Characterization of higher-order topological superconductors using Bott indices

The abundance of bulk and boundary topologies in higher-order topological phases offer remarkable tunability and diversity to boundary states but also pose a challenge to their unified topological characterization. In this work, we propose a theoretical framework to characterize time-reversal invariant topological superconductors hosting Majorana Kramers pairs (MKP) of corner states by using a series of spin Bott indices, which capture both bulk and boundary states topology. The developed invariants can characterize MKP in arbitrarily shaped systems and all distinct spatial distribution patterns of MKP. As an illustrative example, we apply our theory to analyze the Kane-Mele model with sublattice-dependent superconducting pairing potentials. In this representative model, both intrinsic and extrinsic higher-order topological superconductors can be realized and various patterns of MKP can be engineered through edge cleavage. Despite their high sensitivity to boundary terminations, MKP can be faithfully characterized by the proposed topological invariants. We further demonstrate the characterization of higher-order topological superconductors in the BDI symmetry class using Bott indices without resolving the spin degree of freedom.