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

TL;DR

This paper explores the coexistence of gapless vortex modes and Majorana corner states in a 2D second-order topological superconductor, revealing conditions for their emergence and interactions.

Contribution

It demonstrates the simultaneous presence of vortex and Majorana corner modes in a 2D second-order topological superconductor model, highlighting the role of the normal state's gapless spectrum.

Findings

Gapless vortex modes appear with Dirac cones in the normal state.

Majorana corner modes exist when the bulk is gapped and topologically nontrivial.

Vortex modes interact with edge and corner states near boundaries.

Abstract

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.