Boundary-diagnosing topological invariants beyond symmetry indicators: A case study of two-fold rotational symmetric superconductors

Topological crystalline superconductors are known to have possible higher-order topology, which results in Majorana modes on $d-2$ or lower dimensional boundaries. It is desirable to have topological invariants that can predict Majorana boundary types from band structures. Although symmetry indicators have been proposed for certain crystalline superconductors, there exist symmetry classes where symmetry indicators fail to distinguish superconductors with different Majorana boundaries. In this talk, I will focus on an example of this kind, the 2-D time-reversal symmetric superconductors with two-fold rotational symmetry, for which we systematically obtain topological invariants. First I will discuss a momentum-space classification study, which shows that the nontrivial topology is independent of band data on the high-symmetry points and leads to four $\mathbb{Z}_2$ invariants defined on the high-symmetry lines or general points in the Brillouin zone. Then, with the aid of a real-space classification study, I will show that these invariants can predict Majorana boundary types from band structures.