Fulde-Ferrel-Larkin-Ovchinnikov phase and exponential ground state degeneracy in one-dimensional Fermi gas with attractive interactions

We examine the properties of a one-dimensional Fermi gas with attractive intrinsic (Hubbard) interactions in the presence of spin-orbit coupling and Zeeman fields. Such a system can be realized in the setting of ultracold atoms confined in a 1D optical lattice, and has been proposed to host exotic topological phases and edge modes. In absence of any external fields, this system shows a trivial Bardeen–Cooper–Schrieffer (BCS) phase. Introduction of Zeeman field takes the system to a Fulde-Ferrel-Larkin-Ovchinnikov (FFLO) phase, where the quasi-long range superconducting order co-exists with magnetic order in the system, as indicated by its pair momentum distribution. We explore the effect of spin-orbit coupling in this system. Next, we show that the addition of a smooth parabolic potential yields a phase with exponentially decaying pair binding and excitation energy gaps, which is expected to be associated with topological edge modes in the system. However, we show that this ground state degeneracy is susceptible to local impurities, and argue that the exponential splitting in the clean system is similar to a phase with only conventional order.