Higher-order topological superconductors characterized by Fermi level crossings

We demonstrate that level crossings at the Fermi energy serve as robust indicators for higher-order topology in two-dimensional superconductors of symmetry class D. These crossings occur when the boundary condition in one direction is continuously varied from periodic to open, revealing the topological distinction between opposite edges. The associated Majorana numbers acquire nontrivial values whenever the system supports two Majorana zero modes distributed at its corners. Owing to their immunity to perturbations that break crystalline symmetries, Fermi level crossings are able to characterize a wide range of higher-order topological superconductors. By directly identifying the level-crossing points from the bulk Hamiltonian, we establish the correspondence between gapped bulk and Majorana corner states in higher-order phases. In the end, we illustrate this correspondence using two toy models. Our findings suggest that Fermi level crossings offer a possible avenue for characterizing higher-order topological superconductors in a unifying framework.