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We examine the existence and characteristics of the exoticFulde-Ferrel-Larkin-Ovchinnikov (FFLO) phase in a one-dimensional Fermi gaswith attractive Hubbard interactions, in the presence of spin-orbit coupling(SOC) and Zeeman field. We show that a robust FFLO phase can be created in thepresence of attractive on-site interactions and Zeeman field, and that theaddition of SOC suppresses the FFLO order and enhances the pair formation. Inabsence of SOC, the system shows four phases: Bardeen-Cooper-Schrieffer (BCS),FFLO, multi- mode pairing and fully polarized phases by tuning the Zeeman fieldh, and the quantum transition between these phases is discontinuous withrespect to h. In the presence of SOC, the transition from the BCS to FFLO phasebecomes continuous. We present a complete phase diagram of this model both inthe presence and in the absence of SOC at quarter electron filling and alsoexplore the effect of SOC on the FFLO phase.

Fulde-Ferrel-Larkin-Ovchinnikov phase in one dimensional Fermi gas with attractive interactions and transverse spin-orbit coupling