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Tuning interactions in the spin singlet and quintet channels of two collidingatoms could change the symmetry of the one-dimensional spin-3/2 fermionicsystems of ultracold atoms while preserving the integrability. Here we find anovel $SO(4)$ symmetry integrable point in thespin-3/2 Fermi gas and derive theexact solution of the model using the Bethe ansatz. In contrast to the modelwith $SU(4)$ and $SO(5)$ symmetries, the present model with $SO(4)$ symmetrypreserves spin singlet and quintet Cooper pairs in two sets of $SU(2)\otimesSU(2)$ spin subspaces. We obtain full phase diagrams, including theFulde-Ferrel-Larkin-Ovchinnikov like pair correlations, spin excitations andquantum criticality through the generalized Yang-Yang thermodynamic equations.In particular, various correlation functions are calculated by usingfinite-size corrections in the frame work of conformal field theory. Moreover,within the local density approximation, we further find that spin singlet andquintet pairs form subtle multiple shell structures in density profiles of thetrapped gas.

Exotic pairing in 1D spin-3/2 atomic gases with $SO(4)$ symmetry