Unconventional Order Of Ultracold Fermions With Several Flavors

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Unconventional Order Of Ultracold Fermions With
Several Flavors

• Carsten Honerkamp (MPI Stuttgart)
• Collaboration with Walter Hofstetter
  (MIT/Aachen)
• PRL 2004 PRB 2004
• Cold fermions in traps, optical lattices
• SU(N) Hubbard model
• Repulsive model at half filling
• Attractive case, superfluidity

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Ultracold atoms

• Combination of laser cooling and evaporation
  cooling (?ultracold) techniques
• 1995 Demonstration of Bose-Einstein
  condensation in dilute bosonic gases in magnetic
  traps (Nobel prize 2001 for Cornell, Wiemann
  (Boulder), Ketterle (MIT))

Boulder group, Science 1995
Since then rapid development in many directions
...
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Cold Fermions
Degenerate Fermi gases (MIT group T0.05 TF with
3?107 6Li atoms, cooled with 23Na)
Hadzibabic et al., PRL 2003
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Cooper pairs?
Regal, Jin Greiner
Scattering length gets large and changes sign at
Feshbach resonances
Condensate fraction
Feshbach resonance for 40K
mF-7/2,-9/2
BCS side alt0
BEC side agt0
Feshbach resonance above Fermi level, alt0
Molecules, agt0

• Fast B-field sweep from BCS (scattering length
  alt0) to BEC (agt0) side
• Too fast for molecular condensate formation when
  Binitial above Feshbach resonance at Bres
• At low T more P0 molecules when Binitial near
  Bres
• Cooper pairs on BCS side get projected on P0
  molecules?

Zwierlein et al.
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Optical lattices
Jaksch et al. 1998
Load atomic gas into optical lattice made with
standing light waves ? bosonic/fermionic Hubbard
model
Greiner et al., 2002 bosonic 87Ru, wavelength
852nm, 653 sites Mott-insulator/superfluid
transition observed Köhl et al. (ETH), 2004
Hubbard model with fermionic 40K.
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More than spin up spin down
Cold atoms can have more than 2 internal degrees
of freedom hyperfine spin 6Li F3/2,
attractive scattering length between upper
hyperfine states 40K F9/2, tunable interactions
Low field seekers, same as lt 0

• New many body states?
• consider models with N fermion flavors
• lattice models may be most interesting

Hadzibabic et al., PRL 2004
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SU(N) Hubbard model

• Idealized system
• N flavors of fermions on lattice and
• equal nearest neighbor hopping for all flavors m
• local density-density interaction between
  different flavors
• ? SU(N) Hubbard model

invariant w.r.t. global SU(N) rotations among
m1...N fermion flavors (fundamental
representation) Different from large-spin SU(2)
(e.g. Ho Yip)
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Theoretical approach

• Consider
• 2D square lattice, bandwidth 8t
• weak to moderate coupling, Ult 4t
• repulsive (Ugt0) and attractive (Ult0) case
• Use N-patch functional renormalization group
  method (temperature-flow RG, HonerkampSalmhofer
  2001) in one-loop approximation
• dominant ordering tendencies from
  susceptibilities
• mean fieldRPA analysis for phase suggested by
  RG
• observable (?) collective modes

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Finding Fermi Liquid Instabilities Temperature
Flow
HonerkampSalmhofer 2001

• RG-like strategy Consider flow of interactions
  when temperature is lowered
• Can be derived as approximation to exact
  functional RG equation for T-dependence of
  generating functional
• Avoids some shortcomings of momentum-shell RG in
  many-fermion systems

• General structure

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Three choices for the flow parameter
Fermionic action RG-scheme for 1PI vertex
functions (Wetterich 1993, Salmhofer 2001)
delivers flow for arbitrary parameter in Q
Q, quadratic part contains flow parameter

• Cutoff RG Q ? QL with cutoff function C(e/L)
• Temperature-Flow T as flow parameter (CH
  Salmhofer 2001)
• Interaction flow coupling strength grows (CH et
  al. 2004, Meden)
• ? different perspectives on Fermi surface
  instabilities

 
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Implementation for 2D Hubbard model N-patch
scheme
ZanchiSchulz 1997

• Coupling function V(k1, k2, k3 ) with incoming
  k1,k2 and outgoing k3
• Discretize take V(k1, k2, k3 ) constant for
  k1,k2 and k3 in same patch.
• Neglect frequency dependence

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T-flow RG for half-filled Hubbard model N2
nesting processes
N2 SU(2)-breaking strongest instability Interpr
etation AF spin density wave ground state
AF spin density wave
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T-flow RG for the half filled band N3
N3 SU(3)-breaking wins generalized
antiferromagnet with unit cell doubling
flavor density wave acc. by small CDW (meanfield
theory)
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RG for the half filled case Ngt6
Ngt6 strongest flow in in d-wave-charge-channel
V2c Staggered flux (d-density wave) instability!
d-density wave, staggered flux
Marston et al. 1988, 1991 Chakravarty et al.
2001 Maki et al. 2004
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SU(N) Hubbard-Heisenberg model Large N
Marston and Affleck 1988 Large-N Hubbard
Heisenberg model Saddle point theory
becomes exact for N??, corrections smaller by
O(1/N) For small J and half-filling staggered
flux phase
(cp. CappellutiZeyher 1999)
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Half-Filled SU(4) Hubbard-Heisenberg Model
N4

• Assaad 2004 (T0 QMC) SU(4) model with U0
• JltJc DDW
• JltJc no long range order, gapless spin liquid
  (with gap for fermions)
• Perturbative RG for Jgt0
• U0 DDW state for N gt 2
• SU(4) Transition from SDW to DDW as function of
  U
• Details of the transition? Fate of LRO?

?
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Attractive model BCS pairing for N3
Decouple interaction in s-wave even parity
Cooper-channel with onsite order parameter Dab
Even parity order parameter has 3 components
D12, D13, D23
What happens for 3 flavors? Do only 2 flavors
form condensate, or all 3 ?
Similar model ModawiLeggett 1997
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BCS pairing for SU(3)
Order parameter transforms non-trivially wrt
SU(3) no singlet! (cf. color-QCD pions
qq) take SU(3)-transf. U (3D fundamental
repres. of SU(3))
Decompose product of 3dim representations

• even parity order parameter transforms acc. to
  3D representation
• large ground state degeneracy
• can always rotate onto (1,0,0), i.e. D12D0,
  D13D230 !

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Superfluid state with large Fermi surface

• flavors 1 and 2 have gap, flavor 3 is gapless
• coexistence of pairing with large Fermi surface

Mean field solutions for the ground state with
N3 degeneracy of gap functions with fixed
5 Goldstone modes! SU(2) ? U(1) unbroken
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Signatures of the N3 paired state?

• RPA analysis of collective modes (Bragg
  scattering measures density response)
• T gt Tc no zero sound (Ult0)
• T lt Tc Anderson-Bogoliubov phase mode of paired
  flavors shows up in density response ? signature
  of pairing
• Signatures of gapless 3rd flavor
• Damping of phase mode for all frequencies
• Additional flavor mode above particle-hole
  continuum

Ketterle et al.
Dynamical structure factor S(q,w)
phase
Phase mode
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Conclusions

• Cold femions with several (gt2) flavors on optical
  lattices may give rise to interesting phenomena
  that do not occur easily in conventional solid
  state systems
• Flavor density waves
• Staggered flux phases
• Incompressibility and commensurability away from
  half-filling
• Superfluid states with large Fermi surface and
  additional collective modes

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Attractive 2D Hubbard model SU(2)

• At half-filling long range charge density wave
  and s-wave superconducting correlations in ground
  state, Tc0.
• Power-law superconductor away from half filling
  below Tcgt0.

Scalettar et al. 1989
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N3 away from half filling ultracold cannibals
Hartree-Fock with flavor-density meanfields
?nm? near half filling 2 flavors remain
half-filled, 3rd flavor gets depleted ? parts of
system commensurate and incompressible away from
half-filling
cp. 3-leg ladder, 2D Hubbard model?