Resilience of Majorana Fermions in the face of Disorder

TL;DR

This paper investigates how disorder affects Majorana fermions in a topological model, identifying thresholds for their collapse and linking localization length to topological resilience, thereby advancing methods to determine topological indices in disordered systems.

Contribution

It introduces a transfer matrix-based approach to quantify Majorana fermion resilience thresholds and relates localization length behavior to topological phase transitions in disordered systems.

Findings

Majorana edge modes become critically extended before collapsing into Anderson localized states.

Resilience thresholds are related to the localization length of Majorana fermions.

The transfer matrix method can determine the topological index in disordered systems.

Abstract

We elucidate the reduction of the winding number (WN) caused by the onsite disorder in a higher WN 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 (TAI). We identify a resilience threshold $W_t$ for every pair of Majorana fermions (MFs). In response to increasing disorder every pair of MFs collapse into AL bulk at 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 deeply 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, localization length of the zero modes diverges and the system becomes scale invariant. The number of peaks in the localization length as the 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 determination of the topological index of both clean and disordered systems.