Equivalence of topological mirror and chiral superconductivity in one dimension

TL;DR

This paper demonstrates the exact equivalence between topological mirror superconductivity and chiral topological superconductivity in 1D, linking different symmetry classes and providing insights into Majorana modes in TR-invariant systems.

Contribution

It establishes a formal equivalence between topological mirror and chiral superconductivity in 1D, clarifying their relationship within the BDI symmetry class and the Altland-Zirnbauer classification.

Findings

Mirror Berry phase equals chiral winding invariant in BDI class

Topological mirror superconductivity is a special case of BDI class

Examples include spin-orbit coupled wires and ferromagnetic atom chains

Abstract

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 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 co-existing 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_{\pm} superconductivity and the recently discussed experimental system of ferromagnetic atom (Fe) chains embedded on a lead (Pb) superconductor.