Abstract
We classify gapped topological superconducting (TSC) phases of one-dimensional quantum wires with local magnetic symmetries (LMSs), in which the time-reversal symmetry $\mathcal {T}$ is broken but its combinations with certain crystalline symmetry such as $M_x \mathcal {T}$, $C_{2z} \mathcal {T}$, $C_{4z}\mathcal {T}$, and $C_{6z}\mathcal {T}$ are preserved. Our results demonstrate that an equivalent BDI class TSC can be realized in the $M_x \mathcal {T}$ or $C_{2z} \mathcal {T}$ superconducting wire, which is characterized by a chiral Zc invariant. More interestingly, we also find two types of totally new TSC phases in the $C_{4z}\mathcal {T}$, and $C_{6z}\mathcal {T}$ superconducing 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 MZMs are protected at each end. In the $C_{6z}\mathcal {T}$ case, the MZMs can be either chiral or helical, and even helical-chiral coexisting. The minimal models preserving $C_{4z}\mathcal {T}$ or $C_{6z}\mathcal {T}$ symmetry are presented to illustrate their novel TSC properties and MZMs.
Entangled end states with fractionalized spin projection in a time-reversal-invariant topological superconducting wire