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.
Real fermion modes, impurity entropy, and nontrivial fixed points in the phase diagram of junctions of interacting quantum wires and topological superconductors
