Demonstrating lattice symmetry protection in topological crystalline superconductors

We propose to study the lattice symmetry protection of Majorana zero bound modes in topological crystalline superconductors (SCs). With an induced s- wave superconductivity in the (001) surface of the topological crystalline insulator Pb sub(1-x) Sn sub(x) Te, which has a C sub(4) rotational symmetry, we show a class of two-dimensional topological SCs with four Majorana modes obtained in each vortex core, while only two of them are protected by the cyclic symmetry. Furthermore, applying an in-plane external field can break the fourfold symmetry and lifts the Majorana modes to finite energy states in general. Surprisingly, we show that even the C sub(4) symmetry is broken; two Majorana modes are restored exactly one time whenever the in-plane field varies [pi]/2, i.e., 1/4-cycle in the direction. This phenomenon has a profound connection to the fourfold cyclic symmetry of the original crystalline SC and uniquely demonstrates the lattice-symmetry protection of the Majorana modes. We further generalize these results to the system with generic C sub(2N) symmetry, and show that the symmetry class of the topological crystalline SC can be demonstrated by the 2N times of restoration of two Majorana modes when the direction of the external symmetry-breaking field varies one cycle.