Topological phases of quasi-one-dimensional fermionic atoms with a synthetic gauge field

We theoretically investigate the effect of intertube tunneling in topological superfluid phases of a quasi-one-dimensional Fermi gas with a Rashba-type spin-orbit interaction. It is shown that the effective Hamiltonian is analogous to that of a nanowire topological superconductor with multibands. Using a hidden mirror symmetry in the system, we introduce a new topological number that ensures the existence of non-Abelian Majorana zero modes even in the presence of intertube tunneling. It is demonstrated from the full numerical calculation of self-consistent equations that some of Majorana modes survive against the intertube tunneling, when the number of one-dimensional tubes is odd in the $y$-direction. We also discuss a generalization of our consideration to nanowire topological superconductors.