Chiral Dirac superconductors: Second-order and boundary-obstructed topology

We analyze the topological properties of a chiral p + ip superconductor for a two-dimensional metal and semimetal with four Dirac points. Such a system has been proposed to realize second-order topological superconductivity and host corner Majorana modes. We show that with an additional C-4 rotational symmetry, the system is in an intrinsic higher-order topological superconductor phase, and with a lower C-2 symmetry, is in a boundary-obstructed topological superconductor phase. The boundary topological obstruction is protected by a bulk Wannier gap. However, we show that the well-known nested Wilson loop is in general unquantized despite the particle-hole symmetry, and thus fails as a topological invariant. Instead, we show that the higher-order topology and boundary-obstructed topology can be characterized using an alternative defect classification approach, in which the corners of a finite sample are treated as a defect of a space-filling Hamiltonian. We establish Dirac+(p + ip) as a sufficient condition for second-order topological superconductivity.