We theoretically investigate in-medium two- and three-body correlations in one-dimensional two-component Fermi gases with coexistent even-parity
s-wave and odd-parity
p-wave interactions. We find the solutions of the stable in-medium three-body cluster states such as the Cooper triple by solving the corresponding in-medium variational equations. We further feature a phase diagram consisting of the
s- and
p-wave Cooper pairing phases, and Cooper tripling phase, in a plane of
s- and
p-wave pairing strengths. The Cooper tripling phase dominates over the pairing phases when both
s- and
p-wave interactions are moderately strong.
Novel competing orders are found in spin-3/2 cold atomic systems in one-dimensional optical traps and lattices. In particular, the quartetting phase, a four-fermion counterpart of Cooper pairing, exists in a large portion of the phase diagram. The transition between the quartetting and singlet Cooper pairing phases is controlled by an Ising symmetry breaking in one of the spin channels. The singlet Cooper pairing phase also survives in the purely repulsive interaction regime. In addition, various charge and bond ordered phases are identified at commensurate fillings in lattice systems.
The Cooper pair eigenvalue equation is solved exactly for the free-electron density of states in one, two, and three dimensions. The two-dimensional (2-D) case is essentially the original derivation of Cooper, as is the three-dimensional case if the interaction is restricted to the vicinity of the Fermi surface. The essential singularity in coupling, familiar in 2-D quantum binding in a well, and sometimes erroneously attributed to the two-dimensional character of the BCS model interaction which is nonzero only very near the Fermi surface, appears in all three cases.