Interacting Majorana modes at surfaces of noncentrosymmetric superconductors

Noncentrosymmetric superconductors with line nodes are expected to possess topologically protected flat zero-energy bands of surface states, which can be described as Majorana modes. We here investigate their fate if residual interactions beyond BCS theory are included. For a minimal square-lattice model with a plaquette interaction, we find string-like integrals of motion that form Clifford algebras and lead to exact degeneracies. These degeneracies strongly depend on whether the numbers of sites in the $x$ and $y$ directions are even or odd, and are robust against disorder in the interactions. We show that the mapping of the Majorana model onto two decoupled spin compass models [Kamiya et al., Phys. Rev. B 98, 161409 (2018)] and extra spectator degrees of freedom only works for open boundary conditions. The mapping shows that the three-leg and four-leg Majorana ladders are integrable, while systems of larger width are not. In addition, the mapping maximally reduces the effort for exact diagonalization, which is utilized to obtain the gap above the ground states. We find that this gap remains open if one dimension is kept constant and even, while the other is sent to infinity, at least if that dimension is odd. Moreover, we compare the topological properties of the interacting Majorana model to those of the toric-code model. The Majorana model has long-range entangled ground states that differ by $mathbb{Z}_2$ fluxes through the system on a torus. The ground states exhibit string condensation similar to the toric code but the topological order is not robust. While the spectrum is gapped - due to spontaneous symmetry breaking inherited from the compass models - states with different values of the $mathbb{Z}_2$ fluxes end up in the ground-state sector in the thermodynamic limit. Hence, the gap does not protect these fluxes against weak perturbations.