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Topological invariants beyond symmetry indicators: Boundary diagnostics for twofold rotationally 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. Given the rich possibilities of boundary signatures, it is desirable to have topological invariants that can predict the type of Majorana modes from band structures. Although symmetry indicators, a type of invariant that depends only on the band data at high-symmetry points, have been proposed for certain crystalline superconductors, there exist symmetry classes in which symmetry indicators fail to distinguish superconductors with different Majorana boundaries. Here, we systematically obtain topological invariants for an example of this kind, two-dimensional time-reversal symmetric superconductors with twofold rotational symmetry $C_2$. First, we show that the nontrivial topology is independent of band data on the high-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 in the 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 band structures. Our proposed invariants can fuel practical material searches for $C_2$-symmetric topological superconductors featuring Majorana edge and corner modes.