing phases besides SC, and
quantum critical points controlling the
physical properties of this material.
Approach
•
•
•
•
•
Understand the physics resulting from the proximity to a
Mott insulator in the context of the simplest models.
[ Leave out disorder, electronic structure …]
Follow different “states” as a function of parameters.
[Second step compare free energies which will depend
more on the detailed modelling…..]
Mean Field Approach. No R.G. analysis of QCP.
Approach the problem directly from finite temperatures,not
from zero temperature. Address issues of finite frequency –
temperature crossovers, coherent QP incoherent Hubbard
bands.
Work in progress. The framework and the resulting
equations are very non trivial to solve and to interpret.
Hubbard
HubbardHamiltonians
Hamiltonian

H  ijt i,j c 
c

c
U  i n in i
i j
jc i 
t-J Hamiltonian
Slave Boson Formulation: Baskaran Zhou Anderson (1987)
Ruckenstein Hirschfeld and Appell (1987)
b+i bi +f+si fsi = 1
Perspective
U/t
Doping Driven Mott
Transition . Cuprates
Pressure Driven Mott
transtion k-organics
d
t’/t
A Tale of Two Phase Diagrams:
G. Kotliar, J. of Low Temp. Phys.
126, pp.1009-27.
• 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, 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 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). .
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
continously into the
superconducting state.
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)
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.
• Mean field is too uniform on the Fermi surface,
in contradiction with ARPES.[Penetration depth,
Wen and Lee , Ioffe and Millis, Photoemission ]
• Description of the incoherent regime.
Fluctuations.
Development of cluster DMFT may solve some of
problems.!!
these
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 - h å Si
i, j

i , j  ,
i
H MF = - heff So
Easy!!!
s (t
b
ij
 d ij )(ci†s c js  c †js cis )  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)
h
áS ñ=eff
th[b heff ]
PT : Yoshida and Yamada (1970)
NRG: Wilson (1980)
D (w)
0
m0 = áS0 ñHMF ( heff ) IPT: Georges Kotliar
heff =
å
j
J ij m j +
(1992). .
†
G
(
i
w
)
=
­
á
c
os
n
os (iwn )cos (iwn )ñSMF ( D )
QMC: M. Jarrell, (1992),
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)
CDMFT: removes limitations of single site DMFT
•No k dependence of the self energy.
Various cluster approaches,
DCA
momentum
spcace. Cellular DMFT G. Kotliar et.al.
•No
d-wave
superconductivity.
PRL (2004). O Parcollet G. Biroli and G. Kotliar B 69, 205108 (2004)
•No Peierls
dimerization.
T. D. Stanescu
and G. Kotliar
cond-mat/0508302
•No (R)valence bonds.
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
•
•
•

•
•
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
w-S(k,w)+= w/b2 -(D+b2 t) (cos kx + cos ky)/b2 +l
b--------> 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.
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)
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)
Evolution of the spectral function
at low frequency.
A(w  0, k )vs k
Ek=t(k)+ReS( k , w  0)  
 k = ImS( k , w  0)
k
A( k , w  0)  2
2
 k  Ek
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. (Parcollet Biroli and GK
PRL, 92, 226402. (2004)) )
U/D=2
U/D=2.25
Uc=2.35+-.05, Tc/D=1/44. Tmott~.01 W
A(w  0, k )vs k
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)
Larger frustration: t’=.9t U=16t
n=.69 .92 .96
M. Civelli M. CaponeO. Parcollet and GK
PRL (20050
Nodal Antinodal Dichotomy and pseudogap. T.
Stanescu and GK cond-matt 0508302
Finite temperature view of the phase
diagram t-J model.
K. Haule (2006)
Lower Temperature, AF and SC
M. Capone and GK, Kancharla et. al.
SC
AF
SC
AF
AF+SC
d
d
M. Capone and GK cond-mat 0511334 . Competition fo
superconductivity and antiferromagnetism.
• Can we continue the superconducting state
towards the Mott insulating state ?
For U > ~ 8t
YES.
For U ~ < 8t NO, magnetism really
gets in the way.
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. M. Capone (2006).
Energetics and phase separation. Right
U=16t Left U=8t
Optics and RESTRICTED SUM RULES
H hamiltonian, J electric current , P polarization


0
s (w )dw 
Below energy


iV


0
H eff , J eff , Peff
 2 k
 nk 2
k
k
  P, J  
s (w )dw 
 ne 2
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
E Energy difference between the normal
and superconducing state of the t-J model.
K. Haule (2006)
. 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.
.