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We analyze the topological properties of a chiral ${p}+i{p}$ superconductorfor a two-dimensional metal/semimetal with four Dirac points. Such a system hasbeen proposed to realize second-order topological superconductivity and hostcorner Majorana modes. We show that with an additional $\mathsf{C}_4$rotational symmetry, the system is in an intrinsic higher-order topologicalsuperconductor phase, and with a lower and more natural $\mathsf{C}_2$symmetry, is in a boundary-obstructed topological superconductor phase. Theboundary topological obstruction is protected by a bulk Wannier gap. However,we show that the well-known nested-Wilson loop is in general unquantizeddespite the particle-hole symmetry, and thus fails as a topological invariant.Instead, we show that the higher-order topology and boundary-obstructedtopology can be characterized using an alternative defect classificationapproach, in which the corners of a finite sample is treated as a defect of aspace-filling Hamiltonian. We establish "Dirac+$({p}+i{p})$" as a sufficientcondition for second-order topological superconductivity.

Chiral Dirac Superconductors: Second-order and Boundary-obstructed Topology