Coexistence of gapless and gapped vortex modes with Majorana corner states in a 2D second-order topological superconductor

Although the appearance of vortex-localized states with zero energy in first-order topological superconductors is well known, their possibility to form in the higher-order topological phase of 2D systems has not been completely uncovered yet. Here we demonstrate the coexistence of zero-energy vortex modes and Majorana corner modes in the model of a 2D second-order topological superconductor. The model describes an interface between a normal layer supporting the topological insulating phase and a superconducting layer, for which different symmetries of the spin-singlet superconducting order parameter are considered. We show that the gapless vortex modes can appear under certain conditions in the superconducting state with a vortex if the bulk energy spectrum of the normal (non-superconducting) state is gapless and has Dirac cones. The number of pairs of such vortex modes corresponds to the number of Dirac cones. It is essential that if the normal bulk spectrum becomes gapped and the system is in the state of a topological insulator, then the zero-energy vortex modes can not be realized, while Majorana corner modes hold in the superconducting state. The interaction of the vortex modes with the edge and topological corner modes is studied when the vortex appears near the boundaries.