No-go theorem for a time-reversal invariant topological phase in noninteracting systems coupled to conventional superconductors

We prove that a system of noninteracting electrons proximity coupled to a conventional s-wave superconductor cannot realize a time-reversal invariant topological phase. This is done by showing that for such a system, in either one or two dimensions, the topological invariant of the corresponding symmetry class (DIII) is always trivial. Our results suggest that the pursuit of Majorana bound states in time-reversal invariant systems should be aimed at interacting systems or at proximity to unconventional superconductors.