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Topological crystalline superconductors are known to have possiblehigher-order topology, which results in Majorana modes on $d-2$ orlower-dimensional boundaries. Given the rich possibilities of boundarysignatures, it is desirable to have topological invariants that can predict thetype of Majorana modes from band structures. Although symmetry indicators, atype of invariant that depends only on the band data at high-symmetry points,have been proposed for certain crystalline superconductors, there existsymmetry classes in which symmetry indicators fail to distinguishsuperconductors with different Majorana boundaries. Here, we systematicallyobtain topological invariants for an example of this kind, two-dimensionaltime-reversal symmetric superconductors with twofold rotational symmetry $C_2$.First, we show that the nontrivial topology is independent of band data on thehigh-symmetry points by conducting a momentum-space classification study. Then,from the resulting K groups, we derive calculable expressions for four$\mathbb{Z}_2$ invariants defined on high-symmetry lines or general points inthe Brillouin zone. Finally, together with a real-space classification study,we establish the bulk-boundary correspondence and show that the four$\mathbb{Z}_2$ invariants can predict Majorana boundary types from bandstructures. Our proposed invariants can fuel practical material searches for$C_2$-symmetric topological superconductors featuring Majorana edge and cornermodes.

Topological invariants beyond symmetry indicators: Boundary diagnostics for twofold rotationally symmetric superconductors