Surface-dependent Majorana vortex phases in topological crystalline insulators

The topological crystalline insulator SnTe exhibits surface-dependent Dirac cones, which are located at non-time-reversal-invariant momenta on the (001) and (110) surfaces, but at time-reversal-invariant momenta on the (111) surface. Motivated by the recent experimental evidence of Majorana vortex end modes (MVEMs) and their hybridization on the (001) surface [Liu et al., Nature (London) 633, 71 (2024)], we present a comprehensive investigation of Majorana vortex phases in SnTe with proximity-induced superconductivity, including topological classification, surface-state Hamiltonians analysis, and lattice model calculations. By utilizing rotational and magnetic mirror symmetries, we present two equivalent methods to reveal the topology of Majorana phases on different surfaces. We find that the MVEMs on the (001) and (110) surfaces are protected by both magnetic group and rotational symmetries. In contrast, the MVEMs on the (111) surface are protected by magnetic group or particle-hole symmetry. Due to the different properties of Dirac fermions in the $\overline{\mathrm{\ensuremath{\Gamma}}}$ and $\overline{M}$ valleys on the (111) surfaces, including Fermi velocities and energy levels, we find that abundant vortex phase transitions can occur for the [111]-direction vortex. As the chemical potential shifts from the surface to bulk states, the number of robust MVEMs can change from $2\ensuremath{\rightarrow}1\ensuremath{\rightarrow}0$. These vortex transitions are characterized by both $Z$ winding number and ${Z}_{2}$ pfaffian topological invariants.