Superconductivity%20near%20the%20Mott%20transition:%20what%20can%20we%20learn%20from%20plaquette%20DMFT?

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Superconductivity near the Mott transition what
can we learn from plaquette DMFT?

• K Haule
• Rutgers University

2
References and Collaborators

• Strongly Correlated Superconductivity a
  plaquette Dynamical mean field theory study,
• K. H. and G. Kotliar, cond-mat/0709.0019 (37
  pages and 42 figures)
• Nodal/Antinodal Dichotomy and the Energy-Gaps of
  a doped Mott Insulator,
• M. Civelli, M. Capone, A. Georges, K. H., O.
  Parcollet, T. D. Stanescu, G. Kotliar,
  cond-mat/0704.1486.
• Quantum Monte Carlo Impurity Solver for Cluster
  DMFT and Electronic Structure Calculations in
  Adjustable Base,
• K. H., Phys. Rev. B 75, 155113 (2007).
• Optical conductivity and kinetic energy of the
  superconducting state a cluster dynamical mean
  field study,
• K. H., and G. Kotliar, Europhys Lett. 77, 27007
  (2007).
• Doping dependence of the redistribution of
  optical spectral weight in Bi2Sr2CaCu2O8delta
• F. Carbone, A. B. Kuzmenko, H. J. A. Molegraaf,
  E. van Heumen, V. Lukovac, F. Marsiglio, D. van
  der Marel, K. H., G. Kotliar, H. Berger, S.
  Courjault, P. H. Kes, and M. Li, Phys. Rev. B 74,
  064510 (2006).
• Avoided Quantum Criticality near Optimally Doped
  High Temperature Superconductors,
• K.H. and G. Kotliar, cond-mat/0605149

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Approach

• Understand the physics resulting from the
  proximity to a Mott insulator in the context of
  the simplest models.

• Construct mean-field type of theory and follow
  different states as a function of parameters
  superconducting normal state.
• Second step compare free energies which will
  depend more on the detailed modeling and long
  range terms in Hamiltonian..

• Approach the problem from high temperatures where
  
• physics is more local. Address issues of finite
  frequency and finite temperature crossovers.

• Leave out disorder, electronic structure, phonons
  
• CDMFTLDA second step, under way

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Cluster DMFT approach
Exact Baym Kadanoff functional of two variables
GS,G. Restriction 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

• Impurity solvers
• ED
• NCA
• Continuous time QMC

FGplaquette
periodization
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Momentum versus real space
In plaquette CDMFT cluster quantities are
diagonal matrices in cluster momentum base
In analogy with multiorbital Hubbard model exist
well defined orbitals
But the inter-orbital Coulomb repulsion is
nontrivial and tight-binding Hamiltonian in this
base is off-diagonal
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S(iw) with CTQMC
next nearest neighbor important in underdoped
regime
on-site largest
nearest neighbor smaller
Hubbard model, T0.005t
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Momentum space differentiation

• (0,0) orbital reasonable coherent Fermi liquid

• (p,0) very incoherent around optimal doping
• (d20.16 for t-J and d20.1 for Hubbard U12t)

• (p,p) most incoherent and diverging at another
  doping
• (d10.1 for t-J and d10 for Hubbard U12t)

t-J model, T0.01t
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Momentum space differentiation
gets replaced by coherent SC state
with large anomalous self-energy
t-J model, T0.005t
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Fermi surface
d0.09
Cumulant is short in ranged
Arcs FS in underdoped regime pocketslines of
zeros of G arcs
Single site DMFT PD
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Nodal quasiparticles
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Nodal quasiparticles
Vnod almost constant up to 20
the slopevnod almost constant
vD dome like shape
Superconducting gap tracks Tc!
M. Civelli, cond-mat 0704.1486
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Antinodal gap two gaps
M. Civelli, using ED, cond-mat 0704.1486
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Superfluid density at low T
Low T expansion using imaginary axis QMC data.
Current vertex corrections are neglected
In RVB the coefficient bd2 at low d WenLee,
IoffeMillis
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Superfluid density close to Tc
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Anomalous self-energy and order parameter

• Anomalous self-energy
• Monotonically decreasing with iw
• Non-monotonic function of doping
• (largest at optimal doping)
• Of the order of t at optimal doping
• at T0,w0

Order parameter has a dome like shape and is
small (of the order of 2Tc)
Hubbard model, CTQMC
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Anomalous self-energy on real axis

• Many scales can be identified
• J,t,W

• It does not change sign at certain
• frequency wD-gtattractive for any w

• Although it is peaked around J, it
• remains large even for wgtW

Computed by the NCA for the t-J model
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SC Tunneling DOS
Large asymmetry at low doping
Gap decreases with doping DOS becomes more
symmetric
Normal state has a pseudogap with the same
asymmetry
SC d0.08
SC d0.20
NM d0.08
NM d0.20
Approximate PH symmetry at optimal doping
also B. Kyung et.al, PRB 73, 165114 2006
Computed by the NCA for the t-J model
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Optical conductivity

• Low doping two components
• Drude peak MIR peak at 2J

• For xgt0.12 the two components merge

• In SC state, the partial gap opens causes
• redistribution of spectral weight up to 1eV

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Optical spectral weight - Hubbard versus t-J model
f-sumrule
Excitations into upper Hubbard band
Drude
t-J model
J
no-U
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Optical spectral weight Optical mass
mass does not diverge approaches 1/J
Bi2212
F. Carbone,et.al, PRB 74,64510 (2006)
Weight increases because the arcs increase and Zn
increases (more nodal quasiparticles)
Basov et.al., PRB 72,60511R (2005)
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Temperature/doping dependence of the optical
spectral weight
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Optical weight, plasma frequency
Weight bigger in SC, K decreases (non-BCS)
Weight smaller in SC, K increases (BCS-like)
A.F. Santander-Syro et.al, Phys. Rev. B 70,
134504 (2004)
F. Carbone,et.al, PRB 74,64510 (2006)
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Kinetic energy change
Kinetic energy increases
cluster-DMFT, Eu. Lett. 77, 27007 (2007).
Kinetic energy decreases
Phys Rev. B 72, 092504 (2005)
Kinetic energy increases
Exchange energy decreases and gives largest
contribution to condensation energy
same as RVB (see P.W. Anderson Physica C, 341, 9
(2000)
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Origin of the condensation energy

• Resonance at 0.16t5Tc (most pronounced at
  optimal doping)
• Second peak 0.38t120meV (at opt.d)
  substantially contributes to condensation energy

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Conclusions

• Plaquette DMFT provides a simple mean field
  picture of the
• underdoped, optimally doped and overdoped regime
• One can consider mean field phases and track them
  even in the
• region where they are not stable (normal state
  below Tc)
• Many similarities with high-Tcs can be found in
  the plaquette DMFT
• Strong momentum space differentiation with
  appearance of arcs in UR
• Superconducting gap tracks Tc while the PG
  increases with underdoping
• Nodal fermi velocity is almost constant
• Superfluid density linear temperature coefficient
  approaches constant at low doping
• Superfuild density close to Tc is linear in
  temperature
• Tunneling DOS is very asymmetric in UR and
  becomes more symmetric at ODR
• Optical conductivity shows a two component
  behavior at low doping
• Optical mass 1/J at low doping and optical weigh
  increases linearly with d
• In the underdoped system -gt kinetic energy saving
  mechanism
• overdoped system -gt kinetic energy loss
  mechanism
• exchange energy is always optimized in SC
  state

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Issues

• The mean field phase diagram and finite
• temperature crossover between underdoped and
  overdoped regime
• Study only plaquette (2x2) cluster DMFT in the
• strong coupling limit (at large U12t)
• Can not conclude if SC phase is stable in the
  exact solution of the model. If the mean field
  SC phase
• is not stable, other interacting term in H could
  
• stabilize the mean-field phase (long range U, J)

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Doping dependence of the spectral weight
Comparison between CDMFTBi2212
F. Carbone,et.al, PRB 74,64510 (2006)
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RVB phase diagram of the t-J m.

• Problems with the RVB slave bosons
• Mean field is too uniform on the Fermi surface,
  in contradiction with ARPES.
• Fails to describe the incoherent finite
• temperature regime and pseudogap regime.
• Temperature dependence of the penetration
• depth.
• Theory rTx-Ta x2 , Exp rT x-T a.
• Can not describe two distinctive gaps
• normal state pseudogap and superconducting gap

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Similarity with experiments
On qualitative level consistent with
de Haas van Alphen small Fermi surface
Louis Taillefer, Nature 447, 565 (2007).
Shrinking arcs
A. Kanigel et.al., Nature Physics 2, 447 (2006)
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Fermi surface
d0.09
Cumulant is short in ranged
Arcs FS in underdoped regime pocketslines of
zeros of G arcs
Arcs shrink with T!
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Insights into superconducting state
(BCS/non-BCS)?
BCS upon pairing potential energy of electrons
decreases, kinetic energy increases (cooper pairs
propagate slower) Condensation energy is the
difference
non-BCS kinetic energy decreases upon
pairing (holes propagate easier in superconductor)
J. E. Hirsch, Science, 295, 5563 (2002)
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Origin of the condensation energy

• Resonance at 0.16t5Tc (most pronounced at
  optimal doping)
• Second peak 0.38t120meV (at opt.d)
  substantially contributes to condensation energy

local susceptibility
YBa2Cu3O6.6 (Tc62.7K)
Pengcheng et.al., Science 284, (1999)
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Similarity with experiments
On qualitative level consistent with
de Haas van Alphen small Fermi surface
Louis Taillefer, Nature 447, 565 (2007).