Non-Abelian statistics of vortices with multiple Majorana fermions

TL;DR

This paper explores the non-Abelian exchange statistics of vortices trapping multiple Majorana fermions transforming under SO(N), revealing a decomposition into Coxeter group and single-fermion exchange parts, generalizing previous N=3 results.

Contribution

It generalizes the non-Abelian exchange statistics of vortices with multiple Majorana fermions to arbitrary odd N, showing a decomposition into Coxeter and exchange operators.

Findings

Exchange of vortices with multiple Majorana fermions is non-Abelian.

The exchange operator decomposes into Coxeter group and single-fermion parts.

Matrix representation exhibits tensor product structure.

Abstract

We consider the exchange statistics of vortices, each of which traps an odd number ($N$) of Majorana fermions. We assume that the fermions in a vortex transform in the vector representation of the SO(N) group. Exchange of two vortices turns out to be non-Abelian, and the corresponding operator is further decomposed into two parts: a part that is essentially equivalent to the exchange operator of vortices having a single Majorana fermion in each vortex, and a part representing the Coxeter group. Similar decomposition was already found in the case with N=3, and the result shown here is a generalization to the case with an arbitrary odd $N$. We can obtain the matrix representation of the exchange operators in the Hilbert space that is constructed by using Dirac fermions non-locally defined by Majorana fermions trapped in separated vortices. We also show that the decomposition of the exchange operator implies tensor product structure in its matrix representation.