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Cellular DMFT studies of the doped
Mott insulator
• Gabriel Kotliar
• Center for Materials Theory Rutgers University
• CPTH Ecole Polytechnique Palaiseau, and
CEA Saclay , France
Collaborators: M. Civelli, K. Haule, M. Capone, O.
Parcollet, T. D. Stanescu, (Rutgers) V. Kancharla
(Rutgers+Sherbrook) A. M Tremblay, D. Senechal B.
Kyung
(Sherbrooke)
Discussions: A. Georges, N. Bontemps, A. Sacuto.
$$Support : NSF DMR . Blaise Pascal Chair Fondation de l’Ecole Normale.
More Disclaimers
• Leave out inhomogeneous states and ignore disorder.
• What can we understand about the evolution of the
electronic structure from a minimal model of a doped
Mott insulator, using Dynamical Mean Field Theory ?
• Approach the problem directly from finite
temperatures,not from zero temperature. Address
issues of finite frequency –temperature crossovers. As
we increase the temperature DMFT becomes more and
more accurate.
• DMFT provides a reference frame capable of describing
coherence-incoherence crossover phenomena.
RVB physics and Cuprate
Superconductors
• 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.
• Slave boson approach.
<b>
coherence order parameter. k, D singlet formation
order parameters.Baskaran Zhou Anderson ,
Ruckenstein et.al (1987) .
Other states flux phase or s+id ( G. Kotliar (1988) Affleck and
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.
• Neel order. How to continue a Neel insulating state ?
Need to treat properly finite T.
• Temperature dependence of the penetration depth [Wen
and Lee , Ioffe and Millis ] . Theory:r[T]=x-Ta x2 , Exp:
r[T]= x-T a.
• Mean field is too uniform on the Fermi surface, in
contradiction with ARPES.
• No quantitative computations in the regime where there
is a coherent-incoherent crossover,compare well with
experiments. [e.g. Ioffe Kotliar 1989]
The development of DMFT solves may solve many of these
problems.!!
Impurity Model-----Lattice Model
Parametrizes the physics in
terms of a few functions .
D , Weiss Field
Alternative (T. Stanescu and
G. K. ) periodize the cumulants
rather than the self energies.
Cluster DMFT schemes
• Mapping of a lattice model onto a quantum impurity model (degrees
of freedom in the presence of a Weiss field, the central concept in
DMFT). Contain two elements.
• 1) Determination of the Weiss field in terms of cluster quantities.
• 2) Determination of lattice quantities in terms of cluster quantities
(periodization).
Controlled Approximation, i.e. theory can tell when the it is reliable!!
Several methods,(Bethe, Pair Scheme, DCA, CDMFT, Nested
Schemes, Fictive Impurity Model, etc.) field is rapidly developing.
For reviews see: Georges et.al. RMP (1996) Maier et.al RMP (2005),
Kotliar et.al cond-mat 0511085. Kyung et.al cond-mat 0511085
About CDMFT
• Reference frame (such as FLT-DFT ) but is
able describe strongly correlated
electrons at finite temperatures, in a
regime where the quasiparticle picture is
not valid.
• It easily describes a Fermi liquid state
when there is one, at low temperatures
and the coherence incoherence crossover.
Functional mean field!
•
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 U’s
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 .
• DYNAMICAL GENERALIZATION OF SLAVE BOSON ANZATS
 w-S(k,w)+m= w/b2 -(D+b2 t) (cos kx + cos ky)/b2 +l
• b--------> b(k), D ----- D(w), l ----- l (k )
• Extends the functional form of self energy to finite T and higher frequency.
Testing
CDMFT (G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev.
Lett. 87, 186401 (2001) ) with two sites in the Hubbard model in one
dimension V. Kancharla C. Bolech and GK PRB 67, 075110 (2003)][[M.Capone
M.Civelli V Kancharla C.Castellani and GK PR B 69,195105 (2004) ]
U/t=4.
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
K.M. Shen et.al. 2004
A(w  0, k )vs k
Ek=t(k)+ReS( k , w  0) - m
 k = ImS( k , w  0)
A( k , w  0) 
k
 k 2  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
Approaching the Mott transition:
CDMFT Picture
• Fermi Surface Breakup. Qualitative effect,
momentum space differentiation. Formation of
hot –cold regions is an unavoidable
consequence of the approach to the Mott
insulating state!
• D wave gapping of the single particle spectra as
the Mott transition is approached. Real and
Imaginary part of the self energies grow
approaching half filling. Unlike weak coupling!
• Similar scenario was encountered in previous
study of the kappa organics. O Parcollet G.
Biroli and G. Kotliar PRL, 92, 226402. (2004) .
Spectral shapes. Large Doping
Stanescu and GK cond-matt
0508302
Small Doping. T. Stanescu and GK
cond-matt 0508302
Interpretation in terms of lines of zeros and lines of poles of G T.D.
Stanescu and G.K cond-matt 0508302
Lines of Zeros and Spectral Shapes.
Stanescu and GK cond-matt 0508302
Conclusion
• CDMFT delivers the spectra.
• Path between d-wave and insulator. Dynamical RVB!
• Lines of zeros. Connection with other work. of A. Tsvelik
and collaborators. (Perturbation theory in chains , see
however Biermann et.al). T.Stanescu, fully self
consistent scheme.
• Weak coupling RG (T. M. Rice and collaborators).
Truncation of the Fermi surface.
CDMFT presents it as a strong coupling instability that
begins FAR FROM FERMI SURFACE.
Superconducting State t’=0. How
does the superconductor relate to
the Mott insulator
• Does the Hubbard model superconduct ?
• Is there a superconducting dome ?
• Does the superconductivity scale with J ?
• How does the gap and the order
parameter scale with doping ?
Superconductivity in the Hubbard model role of
the Mott transition and influence of the superexchange. ( M. Capone et.al. V. Kancharla et. al.
CDMFT+ED, 4+ 8 sites t’=0) .
Evolution of DOS with doping U=8t. Capone et.al. :
Superconductivity is driven by transfer of spectral weight ,
slave boson b2 !
Order Parameter and Superconducting Gap do not
always scale! Capone et.al.
Superconducting State t’=0
• Does it superconduct ?
• Yes. Unless there is a competing phase, still
question of high Tc is open. See however Maier
et. al.
•
•
•
•
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 .
Superconductivity is destroyed by transfer
of spectral weight. M. Capone et. al.
Similar to slave bosons d wave RVB .
Notice the particle hole asymmetry
(Anderson and Ong)
Anomalous Self Energy. (from Capone et.al.)
Notice the remarkable increase with decreasing
doping! True superconducting pairing!! U=8t
Significant Difference with Migdal-Eliashberg.
Connection between
superconducting and normal state.
• Origin of the pairing. Study optics!
• K. Haule development of an
ED+DCA+NCA approach to the problem.
• New tool for addressing the neighborhood
of the dome.
RESTRICTED SUM RULES
H hamiltonian, J electric current , P polarization


0
 (w )dw 
Below energy


iV


0
H eff , J eff , Peff
 2 k
- nk 2
k
k
  P, J  
 (w )dw 

iV
 ne2
m
  Peff , J eff  
Low energy sum rule can
have T and doping
dependence . For nearest
neighbor it gives the
kinetic energy.
Treatement needs refinement
• The kinetic energy of the Hubbard model
contains both the kinetic energy of the
holes, and the superexchange energy of
the spins.
• Physically they are very different.
• Experimentally only measures the kinetic
energy of the holes.
Conclusion
• There is still a lot to be understood about
the homogenous problem.
• CDMFT is a significant extension of the
slave boson approach.
• It offers an exceptional opportunity to
advance the field by having a close
interaction of the “theoretical
spectroscopy” and experiments.
The “healing power “ of
superconductivity
• PSEUDOPARTICLES
Optical conductivity t-J . K. Haule
What is the origin of the asymmetry
? Comparison with normal state
near Tc. K. Haule
Early slave boson work, predicted the asymmetry, and some features of the
spectra.
Notice that the superconducting gap is smaller than pseudogap!!
Kristjan Haule: there is an avoided
quantum critical point near optimal
doping.
Optical Conductivity near optimal
doping. [DCA ED+NCA study, K.
Haule and GK]
Behavior of the
optical mass and the
plasma frequency.
Backups
Magnetic Susceptibility
Outline
•Theoretical Point of View, and Methodological Developments. :
•Local vs Global observables.
•Reference Frames. Functionals. Adiabatic Continuity.
•The basic RVB pictures.
•CDMFT as a numerical method, or as a boundary condition.Tests.
•The superconducting state.
•The underdoped region.
•The optimally doped region.
•Materials Design. Chemical Trends. Space of Materials.
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). .
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
usingSelf
Consistency.
Extraction of
lattice quantities.
G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87,
186401 (2001)
Cumulant Periodization: 2X2 cluster
Self energy and Greens function
Periodization .
Comparison of 2 and 4 sites
Also, one would like to be able to evaluate from the theory itself when the
approximation is reliable!! For reviews see: Georges et.al. RMP (1996) Maier et.al
RMP (2005), Kotliar et.al cond-mat 0511085. Kyung et.al cond-mat 0511085
Loesser et.al PRL
Connection with large N studies.
Cluster Extensions of Single Site DMFT
Slatt (k ,w ) S0 (w ) 
S1 (w )(cos kx  cos ky )  S 2 (w )(cos kx.cos ky )  .......
Testing
CDMFT (G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev.
Lett. 87, 186401 (2001) ) with two sites in the Hubbard model in one
dimension V. Kancharla C. Bolech and GK PRB 67, 075110 (2003)][[M.Capone
M.Civelli V Kancharla C.Castellani and GK PR B 69,195105 (2004) ]
U/t=4.
References
o Dynamical Mean Field Theory and a cluster
extension, CDMFT: G.. Kotliar,S. Savrasov, G.
Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401
(2001)
o Cluser Dynamical Mean Field Theories: Causality and
Classical Limit.
G. Biroli O. Parcollet G.Kotliar Phys. Rev. B 69 205908
• Cluster Dynamical Mean Field Theories a Strong
Coupling Perspective. T. Stanescu and G. Kotliar
( 2005)
Evolution of the normal state:
Questions.
• Origin of electron hole asymmetry in
electron and doped cuprates.
• Detection of lines of zeros and the
Luttinger theorem.
ED and QMC
Testing
CDMFT (G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev.
Lett. 87, 186401 (2001) ) with two sites in the Hubbard model in one
dimension V. Kancharla C. Bolech and GK PRB 67, 075110 (2003)][[M.Capone
M.Civelli V Kancharla C.Castellani and GK PR B 69,195105 (2004) ]
U/t=4.
Electron Hole Asymmetry Puzzle
What about the electron doped semiconductors ?
Spectral Function A(k,ω→0)= -1/π G(k, ω →0) vs k
electron doped
Momentum space differentiation
a we approach the Mott
transition is a generic
phenomena.
Location of cold and hot regions
depend on parameters.
P. Armitage et.al. 2001
Civelli et.al. 2004
Approaching the Mott transition:
CDMFT Picture
• Qualitative effect, momentum space
differentiation. Formation of hot –cold regions is
an unavoidable consequence of the approach to
the Mott insulating state!
• D wave gapping of the single particle spectra as
the Mott transition is approached.
• Similar scenario was encountered in previous
study of the kappa organics. O Parcollet G.
Biroli and G. Kotliar PRL, 92, 226402. (2004) .
Antiferro and Supra
Competition of AF and SC
or
SC
AF
SC
AF
AF+SC
d
d
D wave Superconductivity and
Antiferromagnetism t’=0 M. Capone
V. Kancharla (see also VCPT Senechal and
Tremblay ).
Antiferromagnetic (left) and d wave superconductor (right) Order Parameters
Competition of AF and SC
U /t << 8
U f 8t
or
SC
AF
AF
SC
AF+SC
d
d
Conclusion
OPTICS
Differences and connections
between the methods presented.
• Variational approaches T=0, similar to
slave boson mean field. Finite T ?
• QMC small U. Is there a qualitative
difference for large U ?
• Weak coupling RG. Flows to strong
coupling. Combine with CDMFT ?
Superconductivity is destroyed by
transfer of spectral weight. M. Capone
et. al. Similar to slave bosons d wave
RVB.