Stability against contact interactions of a topological superconductor in two-dimensional space protected by time-reversal and reflection symmetries

TL;DR

This paper investigates the stability of two-dimensional topological crystalline superconductors with various numbers of edge modes under contact interactions, revealing conditions for gap opening and symmetry breaking.

Contribution

It provides an analytical derivation of topological terms for edge theories with different modes and clarifies the conditions under which edge states remain gapless or become gapped.

Findings

Eight edge modes can be gapped without symmetry breaking.

Two edge modes can be gapped via spontaneous symmetry breaking.

One edge mode remains stable against interactions.

Abstract

We study the stability of topological crystalline superconductors in the symmetry class DIIIR and in two-dimensional space when perturbed by quartic contact interactions. It is known that no less than eight copies of helical pairs of Majorana edge modes can be gapped out by an appropriate interaction without spontaneously breaking any one of the protecting symmetries. Hence, the noninteracting classification $\mathbb{Z}$ reduces to $\mathbb{Z}^{\,}_{8}$ when these interactions are present. It is also known that the stability when there are less than eight modes can be understood in terms of the presence of topological obstructions in the low-energy bosonic effective theories, which prevent opening of a gap. Here, we investigate the stability of the edge theories with four, two, and one edge modes, respectively. We give an analytical derivation of the topological term for the first case, because of which the edge theory remains gapless. For two edge modes, we employ bosonization methods to derive an effective bosonic action. When gapped, this bosonic theory is necessarily associated to the spontaneous symmetry breaking of either one of time-reversal or reflection symmetry whenever translation symmetry remains on the boundary. For one edge mode, stability is explicitly established in the Majorana representation of the edge theory.