1. 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, (2005).
It is my great pleasure to introducing my work on the “quartetting and pairing instabilities in
1D spin 3/2 systems.
Many thanks to S. C. Zhang, E. Demler, Y. P. Wang, A. J. Leggett for helpful discussions.
March meeting, 03/16/2006 (10:24)

2. 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.
The motivation of this work is driven by the rapid progress in the cold atomic physics.
In particular, the Feshbach resonances provide a wonderful opportunity to study the
Cooper pairing superfluidity.
On the other hand, in fermionic systems with multiple component, Pauli’s exclusion
principle allows the multi-particle clustering instability. It means that more than two particles
can come together to form a bound state. For example, baryons are three-quark bound states,
alpha particles are 4-body bound states of 2 protons + 2 neutrons,
bi-excitions are bounds state of 2e+2h.
Taking into account the dramatic success of the Feshbach type experiments,
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.

3. Quartetting order in spin 3/2 systems
4-fermion counterpart of Cooper pairing.
SU(4) singlet:
4-body maximally entangled states
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/
For this purpose, spin 3/2 fermions provide a good starting point, which
are the simplest high spin fermions. We have a number of this kind
of candidate atoms of spin 3/2.
Spin 3/2 systems allow the quaretteting order. 4-four fermions with all the spin
Components form bound states and then condense. The quartet is an SU(4) singlet,
which is also a 4-body maximally entangled state. The order parameter can be written like this.
This kind order is very difficult to analyze because the order paramter itself already
contains four fermions. We do not have a controllable BCS type mean field theory.
In 3D, there is an interesting work by M. Gunn, who used trial WF method, they
showed quartetting order always win over pairing order.
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, (2005).

4. Generic spin 3/2 Hamiltonian in the continuum model
The s-wave scattering interactions and spin SU(2) symmetry.
Pauli’s exclusion principle: only Ftot=0, 2
are allowed; Ftot=1, 3 are forbidden.
1:30”
Let us look at the Hamiltonian in the continuum model. The only assumption we make is
the s-wave scattering interaction and spin SU(2) symmetry. This is the Kinetic energy part,
we have four spin components. Below I will use these long and short arrows to denote the
four spin components.
Let us look at the interaction. For two spin 3/2 particles, their WF has to be antisymmetric.
In the s-wave channel, their orbital WF is already symmetric, thus the spin WF has to be
antisymmetric. As a result, their total spin can only be either singlet or quintet. The total
spin 1 and 3 channels are forbidden.
g0, g2 are two independent coupling constant in this two channels.
Eta and kai, pairing operators are related to fermion operators
through CG coefficent. .
singlet:
quintet:

5. Phase diagram: bosonization+RG
SU(4)
B: Quartetting
C: Singlet pairing
Gapless charge sector.
Spin gap phases B and C: pairing v.s.quartetting.
A: Luttinger liquid
Ising transition between B and C.
Let us skip the technical detail and directly show the phase diagram at
incommensurate fillings. The method I used is the standard Bosonization
plus renormalization group.
G0 is the coupling constant in the singlet channel, g2 is the coupling
Constant in the quintet channel.
First the charge sector remains gapless and decouples with the
spin sectors as usual.
In the spin sector, we find three phases. Phase A is the Luttinger liquid phase,
it lies in the repulsive interaction region where g2<g0.
This phase is controlled by this black dot, the non-interacting fixed point.
Every charge and spin channel is gapless. It exhibits a spin-charge
separation behavior.
Phase B is controlled by the strong coupling fixed point in the infinity as
marked by the red line, and a spin gap opens. This is the quartetting phase.
Similarly, phase C is also controlled by the another strong coupling fixed point
in the infinity as marked by the blue line. It is the singlet Cooper pairing phase.
Now let us translate it into the more physical phase diagram using the coupling constants
in the particle-particle singlet and quintet channels.
The quartetting phase lies in the regime where attractive interactions dominates.
The singlet paring phase is also interesting, it even survives in the purely repulsive interaction
Regime. This is similar to the high Tc, we can have Cooper pairs in the repulsive interaction regime.
Singlet pairing in purely repulsive regime.

6. Phase B: the quartetting phase
Quartetting superfluidity v.s. CDW of quartets (2kf-CDW).
Kc: the Luttinger parameter in the charge channel.
Now let us look at the quartetting phase.
In phase B, there are two competing orders, the quartetting superfluidity and the 2kf charge density wave order.
If we check the periodicity of the 2kf cdw, it is just the density wave of quartets.
So in this phase, four fermions first form quartets, then the quartets can either undergo
superfluid instability or charge density wave instablity.
Because the charge sector is gappless, both orders are power law fluctuating.
The superfluid order wins over the CDW order at Kc>2.
Check Ogatta, Shiba.

7. Phase C: the singlet pairing phase
Singlet pairing superfluidity v.s CDW of pairs (4kf-CDW).
Similarly in the phase C, two particles first form Cooper pairs, and
Coopers can undergo superfluidity or CDW instablities.
Both of them are also powerlaw fluctuating. The superfluid instability
dominates over the cdw at Kc>1/2.
Check Ogatta, Shiba.
Existence of singlet Cooper pair superfluidity at 1>Kc>1/2.

8. Competition between quartetting and pairing phases
Two-component superfluidity
overall phase;
relative phase.
The relative phase channel determines the transition.
the relative phase is locked: pairing order;
the dual field is locked: quartetting order.
The competition between quartetting and pairing can be mapped to
a phase locking problem of two-component superfludity. This
Two component superfluidity was first studied by Prof. Leggett.
One component is the pairing between 3/2,-3/2; the other one is pairing
between ½, -1/2. The pairing operator can be written as two parts. It
contains one overall phase in the charge channel theta_c, and a
relative phase theta_v, which is a variable in the spin channel .
The quartetting order contains the overal phase theta_c, but
it also contains. The dual field phi_v to the relative phase.
The overal phase theta_c is always powerlaw fluctuating, and does
play a role in the transition. On the other hand, it is the relative phase that controls the
transition. The effective Hamiltonian to describe this competition is a sine-Gordon theory in the
relative phase channel. It contains cosine terms of both relative phase and its dual
field. If the coefficient lamba_1 is larger than lambda_2, then the relative phase is locked, pairing order;
Otherwise, the dual field is locked, it is the quartetting order.
This phase transition can be mapped to two majorana fermions with different masses.
One fermions is always gapped, the other one is gapless at lamda_1=lambda_2, which is
The critical point.
Ising transition: two Majorana fermions with masses:
A. J. Leggett, Prog, Theo. Phys. 36, 901(1966); H. J. Schulz, PRB 53, R2959 (1996).

9. Experiment setup and detection
Array of 1D optical tubes.
RF spectroscopy to measure the excitation gap.
pair breaking:
quartet breaking:
Now let us discuss experiment.
First, the 1D systems and quasi 1D systems have been realized as arrays
of 1D optical tubes.
Second, one popular method to confirm the single Cooper pairing in Feshbach
Resonace experiments, is to use the RF spectroscopy to break the pair to
measure the excitation gap.
For quartetting, we have a number of quartett breaking process, 4->1_3, ..
Thus it can be disctinguished from pairing.
Another method for the high dimensional quarttetting systems is to count the
Number of vortices. In the quartetting phase, flux quanta is only half of the
Regular vortex, the number of vortices is doubled.
M. Greiner et. al., PRL, 2001.

10. The phase transition between them is Ising-like at 1D.
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.

11. 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, (2003);
C. Wu, Phys. Rev. Lett. 95, (2005);
S. Chen, C. Wu, S. C. Zhang and Y. P. Wang, Phys. Rev. B 72, (2005);
C. Wu, J. P. Hu, and S. C. Zhang, cond-mat/