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We study the stability of topological crystalline superconductors in thesymmetry class DIIIR and in two-dimensional space when perturbed by quarticcontact interactions. It is known that no less than eight copies of helicalpairs of Majorana edge modes can be gapped out by an appropriate interactionwithout spontaneously breaking any one of the protecting symmetries. Hence, thenoninteracting classification $\mathbb{Z}$ reduces to $\mathbb{Z}^{\,}_{8}$when these interactions are present. It is also known that the stability whenthere are less than eight modes can be understood in terms of the presence oftopological obstructions in the low-energy bosonic effective theories, whichprevent opening of a gap. Here, we investigate the stability of the edgetheories with four, two, and one edge modes, respectively. We give ananalytical derivation of the topological term for the first case, because ofwhich the edge theory remains gapless. For two edge modes, we employbosonization methods to derive an effective bosonic action. When gapped, thisbosonic theory is necessarily associated to the spontaneous symmetry breakingof either one of time-reversal or reflection symmetry whenever translationsymmetry remains on the boundary. For one edge mode, stability is explicitlyestablished in the Majorana representation of the edge theory.

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