New types of topological superconductors under local magnetic symmetries

TL;DR

This paper classifies new topological superconducting phases in one-dimensional wires with local magnetic symmetries, revealing novel phases characterized by unique invariants and Majorana zero modes beyond known classifications.

Contribution

It introduces and characterizes new topological superconductor phases under local magnetic symmetries, expanding the classification beyond the AZ classes with novel invariants and Majorana modes.

Findings

Equivalent BDI class TSC realized in $M_x T$ or $C_{2z} T$ wires

Discovery of new TSC phases with helical $Z^h$ and combined invariants

Majorana zero modes can be chiral, helical, or coexisting in these phases

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 $Z^c$ 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 $Z^h$ invariant and $Z^h\oplus Z^c$ invariants, respectively. In the $Z^h$ 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.