Topological superfluid in one-dimensional ultracold atomic system with spin-orbit coupling

We propose a one-dimensional Hamiltonian H1D which supports Majorana fermions when dx² − y²-wave superfluid appears in the ultracold atomic system and obtain the phase diagrams both for the time-reversal-invariant (TRI) case and time-reversal-symmetry-breaking (TRSB) case. From the phase diagrams, we find that the Majorana doublets and the single Majorana fermions exist in the topological superfluid (TSF) regions for the TRI case and the TRSB case, respectively, and we can reach these regions by tuning the chemical potential μ and spin-orbit coupling αR. Importantly, the spin-orbit coupling has been realized in ultracold atoms by the recent experimental achievement of synthetic gauge field, therefore, our one-dimensional ultra-cold atomic system described by H1D is a promising platform to find the mysterious Majorana fermions.