Hidden symmetry and quantum phases in spin-3/2 cold atomic systems


High symmetry groups are often studied in high energy physics. Their application in condensed matter is typically as a mathematical convenience for applying the 1/N-expansion to handle strong-correlation effects when the system has no natural small parameter.

Since 2003, we have proposed a new perspective from high symmetries (e.g. SU(2N) and Sp(2N)) to study the alkali and alkaline-earth fermion systems Ref.[1], where 2N is the fermion component number and hence typically even. We have explored, in atomic systems, complex and beautiful many-body physics difficult to realize in usual solids. (For a brief review, please see Ref. [2]). It also sheds new light on explorations of novel states of matter in ultra-cold atom experiments. This direction since 2010 has attracted considerable attentions in cold atom research by various groups, and has also become an active experiment focus pioneered by Y. Takahashi's group at Kyoto Univ. Non-technical perspective articles introducing the experimental progress can be found in Physics Ref. [3].

What is new?

The large-spin cold fermion systems are fundamentally different from the large-spin solid state systems. In solids, there also exist large spin objects: Hund's rule coupling aligns spins of several electrons on the same cations forming a large spin. However, quantum fluctuations are suppressed by large spin due to the 1/S-effect. The intersite coupling is dominated by the exchange of a single pair of electrons, which suppresses quantum fluctuations as S goes larger. In contrast, this restriction does not exist in cold atom systems because each large-spin atom moves as an entire object. The exchange of a single pair of large-spin atoms can completely flip the spin configuration, which enhances quantum spin fluctuations. (See Ref. [3] for details.)

Exact Sp(4) symmetry

We proved an exact and generic hidden symmetry of Sp(4), or, isomorphically SO(5) symmetry in spin-3/2 fermion systems (e.g.,132Cs, 9Be, 135Ba, 137Ba, 201Hg) Ref.[1]. Its exactness is independent of dimensionality, lattice geometry and external potentials. Such a high symmetry without fine-tuning is rare, whose role in spin-3/2 systems is analogous to that of SU(2) symmetry in spin-1/2 systems.

This exact Sp(4), or, SO(5), symmetry provides an important guidance in studying novel large-spin quantum phases. It protects the hidden degeneracy among collective excitations of Fermi liquids Ref.[1], and gives rise to the non-Abelian defects and the SO(4) Cheshire charge in the quintet Cooper pairing superfluid Ref. [4]. It also unifies different competing orders including antiferromagnetism in different spin-tensor channels, charge-density-wave, and superconductivity Ref.[1]. This work also greatly enriches the physics of large-N quantum magnetism by providing a realistic system.

Baryon condensation and color magnetism

In spite of the huge difference of energy scales, the large-spin cold fermions can also exhibit similar physics to that in QCD -- the multi-particle clustering instabilities. With attractive interactions, Pauli's exclusion principle allows N-fermions to form an SU(N) singlet state, a ``baryon-like'' multiple-fermion instability when N>2. For the super-exchange physics in the Mott-insulating states, if each site is in the fundamental representation, it also needs $N$ sites to form an SU(N) singlet. We have performed the bosonization analysis on the competition between the baryon-type condensation and the pairing condensation in 1D systems Ref. [5] .

Counter-intuitively, magnetic fluctuations in spin-3/2 systems are even stronger than those with s = 1/2 due to the high symmetry. A four-site plaquette order without any site or bond spin orders can be stabilized. We (with S. Chen et al.) constructed an SU(4) Majumdar- Ghosh model, whose solvable ground state exhibits such order in the two-leg ladder systems Ref. [6] . The plquette SU(4) singlet state and its competition with the antiferromagnetic state and the dimerized state are studied numerically Ref. [7] . Further, a resonant 3D SU(4) plaquette phase is studied in Ref. [8] , which could be connected to the recent research of fractons.

Alice string and SO(4) Cheshire charge

The half-quantum vortex in superuids with the spin degree of freedom is an exotic topological defect as a global analogue of the Alice string in gauge theories. The half-quantum vortex loop can possess spin quantum number which is an example of the Cheshire charge phenomenon. An Abelian version of the global Alice string and Cheshire charge exists in the triplet superuid of the 3He-A phase, where the spin SU(2) symmetry is broken into the U(1) symmetry around the z-axis. A remarkable property is that both quasi-particles and spin wave excitations reverse the sign of their spin quantum numbers Sz when going through the half-quantum vortex loop. Meanwhile, the half-quantum vortex loop also changes Sz to maintain spin conservation. However, no entanglement is generated in this process.

We generalize the above picture to the non-Abelian Alice string and the topological generation of quantum entanglement through the non-Abelian Cheshire charge in spin-3/2 systems Ref. [4] . The quintet Cooper pairing order parameters in the polar basis form a 5-vector of the SO(5) symmetry group. The ground state exhibits the polar condensation where the SO(5) symmetry is broken into SO(4). This allows the half-quantum vortex loop to possess the non-Abelian SO(4) Cheshire charge, in contrast to the U(1) Cheshire charge in the 3He-A phase. We also explore the high symmetry ects on collective spin excitations and the structure of the half-quantum vortex line as a \pi-disclination in the spin channel. We show that by driving the fermion quasiparticle (or spin-wave impulse) through the half-quantum vortex loop, quantum entanglement between them is topologically generated. This effect has a potential application in the topological quantum computation.

References and talks

  • 1. Congjun Wu, Jiang-ping Hu, and Shou-cheng Zhang, "Exact SO(5) symmetry in the spin 3/2 fermionic system", Phys. Rev. Lett. 91 , 186402(2003), see pdf file .

  • 2. Congjun Wu, "Hidden symmetry and quantum phases in spin-3/2 cold atomic systems", invited review article, Mod. Phys. Lett. B 20, 1707 (2006), see pdf file.

  • 3. Congjun Wu, "Exotic many-body physics with large-spin Fermi gases" Physics 3, 92 (2010). See pdf file .

  • 4. Congjun Wu, Jiangping Hu, and Shou-Cheng Zhang, "Quintet pairing and non-Abelian vortex string in spin-3/2 cold atomic systems", Int. J. Mod. Phys. B V24, 311 (2010). See pdf file .

  • 5. Congjun Wu, "Competing orders in one dimensional spin 3/2 fermionic systems", Phys. Rev. Lett. 95, 266404 (2005), see pdf file.

  • 6. Shu Chen, Congjun Wu, Shou-Cheng Zhang, and Yupeng Wang, "Exact spontaneous plaquette ground states for spin-3/2 ladder models",
    Phys. Rev. B 72, 214428 (2005) see, pdf file.

  • 7. Hsiang-hsuan Hung, Yupeng Wang, Congjun Wu, Quantum magnetism of ultra-cold fermion systems with the symplectic symmetry, Phys. Rev. B 84, 054406 (2011) . See pdf file .

  • 8. Cenke Xu, and Congjun Wu , "Resonating plaquette phases in large spin cold atom systems" , Phys. Rev. B 77, 134449 (2008) , see pdf file .

  • Talk "Hidden symmetry and exotic quantum magnetism with large spin alkali and alkaline-earth fermions" .

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    Last modified: July 15, 2007.