Optical Conductivity of Cuprates Superconductors: a Dynamical RVB perspective

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Optical Conductivity of CupratesSuperconductors
a Dynamical RVB perspective

• Work with K. Haule (Rutgers)

K. Haule, G. Kotliar, Europhys Lett. 77, 27007
(2007).
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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
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J. Rozenberg, G. Kotliar, H. Kajueter, G. A.
Thomas, D. H. Rapkine, J. M. Honig, and P.
Metcalf, Phys. Rev. Lett. 75, 105 (1995). L.
Baldassarre Poster P1 this conference.
Hubbard model single site DMFT. W(T) is T
dependent near Mott trans.
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Point of view

• Study simple unrealistic models of the doped
  Mott insulator (RVB)
• Capture local physics. Reference frame is a
  plaquette in a medium.
• No vertex corrections yet! (but work in
  progress).

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Optics and RESTRICTED SUM RULES
ltTgtn is only defined for Tgt Tc, while ltTgts
exists only for TltTc Experiment use of this
equation implies extrapolation. Theory use
of this equation implies of mean field picture
to continue the normal state below Tc.
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Optical Spectral Weight Can be Used to infer the
mechanism of superconductivity.
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Kinetic energy upon condensation
underdoped
overdoped
electrons gain energy due to exchange
energy holes gain kinetic energy (move faster)
electrons gain energy due to exchange energy hole
loose kinetic energy (move slower)
BCS like
same as RVB (see P.W. Anderson Physica C, 341, 9
(2000), or slave boson mean field (P. Lee,
Physica C, 317, 194 (1999)
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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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Optical Conductivity
Hubbard model
U
Excitations into upper Hubbard band

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

Drude
t-J model
J-t
no-U
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E Energy difference between the normal and
superconducing state of the t-J model. K. Haule
GK
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. 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) .
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Superexchange Mechanism

Slave Boson Mean Field Theory Phase Diagram.
Formation of Singlets
Coherent Quasiparticles
Re
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CDMFT optics t-J model
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• At very low doping, one can separate two
  components. Coherent and Incoherent
• At large they merge into one Drude-like broad
  frequency range.
• Expected temperature dependence in overdoped
  region. Narrowing of Drude peak. Anomalous
  temperature dependence
• at low doping.

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Cuttoff and temperature dependence of integrated
optial spectral weight
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At which frequency do we recover all the spectral
weight ?
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• Optical weight increases as temperature
  decreases.
• The magnitude is approximately given by single
  site DMFT as first computed by Toschi et.al,
  PRL (2005). .
• Substantial new physics is brought by the cluster
  effects. Existence of d wave superconductivity
  and pseudogap. Avoided criticality.
• Notice that in spite of the opening of a
  pseudogap. The spectral weight does not decrease
  with decreasing temperature for reasonable
  cuttoffs.!!!

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• At very high frequencies. Of the order of 3t. (t,
  .3-.45 ev)
• It is due to the anomalous greens
• function. Not visible in photoemission.

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Optical Mass at low doping
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Optical mass and plasma frequency
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Padilla et.al.
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Conclusion

• Optical anomalies, do NOT rule out
• the proximity to a Mott transition as a basis
  for a theoretical approach to describe the
  cuprates.
• a) temperature and doping dependence of the
  optical spectral weight.
• CDMFT on a plaquette, is a substantial
  improvement over the earlier slave boson
  approach, to describe the optics, and many other
  key experiments. My talk on Wendesday.
• Further work to improve a) our understanding
  of the plaquette CDMFT equations, b) to make
  the models more realistics c) to make CDMFT more
  flexible and d) to incorporate vertex
  corrections are warranted.

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Power laws in optics. A. El Azrak,et.al. PR B
49, 9846 (1994).D. van der Marel, Nature 425,
271 (2003).
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Avoided Quantum Criticality

• Intermediate physics phenomena.
• No analytic understanding of the dimension 2/3.

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Stephan and Horsch Int. Jour Mod Phys B6, 141
(1992) Eskes Oles Meinders and Stephan PRB 50
(1994) 17980
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RVB phase diagram of the Cuprate Superconductors.
Superexchange.
G. Kotliar and J. Liu Phys.Rev. B 38,5412 (1988)

• 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.

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RVB phase diagram of the 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.

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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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Finite T, DMFT and the Energy Landscape of
Correlated Materials
T
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Impurity Model-----Lattice Model
D , Weiss Field
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Conclusion

• More quantitative comparison with
• experiments
• On the theory side. Investigate effects of t
  t and more realistic electronic structure.
• Effects of vertex corrections, periodization.
• More extreme underdoping and overdoping.
  Better impurity solvers.

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RVB phase diagram of the Cuprate Superconductors.
Superexchange.

• Proximity to Mott insulator renormalizes the
  kinetic energy Trvb increases.
• Proximity to the Mott insulator reduce the
  charge stiffness, and QPcoherence scale . T BE
  goes to zero.
• Superconducting dome. Pseudogap evolves
  continuously 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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Hubbard vs t-J
Transition from uper to lower Hubbard band at U
Drude
Incoherent part of the spectra
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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.
41
Optical Spectral Weight Can be Used to infer the
mechanism of superconductivity.
42
RESTRICTED SUM RULES
Below energy
Low energy sum rule can have T and doping
dependence . For nearest neighbor it gives the
kinetic energy.
43
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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RVB phase diagram of the Cuprate Superconductors.
Superexchange.

• Proximity to Mott insulator renormalizes the
  kinetic energy Trvb increases.
• Proximity to the Mott insulator reduce the
  charge stiffness, and QPcoherence scale . T BE
  goes to zero.
• Superconducting dome. Pseudogap evolves
  continuously 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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For reviews of cluster methods see Georges
et.al. RMP (1996) Maier et.al RMP (2005), Kotliar
et.al RMP (2006)
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.
Impurity solver, NCA, ED, CTQMC
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Superexchange mechanism?
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• Near the Mott transition the optical weight has a
  surprising large T dependence.
• M. J. Rozenberg et al., Phys. Rev. Lett. 75,
  105 (1995).
• This phenomena of buildup of spectral weight with
  reducing temperature was found in cuprates, and
  was well accounted
• by single site DMFT. Toschi et. al. Phys. Rev.
  Lett. 95, 097002 (2005)

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