Superconductors near quantum phase transitions BCSBEC crossovers and Fermi arcs

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Superconductors near quantum phase transitions
BCS-BEC crossovers and Fermi arcs
Arun Paramekanti (Toronto)
Collaborator Erhai Zhao (Toronto)
References Erhai Zhao and A. P., 0606470
(PRL, Dec 2006) A. P. and
Erhai Zhao, 0611762 (submitted to PRL)
Discussions H. Moritz (ETH), J. Thywissen
(Toronto), Y.B. Kim (Toronto)
A. Vishwanath (Berkeley), J.C. Davis
(Cornell)
Funding
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Superconductors near quantum phase transitions

• Cold atoms in an optical lattice The
  BCS-BEC crossover in weakly doped semimetals

• High temperature superconductors Why does
  the Fermi surface break down in the normal state?

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Ultracold fermionic atoms in an optical lattice
ETH experiments on 40K (Group of T. Esslinger)

• Fermions in a periodic potential
• Tunable density Fermi surface data
• Tunable interactions Feshbach resonance

• M. Kohl, et al, PRL 94, 080403 (2005)

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Ultracold fermionic atoms in an optical lattice
MIT experiments on 6Li (Group of W. Ketterle)

• Superfluid loaded in opt. lattice
• Destruction of SF with increasing lattice depth
  Mott insulator?

• J. K. Chin, et al (Nature 2006, to appear)

T
TC
Could be sweeping through a thermal phase
transition?
Superfluid
Normal
Vlattice
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Ultracold fermionic atoms in an optical lattice

• Can such experiments access the BCS-BEC
  crossover in a lattice?
• Can they be pushed to realize fermionic quantum
  phase transitions?

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Ultracold fermionic atoms in an optical lattice

• R. Diener, T.-L. Ho, PRL 96, 010402 (2006)
  Fermions interacting in a single well

• Band 3

• Band 2

• Band 1

• Interactions could cause excitations into higher
  oscillator levels
• Appears to explain the ETH observations of
  higher band populations
• Many band problem Multiband superfluidity at
  T0??

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Ultracold fermionic atoms in an optical lattice

• Hard to parametrize and study a general multiband
  Hamiltonian

• Try to consider a simpler one band model with two
  sites per unit cell to get two bands

• 2D honeycomb lattice forms a semimetal for
  1-electron per site
• Graphene lattice, carbon nanotubes
• Recent realization of anomalous quantized Hall
  effect from Dirac excitation spectrum
• K. S. Novoselev, et al, Nature 438, 197
  (2005) Y. Zhang, et al, Nature 438, 201 (2005)

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Ultracold fermionic atoms in an optical lattice

• Motivated by obtaining a superfluid let us
  focus on attractive interactions
• Attractive Hubbard model

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Ultracold fermionic atoms in an optical lattice
Superconductor
QCP
Semimetal

• At 1 electron per site
• Vanishing DOS at zero energy
• No SC for infinitesimal attraction

Superconductor

• For nonzero electron/hole doping
• Tiny Fermi surface
• SC for infinitesimal attraction

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Ultracold fermionic atoms in an optical lattice
What is the BCS-BEC crossover problem?

• Strong attraction
• - Large pairing gap
• - Beyond Tc, remnants of the gap D remain
• - Superfluid density vanishes at Tc
• - BCS theory does not work

• Weak attraction between electrons
• - Small superconducting gap
• - Upon heating to Tc, the gap D -gt 0
• - BCS theory works

• For strong attractive interactions,
  superfluidity is destroyed by phase fluctuations
  and vanishing of the superfluid stiffness

• The BCS-BEC crossover is unrelated to any phase
  transition

D. M. Eagles (1969) A. J. Leggett (1980) K.
Miyake (1983) Nozieres, Schmitt-Rink (1985)
C. A. R. Sa de Melo, M. Randeria and J. R.
Engelbrecht, PRL 71, 3202 (1993)
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Ultracold fermionic atoms in an optical lattice

• A mean field criterion for BCS-BEC crossover
• When does the SC gap become equal to the Fermi
  energy?

D EF VF (dn)1/2
BCS
BEC
VF(dn)1/2
BCS

• These BCS-BEC crossover lines connect with the
  quantum critical point

• The low density crossover at finite T will lie
  in the quantum critical regime

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Ultracold fermionic atoms in an optical lattice

• Behavior of the mean field chemical potential

BCS
BEC
BCS

• The chemical potential shows nonmonotonic
  behavior

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Ultracold fermionic atoms in an optical lattice
Collective fluctuations in the SC

• Amplitude fluctuations Have a gap 2D
• Acoustic phonons (Goldstone mode) and Optical
  phonons (Leggett mode)

A2,F2
A1,F1

• n1n2 lt-gt F1F2 Goldstone mode

• n1-n2 lt-gt F1-F2 Leggett mode

• Can get rs and estimate Tc min(D,rs)

• Use oscillatory (V1-V2) to detect the Leggett
  mode

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Ultracold fermionic atoms in an optical lattice
Collective fluctuations in the semimetal

• SC fluctuations will decay into two Dirac
  quasiparticles
• At small q Decay rate w (critical)
• At large q Can be nearly undamped
• Very unlike SC fluctuations in a FL

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Summary Ultracold fermionic atoms in an optical
lattice

• Experiments with fermionic atoms on the 2D
  honeycomb lattice can
• Access an interesting and nontrivial quantum
  phase transition
• The BCS-BEC crossover lies in the vicinity of
  this transition
• Probe interesting collective modes not found in
  usual SCs and FLs

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Phase Diagram of the high Tc superconductors

• Coupled CuO2 layers
• Doping the AFM insulator -gt SC
• Nonmonotonic Tc versus doping
• Maximum Tc 50-150 K
• Competing phases at low doping

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Phase Diagram of the high Tc superconductors
Repulsive Hubbard model
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Phase Diagram of the high Tc superconductors
Repulsive Hubbard model
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Phase Diagram of the high Tc superconductors
Repulsive Hubbard model
J t2/U
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Phase Diagram of the high Tc superconductors
Repulsive Hubbard model
A variational approach

• Start with a d-wave the BCS wavefunction

• Suppress configurations which contain doubly
  occupied sites

P.W. Anderson (Science, 1987) C. Gros (PRB, 1988)

• Compute interesting observables
• Momentum distribution
• Quasiparticle spectral weight
• Quasiparticle velocities

A.P., M. Randeria, N. Trivedi (PRL 2001, PRB
2004)
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Phase Diagram of the high Tc superconductors
A variational approach
A.P., M. Randeria, N. Trivedi (PRL 2001, PRB
2004)
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Phase Diagram of the high Tc superconductors
A variational approach
P. Bogdanov et al (PRL 2002)
ARPES P. D. Johnson group (BNL) Z.X. Shen group
(Stanford)
A.P., M. Randeria, N. Trivedi (PRL 2001, PRB
2005)
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High temperature superconductivity from
spin-charge separation?
Notion of spin charge separation
If Y is condensed, recover (roughly) variational
results for the SC state
G. Kotliar, J Liu (PRB 1988)
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Normal state of the high Tc superconductors
M. R. Norman, et al, Nature (1998)

• Low energy electrons live on open Fermi arcs
  rather than a closed Fermi surface! Non-Fermi
  liquid physics.

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High temperature superconductors

• Why does the normal state of the underdoped high
  Tc SCs appear anomalous?
• Why is there no closed Fermi surface in the
  normal state?
• In there an underlying Normal State?

• A. Kanigel, et al, Nature Phys.( 2006)

Exotic nodal metal ground state??
A nodal metal at a generic hole density would
violate Luttingers theorem has to be a non
Fermi liquid!
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Could the normal state be quantum critical??
A.P., E. Zhao (cond-mat/0611762)
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High temperature superconductors
Notion of spin charge separation
What if charge fluctuations are critical??
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High temperature superconductors
What if charge fluctuations are critical??
Combine a thermally excited spinon (chargon)
with a chargon (spinon) excited from the vacuum
Get a zero energy electron
A.P., E. Zhao (cond-mat/0611762)
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High temperature superconductors
Combine a thermally excited spinon (chargon)
with a chargon (spinon) excited from the vacuum
Get a zero energy electron

• A. Kanigel, et al, Nature Phys.( 2006)

Probably quantum critical regime!
A.P., E. Zhao (cond-mat/0611762)
Exotic nodal metal ground state??
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Summary High temperature superconductors

• Experiments in underdoped high Tc superconductors
  show non-Fermi
• liquid physics
• Breakup of the Fermi surface into open Fermi
  arcs
• The finite temperature UD regime could be near a
  SC-insulator transition
• Charge fluctuations in the quantum critical
  regime gt Fermi arcs and non Fermi liquid physics?