  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.
