Optical traps and lattices provide a
new opportunity to study strongly correlated high spin systems with cold
atoms. In this article, we review the recent progress on the hidden symmetry
properties in the simplest high spin fermionic systems with hyperfine spin
F=3/2, which may be realized with atoms of 132Cs, 9Be, 135Ba, 137Ba, and
201Hg. A generic SO(5) or isomorphically, Sp(4) symmetry is proved in such
systems with the s-wave scattering interactions in optical traps, or with the
on-site Hubbard interactions in optical lattices. Various important features
from this high symmetry are studied in the Fermi liquid theory, the mean field
phase diagram, and the sign problem in quantum Monte-Carlo simulations. In the
s-wave quintet Cooper pairing phase, the half-quantum vortex exhibits the
global analogue of the Alice string and non-Abelian Cheshire charge properties
in gauge theories. The existence of the quartetting phase, a four-fermion
counterpart of the Cooper pairing phase, and its competition with other orders
are studied in one-dimensional spin-3/2 systems. We also show that
counter-intuitively quantum fluctuations in spin-3/2 magnetic systems are even
stronger than those in spin-1/2 systems.
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