Demonstrating lattice symmetry protection in topological crystalline superconductors

Abstract

We propose to study the lattice symmetry protection of Majorana zero bound modes in topological crystalline superconductors (SCs). With an induced $s\text{\ensuremath{-}}\mathrm{wave}$ superconductivity in the $(001)$ surface of the topological crystalline insulator ${\mathrm{Pb}}_{1\ensuremath{-}x}{\mathrm{Sn}}_{x}\mathrm{Te}$, which has a ${C}_{4}$ 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 ${C}_{4}$ symmetry is broken; two Majorana modes are restored exactly one time whenever the in-plane field varies $\ensuremath{\pi}/2$, 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 ${C}_{2N}$ 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.