Majorana Fermions and Non-Abelian Statistics in Three Dimensions

TL;DR

This paper demonstrates that three-dimensional superconductors can host Majorana fermions with non-Abelian statistics through topologically nontrivial defect orientations, enabling unique groundstate manipulations without braiding.

Contribution

It introduces the concept of non-Abelian Majorana modes in 3D superconductors and describes 'braidless' operations for manipulating groundstates.

Findings

Majorana modes exist at point defects in 3D superconductors.

Non-Abelian statistics are possible despite trivial braid group in 3D.

Groundstate manipulation without moving defects is feasible.

Abstract

We show that three dimensional superconductors, described within a Bogoliubov de Gennes framework can have zero energy bound states associated with pointlike topological defects. The Majorana fermions associated with these modes have non-Abelian exchange statistics, despite the fact that the braid group is trivial in three dimensions. This can occur because the defects are associated with an orientation that can undergo topologically nontrivial rotations. A new feature of three dimensional systems is that there are "braidless" operations in which it is possible to manipulate the groundstate associated with a set of defects without moving or measuring them. To illustrate these effects we analyze specific architectures involving topological insulators and superconductors.