How SU(2)$_4$ Anyons are Z$_3$ Parafermions

TL;DR

This paper demonstrates that the non-abelian braiding statistics of SU(2)$_4$ anyons are equivalent to those of Z$_3$ parafermions, revealing a deep connection between these topological phases.

Contribution

It establishes an explicit equivalence between SU(2)$_4$ anyon braiding and Z$_3$ parafermion statistics, clarifying their relationship in topological quantum computation.

Findings

SU(2)$_4$ anyon braiding matches Z$_3$ parafermion statistics

The equivalence holds up to an abelian phase

Implications for quantum Hall states and topological quantum computing

Abstract

We consider the braid group representation which describes the non-abelian braiding statistics of the spin 1/2 particle world lines of an SU(2)$_4$ Chern-Simons theory. Up to an abelian phase, this is the same as the non-Abelian statistics of the elementary quasiparticles of the $k=4$ Read-Rezayi quantum Hall state. We show that these braiding statistics are identical to those of Z$_3$ Parafermions.