New classification of topological superconductors in 1D magnetic groups

Topological superconductors (TSCs) with Majorana zero modes (MZMs) and their potential application in topological quantum computation have become an important area in condensed matter physics. Their topological classification with crystalline symmetries is an important and useful question for finding the possible TSC materials. We have classified the TSC phase in 1D system with local crystalline time reversal symmetry (LCTRS), where the time reversal symmetry (TRS) is broken, but the combination of TRS with some local crystalline symmetry is preserved. The LCTRS M_xT or C_2zT will lead to an effective BDI symmetry class which has a Z classification. The case of LCTRS C_4zT (C_6zT) are not included in the Altland and Zirnbauer (AZ) symmetry classes, and exhibit the Z (Z + Z) classification TSC phase with the robust MZMs protected by the unitary chiral symmetry, which is different from the previously studied TSCs in BDI, DIII and D classes. We also present a minimal BdG Hamiltonian with C₄T or C₆T symmetry to illustrate the novel TSC phase.