Quasiclassical theory of disordered multi-channel Majorana quantum wires

TL;DR

This paper develops a quasiclassical approach to study disordered multi-channel Majorana quantum wires, revealing how disorder affects topological states and the formation of zero-energy anomalies related to Majorana modes.

Contribution

It introduces a quasiclassical Eilenberger method for topological nanowires with disorder, linking the Majorana number to Pfaffians of Green's functions, and confirms the robustness of Majorana states against disorder.

Findings

Disordered class D wires show zero-energy spectral peaks similar to Majorana states.

The Majorana number is expressed as a product of Pfaffians of Green's functions.

The quasiclassical approach efficiently analyzes disordered topological superconductors.

Abstract

Multi-channel spin-orbit quantum wires, when subjected to a magnetic field and proximity coupled to s-wave superconductor, may support Majorana states. We study what happens to these systems in the presence of disorder. Inspired by the widely established theoretical methods of mesoscopic superconductivity, we develop a la Eilenberger a quasiclassical approach to topological nanowires valid in the limit of strong spin-orbit coupling. We find that the "Majorana number", distinguishing between the state with Majorana fermion (symmetry class B) and no Majorana (class D), is given by the product of two Pfaffians of gapped quasiclassical Green's functions fixed by right and left terminals connected to the wire. A numerical solution of the Eilenberger equations reveals that the class D disordered quantum wires are prone to the formation of the zero-energy anomaly (class D impurity spectral peak) in the local density of states which shares the key features of the Majorana peak. In this way we confirm the robustness of our previous conclusions [Phys. Rev. Lett. 109, 227005 (2012)] on a more restrictive system setup. Generally speaking, we find that the quasiclassical approach provides a highly efficient means to address disordered class D superconductors both in the presence and absence of topological structures.