Adiabatic manipulations of Majorana fermions in a three-dimensional network of quantum wires

TL;DR

This paper explores how to manipulate and track Majorana fermions in a three-dimensional network of quantum wires, extending previous two-dimensional proposals to enable more complex quantum operations.

Contribution

It introduces methods for adiabatic manipulation and sign-tracking of Majorana states in 3D wire networks, enhancing control over their non-Abelian properties.

Findings

Majorana states can be manipulated via external gates and physical reconfigurations.

Sign changes in Majorana states can be tracked by monitoring individual states.

The approach simplifies understanding of Majorana braiding in complex networks.

Abstract

It has been proposed that localized zero-energy Majorana states can be realized in a two-dimensional network of quasi-one-dimensional semiconductor wires that are proximity-coupled to a bulk superconductor. The wires should have strong spin-orbit coupling with appropriate symmetry, and their electrons should be partially polarized by a strong Zeeman field. Then, if the Fermi level is in an appropriate range, the wire can be in a topological superconducting phase, with Majorana states that occur at wire ends and at $Y$ junctions, where three topological superconductor segments may be joined. Here we generalize these ideas to consider a three-dimensional network. The positions of Majorana states can be manipulated, and their non-Abelian properties made visible, by using external gates to selectively deplete portions of the network, or by physically connecting and redividing wire segments. Majorana states can also be manipulated by reorientations of the Zeeman field on a wire segment, by physically rotating the wire about almost any axis, or by evolution of the phase of the order parameter in the proximity-coupled superconductor. We show how to keep track of sign changes in the zero-energy Hilbert space during adiabatic manipulations by monitoring the evolution of each Majorana state separately, rather than keeping track of the braiding of all possible pairs. This has conceptual advantages in the case of a three-dimensional network, and may be computationally useful even in two dimensions, if large numbers of Majorana sites are involved.