  Understanding novel pairings in attractive degenerate Fermi gases is crucial
for exploring rich superfluid physics. In this report, we reveal unconventional
pairings induced by spin-orbit coupling (SOC) in a one-dimensional optical
lattice, using a state-of-the-art density-matrix renormalization group method.
When both bands are partially occupied, we find a strong competition between
the interband Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) and intraband
Bardeen-Cooper-Schrieffer (BCS) pairings. In particular, for the weak and
moderate SOC strengths, these two pairings can coexist, giving rise to a new
phase called the FFLO-BCS phase, which exhibits a unique three-peak structure
in pairing momentum distribution. For the strong SOC strength, the intraband
BCS pairing always dominates in the whole parameter regime, including the half
filling. We figure out the whole phase diagrams as functions of filling factor,
SOC strength, and Zeeman field. Our results are qualitatively different from
recent mean-field predictions. Finally, we address that our predictions could
be observed in a weaker trapped potential.
