Cellular DMFT studies of the doped Mott insulator

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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 (RutgersSherbrook) A. M Tremblay, D.
Senechal B. Kyung (Sherbrooke) Discussions A.
Georges, N. Bontemps, A. Sacuto.
Support NSF DMR . Blaise Pascal Chair
Fondation de lEcole Normale.
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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.

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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. ltbgt
  coherence order parameter. k, D singlet formation
  order parameters.Baskaran Zhou Anderson ,
  Ruckenstein et.al (1987) .

Other states flux phase or sid ( G. Kotliar
(1988) Affleck and Marston (1988) have point
zeors.
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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 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)
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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 .
  TheoryrTx-Ta x2 , Exp rT 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.!!
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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.
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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

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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!

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.
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 .
• 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.

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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/t4.
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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
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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) .

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Spectral shapes. Large Doping Stanescu and GK
cond-matt 0508302
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Small Doping. T. Stanescu and GK cond-matt 0508302
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Interpretation in terms of lines of zeros and
lines of poles of G T.D. Stanescu and G.K
cond-matt 0508302
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Lines of Zeros and Spectral Shapes. Stanescu and
GK cond-matt 0508302
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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.

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Superconducting State t0. 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 ?

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Superconductivity in the Hubbard model role of
the Mott transition and influence of the
super-exchange. ( M. Capone et.al. V. Kancharla
et. al. CDMFTED, 4 8 sites t0) .
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Evolution of DOS with doping U8t. Capone et.al.
Superconductivity is driven by transfer of
spectral weight , slave boson b2 !
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Order Parameter and Superconducting Gap do not
always scale! Capone et.al.
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Superconducting State t0

• 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 .

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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)
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Anomalous Self Energy. (from Capone et.al.)
Notice the remarkable increase with decreasing
doping! True superconducting pairing!! U8t
Significant Difference with Migdal-Eliashberg.
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Connection between superconducting and normal
state.

• Origin of the pairing. Study optics!
• K. Haule development of an EDDCANCA approach
  to the problem.
• New tool for addressing the neighborhood
• of the dome.

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RESTRICTED SUM RULES
Below energy
Low energy sum rule can have T and doping
dependence . For nearest neighbor it gives the
kinetic energy.
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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.

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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.

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The healing power of superconductivity
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• PSEUDOPARTICLES

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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!!
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Kristjan Haule there is an avoided quantum
critical point near optimal doping.
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Optical Conductivity near optimal doping. DCA
EDNCA study, K. Haule and GK
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Behavior of the optical mass and the plasma
frequency.
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Magnetic Susceptibility
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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.

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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). .
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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)
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Cumulant Periodization 2X2 cluster
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Self energy and Greens function Periodization .
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Comparison of 2 and 4 sites
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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
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Loesser et.al PRL
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Connection with large N studies.
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Cluster Extensions of Single Site DMFT
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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/t4.
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References

• Dynamical Mean Field Theory and a cluster
  extension, CDMFT G.. Kotliar,S. Savrasov, G.
  Palsson and G. Biroli, Phys. Rev. Lett. 87,
  186401 (2001)
• 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)

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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.

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ED and QMC
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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/t4.
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Electron Hole Asymmetry Puzzle
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What about the electron doped semiconductors ?
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Spectral Function A(k,??0) -1/p G(k, ? ?0) vs
k
electron doped
P. Armitage et.al. 2001
Momentum space differentiation a we approach the
Mott transition is a generic phenomena.
Location of cold and hot regions depend on
parameters.
Civelli et.al. 2004
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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) .

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Antiferro and Supra
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Competition of AF and SC
or
SC
AF
SC
AF
AFSC
d
d
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D wave Superconductivity and Antiferromagnetism
t0 M. Capone V. Kancharla (see also VCPT
Senechal and Tremblay ).
Antiferromagnetic (left) and d wave
superconductor (right) Order Parameters
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Competition of AF and SC
U /t ltlt 8
or
SC
AF
AF
SC
AFSC
d
d
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Conclusion
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OPTICS
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Differences and connections between the methods
presented.

• Variational approaches T0, 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 ?

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Superconductivity is destroyed by transfer of
spectral weight. M. Capone et. al. Similar to
slave bosons d wave RVB.