Double Majorana vortex zero modes in superconducting topological crystalline insulators with surface rotation anomaly

Multiple Majorana vortex zero modes can emerge when the parent material in proximity to an s-wave superconductor is a 3D topological crystalline insulator (TCI) with surface Dirac cones protected by crystal symmetry. Thus far a variety of TCIs have been predicted theoretically, while little has been known about Majorana vortex zero modes in superconducting TCIs, especially for newly-proposed TCIs having surface rotation anomaly, which realize a new class of topological surface states involving multiple Dirac cones. We show that the proximity-induced s-wave superconductivity on the surface of
these TCIs yields a topological superconducting phase in which two Majorana zero modes are bound to a vortex, and that n-fold rotation symmetry (n = 2, 4, 6) enriches the topological classification of a superconducting vortex from Z₂ to Z₂ ×Z₂. Using a model of a three-dimensional high-spin topological insulator with s-wave superconductivity and two-fold rotation symmetry, we show that, with increasing chemical potential, the number of Majorana zero modes at one end of a vortex changes as 2→1→0 through two topological vortex phase transitions. In addition, we show that additional magnetic-mirror symmetry further enhances the topological classification to Z×Z.