Controlling non-Abelian statistics of Majorana fermions in semiconductor nanowires

TLDR

A semiconductor nanowire proximate to an s‑wave superconductor can enter a topological superconducting phase that hosts zero‑energy Majorana fermions at its ends, yet demonstrating their non‑Abelian exchange statistics is hindered by the wire’s one‑dimensional topology. This work proposes a gate‑controlled tunneling scheme to move Majorana fermions between the ends of different wires. The tunneling‑induced hops exchange the Majorana fermions, yielding a non‑Abelian braid operator uniquely fixed by the microscopic tunneling parameters.

Abstract

Under appropriate external conditions a semiconductor nanowire in proximity to an $s$-wave superconductor can be in a topological superconducting (TS) phase. This phase supports localized zero-energy Majorana fermions at the ends of the wire. However, the non-Abelian exchange statistics of Majorana fermions is difficult to verify because of the one-dimensional topology of such wires. In this paper we propose a scheme to transport Majorana fermions between the ends of different wires using tunneling, which is shown to be controllable by gate voltages. Such tunneling-generated hops of Majorana fermions can be used to exchange the Majorana fermions. The exchange process thus obtained is described by a non-Abelian braid operator that is uniquely determined by the well-controlled microscopic tunneling parameters.