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Quartetting and pairing instabilities in 1D spin 3/2
fermionic systems
Congjun Wu
Kavli Institute for Theoretical Physics, UCSB
Ref: C. Wu, Phys. Rev. Lett. 95, 266404 (2005).
Many thanks to S. C. Zhang, E. Demler, Y. P. Wang, A. J. Leggett for
helpful discussions.
March meeting, 03/16/2006 (10:24)
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Multiple-particle clustering (MPC) instability
• Feshbach resonance: Cooper pairing superfluidity.
• Beyond Cooper pairing: In fermionic systems with multiple
components, Pauli’s exclusion principle allows MPC.
• More two particles form bound states.
baryon (3-quark); alpha particle (2p+2n); bi-exciton (2e+2h)
• Driven by logic, it is natural to expect the MPC as a possible
focus for the future research.
• Spin-3/2 fermions have 4-componets.
132Cs, 9Be, 135Ba, 137Ba, 201Hg.
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Quartetting order in spin 3/2 systems
• 4-fermion counterpart of Cooper pairing.
SU(4) singlet:
k1
k2
4-body maximally
entangled states

Oqt   3/ 2 (r ) 1/ 2 (r ) 1/ 2 (r ) 3/ 2 (r )


k2
k1

• Difficulty: lack of a BCS type well-controlled mean field theory.
trial wavefunction in 3D SU(4) symmetric model:
A. S. Stepanenko and J. M. F
Gunn, cond-mat/9901317.
• Quartetting v.s singlet pairing in the 1D spin 3/2 systems with
the general s-wave scattering interactions.
C. Wu, Phys. Rev. Lett. 95, 266404 (2005).
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Generic spin 3/2 Hamiltonian in the continuum model
• The s-wave scattering interactions and spin SU(2) symmetry.
2


2
H   dx   ( x)(
 x   )  ( x)
2m
  3 / 2 , 1 / 2

g0 
g
 ( x) ( x)  2
2
2


 a ( x)  a ( x)
a 1~ 5
3
2

1
2
1
2

3
2
• Pauli’s exclusion principle: only Ftot=0, 2
are allowed; Ftot=1, 3 are forbidden.
3 3
singlet:   ( x)   00 | 2 2 ;    ( x)  ( x)

3 3



quintet:  a ( x)    2a | 2 2 ;    ( x)  ( x)

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Phase diagram: bosonization+RG
g2
SU(4)
g0  g2
C: Singlet pairing
A: Luttinger liquid
g0
B: Quartetting
SU(4) g 0
 g2
• Gapless charge sector.
• Spin gap phases B and C:
pairing v.s.quartetting.
• Ising transition between
B and C.
• Singlet pairing in purely
repulsive regime.
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Phase B: the quartetting phase
• Quartetting superfluidity v.s. CDW of quartets (2kf-CDW).
Oqt   3/ 2 1/ 2 1/ 2 3 / 2  e 2i
N 2 k f   R L  ei
 c
 c
wins at K c  2;
wins at K c  2.
Kc: the Luttinger parameter in the charge channel.
d  2 /( 2k f )
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Phase C: the singlet pairing phase
• Singlet pairing superfluidity v.s CDW of pairs (4kf-CDW).
    3/ 2 3 / 2  1/ 2 1/ 2  ei
O4 k f ,cdw   R R L L  e 2i
c
 c
wins at K c 
1
;
2
wins at K c  12 .
d  2 /( 4k f )
• Existence of singlet Cooper pair superfluidity at 1>Kc>1/2.
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Competition between quartetting and pairing phases

• Two-component superfluidity 1   3/ 23
/2
   1  2  ei
 c
Oquar  12  ei
4  c
(e i
v
+ e i
v
cos2
   v .

2  1/ 21/
2
c overall phase;
);

v relative phase.

• The relative phase channel determines
the transition.

1
1
H eff  {( x v ) 2  ( xv ) 2 } 
(1 cos 2   v  2 cos 2   v )
2
2a
• 1  2 the relative phase is locked: pairing order;
1  2 the dual field is locked: quartetting order.
Ising transition: two Majorana fermions with masses:
1  2
A. J. Leggett, Prog, Theo. Phys. 36, 901(1966); H. J. Schulz, PRB 53, R2959 (1996).
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Experiment setup and detection
• Array of 1D optical tubes.
• RF spectroscopy to measure
the excitation gap.
pair breaking:
quartet breaking:
M. Greiner et. al., PRL, 2001.
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Summary
• Spin 3/2 cold atomic systems provide a good starting point
to study the quartetting problem.
• Both singlet Cooper pairing and quartetting orders are allowed
in 1D systems.
• The phase transition between them is Ising-like at 1D.
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Hidden symmetry and novel phases in spin 3/2
systems
• The exact Sp(4) or SO(5) symmetry without fine-tuning.
• Quintet Cooper pairing: the Alice string and topological
generation of quantum entanglement.
• Strong quantum fluctuations in spin 3/2 magnetic systems.
Ref: C. Wu, J. P. Hu, and S. C. Zhang, Phys. Rev. Lett. 91, 186402(2003);
C. Wu, Phys. Rev. Lett. 95, 266404 (2005);
S. Chen, C. Wu, S. C. Zhang and Y. P. Wang, Phys. Rev. B 72, 214428 (2005);
C. Wu, J. P. Hu, and S. C. Zhang, cond-mat/0512602.
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