Majorana meets Coxeter: Non-Abelian Majorana Fermions and Non-Abelian Statistics

TL;DR

This paper introduces a novel non-Abelian statistics for vortices with Majorana fermions, combining Abelian Majorana statistics with Coxeter groups, and explores their mathematical and geometrical structure.

Contribution

It derives a new form of non-Abelian statistics involving a tensor product of Abelian Majorana and Coxeter groups, expanding the understanding of vortex statistics.

Findings

Derived a new non-Abelian statistics combining two group structures

Represented Coxeter group actions in high-dimensional Hilbert space

Analyzed the SO(3) symmetry in a three-Majorana vortex system

Abstract

We discuss statistics of vortices having zero-energy non-Abelian Majorana fermions inside them. Considering the system of multiple non-Abelian vortices, we derive a non-Abelian statistics that differs from the previously derived non-Abelian statistics. The new non-Abelian statistics presented here is given by a tensor product of two different groups, namely the non-Abelian statistics obeyed by the Abelian Majorana fermions and the Coxeter group. The Coxeter group is a symmetric group related to the symmetry of polytopes in a high-dimensional space. As the simplest example, we consider the case in which a vortex contains three Majorana fermions that are mixed with each other under the SO(3) transformations. We concretely present the representation of the Coxeter group in our case and its geometrical expressions in the high-dimensional Hilbert space constructed from non-Abelian Majorana fermions.