Majorana Fermions and Non-Abelian Statistics in Three Dimensions

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 feature of three dimensional systems is that there are “braidless” operations in which it is possible to manipulate the ground state associated with a set of defects without moving or measuring them. To illustrate these effects, we analyze specific architectures involving topological insulators and superconductors.

Figure 1

Inverse image paths depicting the interchange of two hedgehogs, as described in the text. (a)–(e) show a sequence of continuous deformations of $\mathbf{n}(\mathbf{r})$ that preserve the orientation of the hedgehogs. In (f), a $2\pi$ rotation is required to return to the original configuration.

Figure 2

Architectures demonstrating 3D Majorana states using SCs and TIs. (a) Braiding 3 around 1 can be deformed into a braidless operation. (b) A geometry for implementing braidless operations with a Josephson junction device [15]. (c), (d) Thin film geometries for interchanging and measuring Majorana states. (c) shows a layered SC-TI-SC structure, while (d) shows a thin film of a SC weakly doped TI.