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Recently it has been shown that multicomponent spin-orbit-coupled fermions inone-dimensional optical lattices can be viewed as spinless fermions moving intwo-dimensional synthetic lattices with synthetic magnetic flux. The quantumHall edge states in these systems have been observed in recent experiments. Inthis paper we study the effect of an attractive Hubbard interaction. Since theHubbard interaction is long-range in the synthetic dimension, it is able toefficiently induce Cooper pairing between the counterpropagating chiral edgestates. The topological class of the resultant one-dimensional superfluid isdetermined by the parity (even/odd) of the Chern number in the two-dimensionalsynthetic lattice. We also show the presence of a chiral symmetry in our model,which implies ${\rm Z}$ classification and the robustness of multiple zeromodes when this symmetry is unbroken.

Topological Superfluid and Majorana Zero Modes in Synthetic Dimension