Anderson localization of a Majorana fermion

TL;DR

This paper investigates how a Majorana fermion, localized at a superconductor boundary, becomes Anderson localized when the superconductor is connected to a disordered quantum wire, using a supersymmetric sigma model.

Contribution

It provides a theoretical analysis of Majorana fermion localization in disordered wires, extending understanding of topological states in complex environments.

Findings

Average local density of states calculated as a function of energy and position.

Majorana states are repelled from the zero-energy level due to disorder.

Localized states hybridize through Mott mechanisms.

Abstract

Isolated Majorana fermion states can be produced at the boundary of a topological superconductor in a quasi-one-dimensional geometry. If such a superconductor is connected to a disordered quantum wire, the Majorana fermion is spread into the wire, subject to Anderson localization. We study this effect in the limit of a thick wire with broken time-reversal and spin-rotational symmetries. With the use of a supersymmetric nonlinear sigma model, we calculate the average local density of states in the wire as a function of energy and of the distance from the interface with the superconductor. Our results may be qualitatively explained by the repulsion of states from the Majorana level and by Mott hybridization of localized states.