Competing Orders in One-Dimensional Spin-$3/2$ Fermionic Systems

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.

Figure 1

RG flows in the spin channels. Various phases are determined as: (A) the gapless Luttinger liquid phase; (B) the quartetting phase with QLRO superfluidity at $K_c>2$ or $2k_f$ CDW at $K_c<2$; (C) the singlet pairing phase with QLRO superfluidity at $K_c<1/2$ or $4k_f$ CDW at $K_c>1/2$. They are controlled by the FPs of (0,0), $(+\infty ,+\infty )$ (line 4), and $(-\infty ,+\infty )$ (line 5), respectively. Phase boundaries (lines 1, 2, 3) are marked with dashed lines.

Figure 2

Phase diagram in terms of $g_0$ and $g_2$. We also show the charge and bond ordered states at quarter filling in lattice systems. Phase (A) is the $SU(4)$ gapless spin liquid. Both phase (B) and (C) split into two parts. With $g_u\to +\infty$, (B.1) represents quartets with both bond and charge orders; (C.1) is dimerization of spin Peierls order. With $g_u\to -\infty$, (B.2) is CDW phases of quartets and (C.2) is CDW of singlet Cooper pairs. Boundaries among (A), (B), and (C) are marked with solid lines, and those between (B.1) and (B.2), and (C.1) and (C.2) are sketched with dashed lines.