Stability of the boundary zero modes in one-dimensional topological superconductors

We calculate the spectrum of the Andreev bound states in a one-dimensional superconductor with a strong Rashba spin-orbit coupling. We focus on the fate of the zero-energy Andreev modes in the presence of time reversal symmetry-breaking perturbations, both at the boundary and in the bulk. It is shown that the zero modes are destroyed by time reversal symmetry-breaking fluctuations, even if the mean-field state of the system is time-reversal invariant and topologically nontrivial.