Benefits of Weak Disorder in One-Dimensional Topological Superconductors

Majorana bound states are zero-energy modes localized at the ends of a one-dimensional (1D) topological superconductor. Introducing disorder usually increases the Majorana localization length, until eventually inducing a topological phase transition to a trivial phase. In this Letter, we show that in some cases weak disorder causes the Majorana localization length to decrease, making the topological phase more robust. Increasing the disorder further eventually leads to a change of trend and to a phase transition to a trivial phase. Interestingly, the transition occurs at ξ_{0}≫l, where l is the disorder mean free path, and ξ_{0} is the localization length in the clean limit. Our results are particularly relevant to 1D topological superconductors formed in planar Josephson junctions.