First- and second-order topological phases in a superconducting multiorbital model*

We investigate the topological phases that appear in an orbital version of the Benalcazar-Bernevig-Hughes (BBH) model in the presence of conventional $s$-wave spin-singlet superconductivity and an in-plane magnetic field. We chart out the phase diagram in the magnetic field vs pairing amplitude plane by considering various boundary conditions, while the topology of the individual phases is examined by considering the Wannier and entanglement spectrum, as well as the Majorana polarization. For small to moderate values of magnetic field and superconducting pairing amplitude, we find a second-order topological superconductor phase with eight zero-energy corner modes. In addition, we find nodal as well as nodeless flat bands, localized only along the mirror-symmetric open edges, with momentum parallel to the mirror symmetry broken direction, which is considered to be the first-order topological superconducting phase. The former arises in a small momentum range when the magnetic field amplitude is stronger than the superconducting pairing amplitude and vice versa. On the other hand, nodeless flat bands span across the whole Brillouin zone when the magnetic field is comparable to the superconducting pairing amplitude. We further find a region on the phase diagram sandwiched between the first-order flat bands and second-order corner modes with a hybrid phase where both corner and partial edge localizations are observed. The half-quantization of Wannier spectrum and mid-gap distribution of eigenvalues in the entanglement spectrum allows us to distinguish between the first-order, second-order, and hybrid topological phases.