The Double-Edged Sword of Weak Disorder in Topological Superconductivity

Majorana bound states are zero-energy modes localized at the ends of a topological superconductor. As long as their localization length is smaller than their spatial separation, they are protected aginst splitting from zero energy. Introducing disorder usually tends to increase the Majorana localization length, until eventually inducing a topological phase transition to a trivial phase. In this work we show that in some cases weak disorder actually causes the Majorana localization length to decrease, making the topological phase more robust. Increaseing the disorder further eventually leads to a change of trend and to a phase transition to a trivial phase, however, interestingly the transition occurs at τ^-1>> E_g, where τ is the disorder mean-free time and E_g is the bulk energy gap. Our results are particularly relevant to a recent experiment [S. Hart et al. Nat. Phys. 13, 87-93 (2017)] in planar Josephson junctions, a setup predicted to host Majorana bound states [F. Pientka et al. Phys. Rev. X 7, 021032 (2017), M. Hell et al. Phys. Rev. Lett. 118, 107701 (2017)].