Equivalence of topological mirror superconductivity and chiral superconductivity in one dimension

Recently it has been proposed that a unitary topological mirror symmetry can stabilize multiple zero energy Majorana fermion modes in one-dimensional (1D) time-reversal (TR) invariant topological superconductors. Here we establish an exact equivalence between 1D "topological mirror superconductivity" and chiral topological superconductivity in the BDI class which can also stabilize multiple Majorana-Kramers pairs in 1D TR invariant topological superconductors. The equivalence proves that topological mirror superconductivity can be understood as chiral superconductivity in the BDI symmetry class coexisting with time-reversal symmetry. Furthermore, we show that the mirror Berry phase coincides with the chiral winding invariant of the BDI symmetry class, which is independent of the presence of the time-reversal symmetry. Thus, the time-reversal invariant topological mirror superconducting state may be viewed as a special case of the BDI symmetry class in the well-known Altland-Zirnbauer periodic table of free fermionic phases. We illustrate the results with the examples of 1D spin-orbit coupled quantum wires in the presence of nodeless s± superconductivity and the recently discussed experimental system of ferromagnetic atom (Fe) chains embedded on a lead (Pb) superconductor.