Direct demonstration of bulk-boundary correspondence in higher-order topological superconductors with chiral symmetry

A higher-order topological superconductor can experience topological phase transitions driven by variations in a bulk parameter without closing the bulk gap. This presents a challenge in establishing a direct bulk-boundary correspondence, as conventional bulk invariants change only upon the closure of the bulk gap. Our study of two-dimensional higher-order phases in the DIII and BDI symmetry classes, both characterized by chiral symmetry, demonstrates that zero-energy crossings facilitate a direct connection between the bulk Hamiltonian and Majorana zero modes at corners. These crossings, emerging as boundary conditions vary, can be identified from the bulk Hamiltonian. For both classes, we introduce a pair of topological invariants derived from these zero-energy crossings to characterize the higher-order topology. Phases in which at least one invariant assumes a nonzero value are anticipated to host Majorana corner modes. Moreover, these invariants may change with the closure of either bulk or edge gaps, thereby providing a clear and direct demonstration of bulk-boundary correspondence in higher-order phases. Our findings offer a promising framework for systematically exploring higher-order topology through boundary condition modulation.