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Topological Uniform Superfluid and FFLO Phases in 3D to 1D crossover of spin-orbit coupled Fermi gases

We consider the quasi-one dimensional system realized by an array of weakly coupled parallel one-dimensional "tubes" in a two-dimensional lattice which permits free motion of atoms in an axial direction in the presence of a Zeeman field, Rashba type spin orbit coupling (SOC), and an s-wave attractive interaction, while the radial motion is tightly confined. We solve the zero-temperature (T=0) Bogoliubov-de Gennes (BdG) equations for the quasi-1D Fermi gas with the dispersion modified by tunneling between the tubes, and show that the T=0 phase diagram hosts the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase with non-zero center of mass momentum Cooper pairs for small values of the SOC while for larger values of the SOC and high Zeeman fields the uniform superfluid phase with zero center of mass momentum Cooper pairs has an instability towards the topological uniform superfluid phase with Majorana fermions at the tube ends. Also, we show that tuning the two-dimensional optical lattice strength in this model allows one to explore the crossover behaviors of the phases during the transition between the 3D and the 1D system and in general the FFLO (for small SOC) and the topological uniform superfluid phase (for large SOC) are favored as the system becomes more one-dimensional. We also find evidence of the existence of a Zeeman tuned topological quantum phase transition (TQPT) within the FFLO phase itself and for large values of the Zeeman field and small SOC the TQPT gives rise to a topologically distinct FFLO phase.