1. Hidden Symmetry and Quantum Phases in Spin 3/2 Cold Atomic Systems
Congjun Wu
Kavli Institute for Theoretical Physics, UCSB
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/
It is my great pleasure to visit OSU, and to discuss physics with people in this
famous place. Let me thank Jason for his kind invitation and organizing this talk for me.
It is also my great pleasure to take this opportunity to introduce our work on the
Spin 3/2 systems. Here are some references.
OSU , 05/09/2006

2. Collaborators L. Balents, UCSB.
S. Chen, Institute of Physics, Chinese Academy of Sciences, Beijing.
J. P. Hu, Purdue.
O. Motrunich, UCSB
Y. P. Wang, Institute of Physics, Chinese Academy of Sciences, Beijing.
Let me thank my collaborators.
S. C. Zhang, Stanford.
Many thanks to E. Demler, M. P. A. Fisher, E. Fradkin, T. L. Ho, A. J. Leggett, D. Scalapino, J. Zaanen, and F. Zhou for very helpful discussions.

3. Rapid progress in cold atomic physics
M. Greiner et al., Nature 415, 39 (2002).
M. Greiner et. al., PRL, 2001.
Magnetic traps: spin degrees of freedom are frozen.
Optical traps and lattices: spin degrees of freedom are released; a controllable way to study high spin physics.
The motivation of this work is inspired from the rapid progress in cold atomic physics.
In the early days, BEC was realized in magnetic traps, where the spin degrees of
freedom are frozen. Later on, important progress has been made. People can trap
atoms by using optical method, including optical lattices and traps.
A major advantage is that the spin degrees of freedom are released in optical traps
and lattices. This provides a controllable way to study high spin physics.
Furthermore, in optical lattices, interaction effects can be easily adjusted from the
weak interaction regime to the strong interaction regime. For example, the superfluid
to Mott insulator transition has been observed. Optical lattices open up a new
direction to study strongly correlated high spin systems.
In optical lattices, interaction effects are adjustable. New opportunity to study strongly correlated high spin systems.

4. High spin physics with cold atoms
Most atoms have high hyperfine spin multiplets.
F= I (nuclear spin) + S (electron spin).
Spin-1 bosons: 23Na (antiferro), 87Rb (ferromagnetic).
Next focus: high spin fermions.
Fukuhara et al., cond-mat/
Spin-3/2: 132Cs, 9Be, 135Ba, 137Ba, 201Hg.
Spin-5/2 : 173Yb (quantum degeneracy has been achieved)
Most atoms have hyperfine spin multiplets. Here spin means the total spin of the atom,
including the electron spin and the nuclear spin. I will follow the convention in atomic
physics. I use F to denote the total spin of the atom.
The cold atomic high spin systems are different from the high spin
condensed matter systems. In many transition metal oxide, several
electrons can be combined by Hund’s rule to form a high spin object.
However, the intersite coupling is still mediated by the virtual hopping
of a single electron. In contrast, in the cold atomic systems, each
atom carries high spin.
The study of cold atomic high spin system was pioneered by Jason about 10 years ago.
A lot of important progress has been made in this field. For example, there is already
extensive research on spin-1 bosons such as Na, Rb, including the spinor condensation,
and the more interesting spin nematic phase in the Mott insulating state in optical lattices.
On the other hand, high spin fermionic systems are also interesting.
In this talk, I will focus on the simplest high spin fermionic system—
spin 3/2 fermions.
High spin fermions: zero sounds and Cooper pairing structures.
D. M. Stamper-Kurn et al., PRL 80, 2027 (1998); T. L. Ho, PRL 81, 742 (1998);
F. Zhou, PRL 87, (2001); E. Demler and F. Zhou, PRL 88, (2002);
T. L. Ho and S. Yip, PRL 82, 247 (1999); S. Yip and T. L. Ho, PRA 59, 4653(1999).

5. Difference from high-spin transition metal compounds
New feature: counter-intuitively, large spin here means large quantum fluctuations. e e
Optical lattice also provides a wonderful opportunity to study high spin physics.
Most atoms have hyperfine spin multiplets. Here spin means the total spin of the atom,
including the electron spin and the nuclear spin. I will follow the convention in atomic
physics. I use F to denote the total spin of the atom.
The cold atomic high spin systems are different from the high spin
condensed matter systems. In many transition metal oxide, several
electrons can be combined by Hund’s rule to form a high spin object.
However, the intersite coupling is still mediated by the virtual hopping
of a single electron. In contrast, in the cold atomic systems, each
atom carries high spin.
high-spin transition metal compounds
high-spin cold atoms

6. Hidden symmetry: spin-3/2 atomic systems are special!
Spin 3/2 atoms: 132Cs, 9Be, 135Ba, 137Ba, 201Hg.
Hidden SO(5) symmetry without fine tuning!
Continuum model (s-wave scattering); the lattice-Hubbard model.
Exact symmetry regardless of the dimensionality, lattice geometry, impurity potentials.
SO(5) in spin 3/2 systems  SU(2) in spin ½ systems
These systems are indeed very special, because it contains a hidden and generic symmetry.
In additional to the obvious spin SU(2) symmetry, we will show it has an hidden SO(5) symmetry,
or isomorphically Sp(4) symmetry in it. This SO(5) symmetry is robust, and free of fine tuning.
It is exact in the continuum model with the s-wave scattering interactions,
and it is also exact in the lattice-Hubbard model, regardless the dimensionality,
Lattice geometry, impurity potentials.
The cold atomic spin 3/2 systems are very different from high spin transition metal compounds,
in which three electrons coupled by Hund’s rule to form a spin 3/2 object.
However, the inter-site coupling is still mediated by the virtual hopping of
a single electron. Thus these systems only have the SU(2) symmetry, and
there are no hidden symmetry.
Before I move on, let me emphasis that this SO(5) symmetry I am talking about
is qualitatively different from the SO(5) theory of high Tc superconductivity.
Only the name of the symmetry group is the same, the physics is quite different.
This SO(5) symmetry is qualitatively different from the SO(5) theory of high Tc superconductivity.
Brief review: C. Wu, Mod. Phys. Lett. B 20, 1707 (2006)

7. What is SO(5)/Sp(4) group?
SO(3) /SU(2) group.
3-vector: x, y, z.
3-generator:
2-spinor:
SO(5) /Sp(4) group.
5-vector:
Now let me briefly explain what is SO(5) group. SO(5) group describes
the rotation in a real 5-dim space. The axis n1 to n5 are orthogonal
to each other, which form the vector representation of this group.
It contains 10 generators. Each geneator Lab is responsible for
the rotation in ab plane.
SO(5)’s spinor Rep is 4-dimensional. The four spin
components just form the SO(5) spinor.
Rigorously speaking, spinors are Rep of the SO(5)’s covering
Group called Sp(4). The relation between SO(5) and Sp(4) is similar
to the relation between SO(3) and SU(2).
10-generator:
4-spinor:

8. Quintet superfluidity and half-quantum vortex
High symmetries do not occur frequently in nature, since every new symmetry brings with itself a possibility of new physics, they deserve attention.
---D. Controzzi and A. M. Tsvelik, cond-mat/
Cooper pairing with Spair=2.
Half-quantum vortex (non-Abelian Alice string) and quantum entanglement.
Why this symmetry is interesting? Let me cite a comment ….…..
Next I will outline the new physics from this symmetry in spin 3/2 systems.
First, spin 3/2 systems support the quintet Cooper pairing phase.
The half-quantum vortex exhibits a non-Abelian Alice string behavior.
After a particle penetrates the HQV loop, quantum entanglement is
generated between the vortex loop and the particle.
C. Wu, J. P. Hu, and S. C. Zhang, cond-mat/

9. Multi-particle clustering order
Quartetting order in spin 3/2 systems.
4-fermion counter-part of Cooper pairing.
Feshbach resonances: Cooper pairing superfluidity.
Driven by logic, it is natural to expect the quartetting order as a possible focus for the future research.
Second, spin 3/2 systems can exhibit quartetting order.
It means that four particles with all spin components form
a bound state and then condense. This is a four-fermion counterpart
to the Cooper pairing.
Taking into account the extensive study of Feshbach resonance, it
is reasonable to expect that the quartetting order is one possible
focus for the future research.
C. Wu, Phys. Rev. Lett. 95, (2005).

10. Strong quantum fluctuations in magnetic systems
Intuitively, quantum fluctuations are weak in high spin systems.
However, due to the high SO(5) symmetry, quantum fluctuations here are even stronger than those in spin ½ systems.
Another interesting aspect of spin 3/2 systems is the strong quantum
fluctuations. Intuitively, one would expect that quantum fluctuation is weak
in spin 3/2 systems because of the high spin. However, due to the high symmetry
of SO(5), quantum fluctuations are actually even stronger
than spin ½ systems. In other words, spin 3/2 system is in the
large N limit instead of large S limit.
This is a cartoon from Auerbach’s book. We expect there is a finite overlap among
the spin 3/2 and the more exotic physics to search spin disordered state and spin liquid state.
large N (N=4) v.s. large S.
From Auerbach’s book:

11. C. Wu, J. P. Hu, and S. C. Zhang, PRL 91, 186402(2003).
Outline
The proof of the exact SO(5) symmetry.
C. Wu, J. P. Hu, and S. C. Zhang, PRL 91, (2003).
Quintet superfluids and non-Abelian topological defects.
Quartetting v.s pairing orders in 1D spin 3/2 systems.
Next I will first prove this symmetry and present its physical consequences.
SO(5) (Sp(4)) Magnetism in Mott-insulating phases.

12. The SO(4) symmetry in Hydrogen atoms
An obvious SO(3) symmetry: (angular momentum) .
The energy level degeneracy n2 is mysterious. H
Not accidental! 1/r Coulomb potential gives rise to a hidden conserved quantity.
Runge-Lentz vector ; SO(4) generators
Before we move on. Let us look at a familiar example of the hidden symmetry in the H atom.
The H atom has an obvious SO(3) symmetry. However, the energy level degeneracy n2
can not be explained by the SO(3) symmetry.
Actually, this high degeneracy is not accidental. The Coulomb potential gives rise to a
hidden conserved quantity, the Runge-Lentz vector. This vector describes the orientation
of the classical orbits, and the eccentricity of the ellipse orbit. The RL vector and L together
form a closed algebra of SO(4). This SO(4) symmetry
is responsible for the n2 Level degeneracy.
Now we ask the question: what are the hidden conserved quantities
in spin 3/2 systems in addition to the obvious angular momentum.
In the following several slides, I will answer this question.
Q: What are the hidden conserved quantities in spin 3/2 systems?

13. Generic spin-3/2 Hamiltonian in the continuum
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:

14. Spin-3/2 Hubbard model in optical lattices
The single band Hubbard model is valid away from resonances.
40”
Now let us look at the lattice system. Within the reasonable experimental parameters,
The single bond Hubbard model is valid.
The hopping term, the chemical potential term. Similarly to the continuum model,
We have two independent Hubbard repulsions, the on-site singlet and on-site quintet
channel interactins..
as: scattering length, Er: recoil energy

15. SO(5) symmetry: the single site problem
24=16 states.
SU(2)
SO(5)
degen
eracy
E0,3,5
singlet
scalar 1
E1,4
quartet
spinor 4 E2
quintet
vector 5
U0=U2=U, SU(4) symmetry.
The best way to see the SO(5) symmetry is to look at the a single site problem.
There are 16 states. They are classified as empty, singly occupied, doubly occupied state,
and so on. For the doubly occupied states, as we explained before
They are either spin singlet, or quintet.
Let us check their energy level degeneracy pattern: 1, 4, 5; it
matches the SO(5) representation perfectly. The spin singlet is also SO(5)
singlet, spin quartet fermionic states form an SO(5) spinor Rep.
The spin quintet states are SO(5) vectors. So we can see the
arrival of SO(5) without fine-tuning.
If we fine tune U0=U2, in this case, the interaction is simply density-density
Interaction, an SU(4) symmetry appears, which means the 4-componets
are equivalent to each other.
Let me clarify a terminology convention: SO(5) is isomorphic to its covering group Sp(4),
Their relation is similar to the relation between SO(3) and SU(2).
In the following, I will use SO(5) or Sp(4) alternatively.

16. Particle-hole bi-linears (I)
Total degrees of freedom: 42=16=
1 density operator and 3 spin operators are not complete.
rank: 0 1 2 3
Spin-nematic matrices (rank-2 tensors) form five-G matrices (SO(5) vector) .
Now we construct the SO(5) algebra. We have four spin components, thus
we have 16 bi-linears One density and three spin operators are not enough.
We need high order symmetric spin tensors. The rank-2 tensor has five degrees
of freedom, rank-3 has 7 degrees of freedoom =16, this is a complete set.
The rank-2 tensors are often denoted as spin-nematics matrices.
They anticommute with each other, and form a basis for the five Dirac matrices.
They are SO(5) vectors.

17. Particle-hole channel algebra (II)
Both commute with Hamiltonian generators of SO(5): 10=3+7.
7 spin cubic tensors are the hidden conserved quantities.
SO(5): 1 scalar + 5 vectors + 10 generators = 16
Time Reversal
1 density:
5 spin nematic:
3 spins + 7 spin cubic tensors:
even
On the other hand, the three spin operators and seven cubit spin operators form the SO(5)
Generators. They commute with the Hamiltonian. These seven cubic spin operators are
the hidden conserved quantities, just like the Rugne-Lenz vectors.
In short, the p-h bilinears can be classified into SO(5) scalar, vectors, and
generators. The SO(5) vectors are just spin-nematics. The 10 generators
either carry spin 1 or spin 3. We can see this SO(5) symmetry purely lies
in the particle-hole channel. In comparison in the high Tc, the SO(5) symmetry
there involves both particle-hole and particle-particle channel.
even
odd

18. C. Wu, J. P. Hu, and S. C. Zhang, cond-mat/0512602.
Outline
The proof of the exact SO(5) symmetry.
Quintet superfluids and Half-quantum vortices.
C. Wu, J. P. Hu, and S. C. Zhang, cond-mat/
Quartetting v.s pairing orders in 1D spin 3/2 systems.
Let us discuss the consequences of the high symmetry. First, I will
discuss the quintet superfluids and non-Abelian topological defects therein.
SO(5) (Sp(4)) Magnetism in Mott-insulating phases.

19. g2<0: s-wave quintet (Spair=2) pairing
BCS theory: polar condensation; order parameter forms an SO(5) vector.
5d unit vector
When the interaction in the quintet channel is negative, we have quintet Cooper pairing.
This problem was studied by Jason Ho before. Their results showed that within the
BCS theory, the ground state exhibits polar condensation.
Now we further conclude that the order parameter forms an SO(5) vector.
In the polar basis, this five component order parameter just forms the
analogy to the five d-orbits in the chemistry and atomic physics.
Ho and Yip, PRL 82, 247 (1999);
Wu, Hu and Zhang, cond-mat/

20. Superfluid with spin: half-quantum vortex (HQV)
Z2 gauge symmetry
-disclination of as a HQV.
remains single-valued.
is not a rigorous vector, but a directionless director.
Generally speaking, superfluid with spin degrees of freedom can support half-quantum vortex.
It is because of a Z2 gauge symmetry, if we change d to –d, and also shift theta by pi,
the order parameter does not change. This feature allows the HQV as a pi disclination
of the d-vector. Let us look at such a vortex. As we wind around the vortex core, d is only
rotated at pi, also the phase winds at pi, thus the order parameter is still single-valued.
In other words, d is not a rigorous vector but a directionless director. Formally, the Goldstone
manifold for d is a Hemisphere. This hemisphere contains a non-contractable loop.
The fundamental group of RP4 is Z2, this allows a line defect of pi disclination.
Fundamental group of the manifold.

21. Stability of half-quantum vortices
Single quantum vortex:
A pair of HQV:
Now let us do some simple energetics, and compare the energy of
a single quantized vortex and a pair of HQV.
The energy functional can be expressed like this, this is the superfluid
density and this is the spin-wave stiffness.
The energy for a single quantum vortex without the distortion in the spin
channel is like this. This is the configuration of a pair of HQV.
In the spin channel, it exhibits a configuration of pi-disclination
and anti pi-disclination.
If this pair is separated at a large distance, their energy can
be written as this.
Thus if the spin-wave stiffness is less than the superfluid density.
Energetically, a pair of HQV is more stable than a single quantum vortex.
Stability condition:

22. Example: HQV as Alice string (3He-A phase)
A particle flips the sign of its spin after it encircles HQV.
Example: 3He-A, triplet Cooper pairing.
The HQV has an interesting property called Alice string behavior.
When a particle encircles vortex core adiabatically, its spin flips sign.
The Alice string physics was extensively studied in high energy physics.
In condensed matter physics, a good example is the He-3 A phase.
The $3$He-A phase is a triplet Cooper pairing phase. Its order parameter
also contains a d-vector, which lies in a 3D hemisphere.
Let us look at this pi disclination. We first set up a frame with x,y, z
axis. We assume at the starting point, the d-vector
is along the z direction. As we move around the vortex, we can rotate the
d-vector pi degree around any axis in the xy plane, say the n-axis.
Thus the configuration space of HQV can map to the U(1) circle of xy plane.
For a particle with spin encircles the vortex, since the rotation angle is only pi.
Thus spin up goes to spin down, and spin down goes to spin up up a U(1) phase.
This U(1) phase is determined by alpha angle of the n-axis.
The phenomenon is not surprising, because the spin rotation symmetry is already broken.
It is just the ordinary U(1) Berry phase.
A. S. Schwarz et al., Nucl. Phys. B 208, 141(1982);
F. A. Bais et al., Nucl. Phys. B 666, 243 (2003);
M. G. Alford, et al. Nucl. Phys. B 384, 251 (1992).
P. McGraw, Phys. Rev. D 50, 952 (1994).
Configuration space U(1)

23. The HQV pair in 2D or HQV loop in 3D
SO(3)SO(2)
Now let us study a pair of HQV or HQV loop.
We start from a defect free uniform group state, where the d-vector is along the z-axis.
The remaining symmetry is SO(2) rotation along the z-axis.
Now we add a pair of HQV. In the region faraway from the
Vortex loop, d is still parallel to the z-axis. The SO(2) rotational
Symmetry is broken, but is only broken within a finite region around the vortex loop.
Again the configuration space of the HQV pair can be
parameterized by the U(1) angle in the xy-plane.
We call this state as phase state alpha.
For a spin transport, if the particle penetrates the loop, its spin flips up
To a U(1) phase. If it goes outside, no spin flip happens.
Phase state.
P. McGraw, Phys. Rev. D, 50, 952 (1994).

24. SO(2) Cheshire charge (3He-A)
HQV pair or loop can carry spin quantum number.
For each phase state, SO(2) symmetry is only broken in a finite region, so it should be dynamically restored.
Cheshire charge state (Sz) v.s phase state.
An even more interesting concept is Cheshire charge. It means a pair of half-quantum vortex
and anti-vortex pair or a vortex loop can carry spin quantum number called Cheshire charge.
Let us start with an order ground state, the d-vector lies along the z-direction.
There is an SO(2) symmetry around the z-axis.
Now we introduce a HQV pair or loop.
Faraway from the vortex, the orientation of d-vector does not change.
The configuration of the vortex pair is also U(1), is determined by the axis n,
around with the d-vector is rotated by pi.
We call it as phase state.
For each phase state alpha, SO(2) symmetry only breaks in a finite region close to the vortex loop,
Thus it should be dynamically restored in quantum mechanics.
We do liner superposition by the Fourier transformation to form the
Cheshire charge eigenstate, Here charge means Sz.
For example, a zero charge state is just an average of the phase state.
P. McGraw, Phys. Rev. D, 50, 952 (1994).

25. Spin conservation by exciting Cheshire charge
Initial state: particle spin up and HQV pair (loop) zero charge.
Final state: particle spin down and HQV pair (loop) charge 1.
Let us we study the spin transport process with a Cheshire quantized vortex loop.
We assume the initial state is a spin up particle with a zero charge vortex loop.
After a particle passing the vortex loop,
Its spin still flips. Spin up goes to spin down, and the spin conservation is realized
by exciting Cheshire charge, the vortex loop gains spin-1.
Let us look at it more carefully at the algebra. We expand the charge eigenstate
In terms of the phase state. For each configuration of alpha, the finial state are
With a different phase, after the liner-superposition, the finial state of vortex loop
Is an excited state carrying Cheshire charge.
This is indeed interesting. However, both the final and initial state are product
state. No entaglement! It is not useful for quantum computation.
Both the initial and final states are product states. No entanglement!
P. McGraw, Phys. Rev. D, 50, 952 (1994).

26. Quintet pairing as a non-Abelian generalization
HQV configuration space , SU(2) instead of U(1).
equator:
Let us move to the quintet pairing, which is a non-Abelian generalization of the
above physics. In this case, the configuration space of a pi-disclination is SU(2).
D is a 5 dimensional d-vector. We set up a frame e1,2,3,4,5.
At the starting point, we assume d//e4. We can rotate d-vector around any axis in the
Equator, which is a S3 phere spaned by e1,2,3,5, and S^3 is group space for SU(2).
p-disclination

27. SU(2) Berry phase
After a particle moves around HQV, or passes a HQV pair:
Now we move a particle around HQV, or through a HQ loop.
3/2,-3/2 goes to ½, -1/2.
The transformation exhibits SU(2) Berry phase.
W can be any SU(2) matrix, which is determined by n vector on the S^3 sphere.

28. Non-Abelian Cheshire charge
Construct SO(4) Cheshire charge state for a HQV pair (loop) through S3 harmonic functions.
SO(5)  SO(4)
Zero charge state.
Similarly, we can introduce SO(4) Cheshire charge eigenstates for HQV loop or pairs
by introducing Fourier transformation to the phase state.
Let me skip the detail. These T’s are SO(4) quantum numbers, Y are harmonic functions
on S^3 sphere.
The simplest case, the zero charge state is just an average of on the phase state.

29. Entanglement through non-Abelian Cheshire charge!
SO(4) spin conservation. For simplicity, only is shown.
Generation of entanglement between the particle and HQV loop!
Now we move a particle through HQV loop which is Cheshire charge eigenstate.
Amazingly, in order to maintain the SO(4) spin, quantum entanglement
is generated in the finial state to ensure the SO(4) spin conservation.
For example, the initial state, particle with Sz=3/2, HQV with zero charge.
The final state is an entangled state determined by the SO(4) C-G coefficentl.
Thus it is useful for topogolical quantum computation.

30. Outline The proof of the exact SO(5) symmetry.
Alice string in quintet superfluids and non-Abelian Cheshire charge.
Quartetting and pairing orders in 1D systems.
C. Wu, Phys. Rev. Lett. 95, (2005).
Next let me move to the quartetting problem.
SO(5) (Sp(4)) Magnetism in Mott-insulating phases.

31. Multiple-particle clustering (MPC) instability
Pauli’s exclusion principle allows MPC. More than two particles form bound states.
baryon (3-quark); alpha particle (2p+2n); bi-exciton (2e+2h)
Spin 3/2 systems: quartetting order.
SU(4) singlet:
4-body maximally entangled states
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.
Spin 3/2 systems allow the quaretteting order. 4-four fermions with all the spin componets
Form a SU(4) singlet, which is also a 4-body EPR 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. There is trial wf study in 3D by M. Gunn, however, we still
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, he
Claimed quartetting order always win over pairing order. But this result
Remains controversial.
Difficulty: lack of a BCS type well-controlled mean field theory.
trial wavefunction in 3D: A. S. Stepanenko and J. M. F Gunn, cond-mat/

32. 1D systems: strongly correlated but understandable
Bethe ansatz results for 1D SU(2N) model:
2N particles form an SU(2N) singlet; Cooper pairing is not possible because 2 particles can not form an SU(2N) singlet.
P. Schlottmann, J. Phys. Cond. Matt 6, 1359(1994).
Competing orders in 1D spin 3/2 systems with SO(5) symmetry.
Both quartetting and singlet Cooper pairing are allowed.
Transition between quartetting and Cooper pairing.
In 1D, fortunately, we are still able to say something clearly.
Scholttmann showed that by Bethe ansatz in 1D SU(2N) model,
2N particles from A SU(2N) singlet in the ground state.
In contrast, the Cooper pairing can not exist because two particles can not
form a SU(2N) singlet. The strong quantum fluctuations in the spin channel in
1D suppress the Cooper pairing.
In the following, I will study the 1D spin 3/2 system which is SO(5) symmetric.
I will show both quartetting and singlet pairing are allowed in different parameter regimes.
But the quintet paring is not allowed, because it is not a SO(5) singlet.
The transtion between quartetting and Cooper pairing is Ising-like.
Check QHE, Read Rezayi
C. Wu, Phys. Rev. Lett. 95, (2005).

33. Phase diagram at incommensurate fillings
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.
Singlet pairing in purely repulsive regime.
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.
Quintet pairing is not allowed.

34. Competition between quartetting and pairing phases
A. J. Leggett, Prog, Theo. Phys. 36, 901(1966); H. J. Schulz, PRB 53, R2959 (1996).
Phase locking problem in the two-band model.
overall phase; relative phase.
dual field of the relative phase
No symmetry breaking in the overall phase (charge) channel in 1D.
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.

35. Ising transition in the relative phase channel
the relative phase is pinned: pairing order;
the dual field is pinned: quartetting order.
Ising transition: two Majorana fermions with masses:
Ising symmetry:
relative phase:
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 ordered phase:
Ising disordered phase:

36. 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.

37. Outline The proof of the exact SO(5) symmetry.
Alice string in quintet superfluids and non-Abelian Cheshire charge.
Quartetting v.s pairing orders in 1D spin 3/2 systems.
SO(5) (Sp(4)) magnetism in Mott-insulating phases.
Under very strong repulsive interactions, the charge fluctuations
is frozen, and the low energy physics is described by the spin
exchange model. In the following, I discuss spin 3/2 magnetic models.
S. Chen, C. Wu, S. C. Zhang and Y. P. Wang, Phys. Rev. B 72, (2005).
C. Wu, L. Balents, in preparation.

38. Spin exchange (one particle per site)
Spin exchange: bond singlet (J0), quintet (J2). No exchange in the triplet and septet channels.
Heisenberg model with bi-linear, bi-quadratic, bi-cubic terms.
SO(5) or Sp(4) explicitly invariant form:
Let us first construct the exchange Hamiltonian at quarter filling, which means
one particle per site. Let us consider a bond, each site carries a spin 3/2, thus the
total spin on the bond can be either 0,1,2,3. By the standard perturbation theory,
we can see the exchange energy only exists in the total spin singlet and quintet
channels. There is no spin exchange in the total spin 1 and 3 channels.
This model can be organized into an explicitly SO(5) symmetric way,
Where Ls are SO(5) generators, which contains spin and spin cubic tensors,
and ns are spin nematic operators, which are SO(5) vector.
Lab: 3 spins + 7 spin cubic tensors; na: spin nematic operators; Lab and na together form the15 SU(4) generators.

39. Two different SU(4) symmetries
singlet
quintet
triplet
septet J
A: J0=J2=J, SU(4) point. *
B: J2=0, the staggered SU’(4) point.
In a bipartite lattice, a particle-hole transformation on odd sites:
singlet
quintet
triplet
septet J0
The above exchange mode contains two special high symmetry points.
First, if exchange constants in the singlet and quintet channels are the same,
then the bond singlet and quintet channels are degenerate.
This mode reduces into an SU(4) symmetric model. Each site is in
the SU(4) fundamental spinor representation.
On the other hand, we set the quintet channel exchange to zero, only allows the
exchange in the bond singlet channel. Then in a bipartite lattice,
we do particle-hole transformation in the odd sites, the Hamiltonian is again SU(4) symmetric.
This SU(4) is different from the first one. The even sites belong to the
fundamental spinor representation, where the odd sites belong to its complex conjugate representation.
The first SU(4) model is often called Kugel-Khomski model, we need 4-site to form a singlet.
The model is even difficult for numerics, QMC has huge sign problem.
The second one is much more friendly, it is the CP(3) generalization to the usual Heisenberg model,
Only 2 sites are needed to form a singlet. There is no sign problem.

40. Construction of singlets
The SU(2) singlet: 2 sites.
The uniform SU(4) singlet: 4 sites.
baryon
The staggered SU’(4) singlet: 2 sites.
These two different SU(4) have dramatically different properties.
For the first SU(4), we need 4-particles in a plaquette to form a singlet.
It is a baryon like state, there is no site and bond spin orders, only
Plaquette order.
On the other hand, the second staggered SU(4) model. We only need two particles
to form a singlet. It is a meson-like state.
meson

41. Phase diagram in 1D lattice (one particle per site)
spin gap dimer
gapless spin liquid
Now let us present the phase diagram for the above exchange model in 1D.
If J2>J0, including the SU(4) line, we obtain the translationally invariant gapless
Spin liquid state.
Otherwise, including the staggered SU(4) line, we have dimerized spin gap phase.
On the SU’(4) line, dimerized spin gap phase.
On the SU(4) line, gapless spin liquid phase.
C. Wu, Phys. Rev. Lett. 95, (2005).

42. SU’(4) and SU(4) point at 2D
At SU’(4) point (J2=0), QMC and large N give the Neel order, but the moment is tiny.
Read, and Sachdev, Nucl. Phys. B 316(1989).
K. Harada et. al. PRL 90, , (2003).
J2>0, no conclusive results!
SU(4) point (J0=J2), 2D Plaquette order at the SU(4) point?
Exact diagonalization on a 4*4 lattice
Now let us look at the 2D phase diagram. Then the situation is much more complicated.
We only have good understanding about the staggered SU(4) point. Both the QMC and
Large N shows the group state is long range Neel ordered. But the Neel moment is tiny.
However, for any other nonzero values of J2, the phase structure is an extremely difficult problem.
Even QMC has sign problem, there is no conclusive answer yet.
Let us look at the Kugel-Komskii SU(4) point, there is an exact diagonalization
on 4*4 lattice, they conjectured that the ground state exhibits the plaqutte order.
As I mentioned before, the plaquette order is an order of 4 sites, no bond order.
Their result is interesting but still no conclusive.
Bossche et. al., Eur. Phys. J. B 17, 367 (2000).

43. Exact result: SU(4) Majumdar-Ghosh ladder
Exact dimer ground state in spin 1/2 M-G model.
SU(4) M-G: plaquette state.
This plaqutette ordered state in 2D is interesting, but is difficult to analyze.
Let us return to 1D systems, we construct an SU(4) M-G like model,
Its ground state exactly exhibits this order.
The usual M-G is defined on a spin ½ Heisenberg chain. If the nnn coupling is ½ of the nn coupling,
the ground state can be exactly solved, which is just a product of dimer state.
Similarly, in the our SU(4) case, the ground state of our model
Is a product of all these plaquette state. The hamiltonian is
A little bit complicated, we need couple all the sites within
every six-site cluster
Again, the excitations have the domain wall structrue.
SU(4) Casimir of the six-site cluster
Excitations as fractionalized domain walls.
S. Chen, C. Wu, S. C. Zhang and Y. P. Wang,
Phys. Rev. B 72, (2005).

44. Conclusion
Spin 3/2 cold atomic systems open up a new opportunity to study high symmetry and novel phases.
Quintet Cooper pairing: the Alice string and topological generation of quantum entanglement.
Quartetting order and its competition with the pairing order.
Strong quantum fluctuations in spin 3/2 magnetic systems.

45. SU(4) plaquette state: a four-site problem
Bond spin singlet:
Plaquette SU(4) singlet:
4-body EPR state; no bond orders
Level crossing:
d-wave
Check Ogatta, Shiba.
d-wave to s-wave
Hint to 2D?
s-wave

46. Speculations: 2D phase diagram ?
J2=0, Neel order at the SU’(4) point (QMC).
K. Harada et. al. PRL 90, , (2003).
SU’(4)
J2>0, no conclusive results!
SU(4)
2D Plaquette order at the SU(4) point?
Exact diagonalization on a 4*4 lattice
Bossche et. al., Eur. Phys. J. B 17, 367 (2000).
Phase transitions as J0/J2? Dimer phases? Singlet or magnetic dimers?
Now let us look at the 2D phase diagram. At the staggered SU(4) point where J2=0,
the QMC simulations conclude that the ground state is Neel ordered.
However, for general values of J2, the phase structure is an extremely difficult problem.
Even QMC has sign problem, there is no conclusive answer yet.
Let us look at the Kugel-Komskii SU(4) point, there is an exact diagonalization
on 4*4 lattice, they conjectured that the ground state exhibits the plaqutte order.
As I mentioned before, the plaquette order is an order of 4 sites, no bond order.
Their result is interesting but still no conclusive.
Now we speculate the general phase structure. Between the Neel order, which is a
Site order, and the plaquette order, there might be a dimer ordered phase.
Different from the usual spin ½ systems, the dimer here can be either singlet or quintet,
So there might be magneticly ordered dimer state.
C. Wu, L. Balents, in preparation.

47. Strong quantum fluctuations: high symmetry
Different from the transition metal high spin compounds!
classical Neel order
SU(2): S=3/2 …
quantum: spin disordered states
Sp(4), SU(4): F=3/2
large N
large S
SU(4) singlet plaquette state: 4 sites order without any bond order.
As we mentioned in the introduction part, the spin 3/2 magnetic systems
have strong quantum fluctuations due to the high symmetry. Their physics
is dramatically different from the high spin transition metal compounds.
In other words, our systems has Sp(4) or SU(4) symmetry, it is in the large N limit, where N=4.
In contrast, the transition metal systems are in the large S limit, only SU(2) symmetry
is present.
As a demonstration of strong quantum fluctuation, we will discuss the
SU(4) plaquette state, where 4 sites within a plaquette form an SU(4) singlet.
Similarly to the quartetting order we discussed before,
this is really an order of 4-sites, there is no any bond order.
S. Chen, C. Wu, Y. P. Wang, S. C. Zhang, cond-mat/ , accepted by PRB.

48. 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.

49. 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.

50. High spin physics with cold atoms
Most atoms have high hyperfine spin multiplets.
F= I (nuclear spin) + S (electron spin).
Different from high spin transition metal compounds. e
Spin-1 bosons: 23Na (antiferro), 87Rb (ferromagnetic).
Most atoms have hyperfine spin multiplets. Here spin means the total spin of the atom,
including the electron spin and the nuclear spin. I will follow the convention in atomic
physics. I use F to denote the total spin of the atom.
The cold atomic high spin systems are different from the high spin
condensed matter systems. In many transition metal oxide, several
electrons can be combined by Hund’s rule to form a high spin object.
However, the intersite coupling is still mediated by the virtual hopping
of a single electron. In contrast, in the cold atomic systems, each
atom carries high spin.
The study of cold atomic high spin system was pioneered by Jason about 10 years ago.
A lot of important progress has been made in this field. For example, there is already
extensive research on spin-1 bosons such as Na, Rb, including the spinor condensation,
and the more interesting spin nematic phase in the Mott insulating state in optical lattices.
On the other hand, high spin fermionic systems are also interesting.
In this talk, I will focus on the simplest high spin fermionic system—
spin 3/2 fermions.
High spin fermions: zero sounds and Cooper pairing structures.
D. M. Stamper-Kurn et al., PRL 80, 2027 (1998); T. L. Ho, PRL 81, 742 (1998);
F. Zhou, PRL 87, (2001); E. Demler and F. Zhou, PRL 88, (2002);
T. L. Ho and S. Yip, PRL 82, 247 (1999); S. Yip and T. L. Ho, PRA 59, 4653(1999).

51. Quintet channel (S=2) operators as SO(5) vectors
Kinetic energy has an obvious SU(4) symmetry; interactions break it down to SO(5) (Sp(4)); SU(4) is restored at U0=U2.
5-polar vector
Having studied the single site problem, let us come back to the
Many body Hamiltonian. The kinetic energy has an obvious SU(4) symmetry.
But the interactions terms break it down to SO(5). The SU(4) symmetry
is restored at U0=U2.
Now let us look at the quintet-channel interaction more carefully.
As we mentioned before, they form an SO(5) vector. To see it
explicitly, we organize them into polar basis as an analogy to
the polar wf of the d-orbits in the quantum chemistry.
The real linear combination of these chi’s span a 5-d space.
The SO(5) symmetry simply means a free rotation on
this 5-d sphere. We start from the North pole, we can arrive at
any point by an SO(5) rotation. If we are only allowed to do spin SU(2)
Operation, we can only move within a sub-manifold is this sphere.
The SO(5) symmetry is indeed more general and it includes SU(2) as its subgroup.
trajectory under SU(2).

52. The SU’(4) model: dimensional crossover
SU’(4) model: 1D dimer order; 2D Neel order.
SU’(4) model in a rectangular lattice; phase diagram as Jy/Jx.
dimer
renormalized
classical
quantum
disorder
Neel order
Competition between the dimer and Neel order.
Now let us look at the staggered SU(4) model more carefully. We know in 1D, it
Is in the dimerized phase, while in 2D it has long range Neel order.
Thus it would be interesting to study its dimensional crossover problem.
Let us put this model in the rectangular lattice, and allow the spin exchange
Along the x-direction Jx to be different from the Jy.
Then we sketch the phase diagram as ration between Jy/Jx.
We expect a transition from dimer order to the Neel order.
The nature of the transition would be interesting, should it be first order
Or second order, is currently under investigation.
The nice thing in this model is that there is no frustration, thus QMC has
no sign problem. In principle, this transition can be accessible to QMC.
No frustration; transition accessible by QMC.
C. Wu, O. Motrunich, L. Balents, in preparation .