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$_{1-x}$Sn$_x$Te, which has a C$_4$ rotational symmetry, we show a new class of 2D topological SC 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 four-fold symmetry and lifts the Majorana modes to finite energy states in general. Surprisingly, we show that even the C$_4$ 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 novel phenomenon has a profound connection to the four-fold 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$_{2N}$ 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 external symmetry-breaking field varies one cycle in direction.