New types of topological superconductors under local magnetic symmetries.

Abstract

We classify gapped topological superconducting (TSC) phases of one-dimensional quantum wires with local magnetic symmetries, in which the time-reversal symmetry n}{}mathcal {T} is broken, but its combinations with certain crystalline symmetries, such as n}{}M_x mathcal {T}, n}{}C_{2z} mathcal {T}, n}{}C_{4z}mathcal {T} and n}{}C_{6z}mathcal {T}, are preserved. Our results demonstrate that an equivalent BDI class TSC can be realized in the n}{}M_x mathcal {T} or n}{}C_{2z} mathcal {T} superconducting wire, which is characterized by a chiral Zc invariant. More interestingly, we also find two types of totally new TSC phases in the n}{}C_{4z}mathcal {T} and n}{}C_{6z}mathcal {T} superinducting wires, which are beyond the known AZ class, and are characterized by a helical Zh invariant and Zh⊕Zc invariants, respectively. In the Zh TSC phase, Z pairs of Majorana zero modes (MZMs) are protected at each end. In the n}{}C_{6z}mathcal {T} case, the MZMs can be either chiral or helical, and even helical-chiral coexisting. The minimal models preserving n}{}C_{4z}mathcal {T} or n}{}C_{6z}mathcal {T} symmetry are presented to illustrate their novel TSC properties and MZMs.