First-, second- and third-order topological superconductivity protected by point-group symmetries

The rich variety of crystalline symmetries of solids greatly enhance tenfold-way classes. Topological crystalline superconductors (TSC) are potential to host first-, second- and third-order topological states with protected surface, edge, and corner Majorana gapless modes. In this work, we establish a complete classification of first-, second- and third-order anomalous surface state in DIII class by stacking multiple copies of first-order states with different winding numbers on suitably chosen surface geometries. We find that under the given symmetry setting of normal state and of the pairing order parameter, there is a global obstruction to achieve a fully gapped surface, which leads to helical Majorana mid-gap modes that propagate along 1-D curves or are localized at some points on the surface. We analyze cases in all 32 point groups and give the "root states" that generate such anomalous surface state upon stacking.