Majorana Fermions in Quasi-One-Dimensional and Higher-Dimensional Ultracold Optical Lattices

We study Majorana fermions (MFs) in quasi-one dimensional (quasi-1D) and higher-dimensional fermionic optical lattices with a strictly 1D spin-orbit coupling, which has already been realized in cold atom experiments. We show that when the superfluid order parameters are homogeneous and are enforced to be identical along different chains, there are multiple MFs at each end with or without an experimentally tunable in-plane Zeeman field V{y}. For V{y} = 0 the multiple MFs are topologically protected by a chiral symmetry; however, for V{y} ≠ 0 the existence of multiple MFs is related to the peculiar spectrum properties of the system despite the broken chiral symmetry. In the generalization to higher dimensions, the multiple MFs form a zero-energy flat band. Furthermore, when the superfluid order parameters are solved self-consistently, the multiple MFs are usually destroyed because of the inhomogeneous order parameters of either Bardeen-Cooper-Schrieffer (V{y} = 0) type or Fulde-Ferrell (V{y} ≠ 0). Our results are useful to guide the experimentalists on searching for MFs in ultracold spin-orbit coupled fermionic superfluids.