Resilience of Majorana fermions in the face of disorder

We elucidate the reduction of the winding number caused by the on-site disorder in a next-nearest-neighbor XY model. When disorder becomes strong enough, Majorana edge modes become critically extended, beyond which they collapse into Anderson localized (AL) states in the bulk, resulting in a topological Anderson insulating state. We identify a resilience threshold Wt for every pair of Majorana fermions (MFs). In response to increasing disorder, every pair of MFs collapse into AL states in the bulk beyond their resilience threshold. For very strong disorder, all Majorana fermions collapse, and a topologically trivial state is obtained. We show that the threshold values are related to the localization length of Majorana fermions, which can be efficiently calculated by an appropriate modification of the transfer-matrix method. At the topological transition point, the localization length of the zero modes diverges, and the system becomes scale invariant. The number of peaks in the localization length as a function of disorder strength determines the number of zero modes in the clean state before disorder is introduced. This finding elevates the transfer-matrix method to the level of a tool for the determination of the topological index of both clean and disordered systems.