Non-Abelian statistics of vortices with non-Abelian Dirac fermions

TL;DR

This paper demonstrates that vortices trapping two Dirac fermions with U(2) symmetry exhibit non-Abelian exchange statistics, extending previous single-fermion results and revealing genuine non-Abelian behavior among identical vortices.

Contribution

It extends the analysis of vortex exchange statistics to systems with doublet Dirac fermions, showing non-Abelian braid group representations and identifying genuine non-Abelian statistics among identical vortices.

Findings

Vortices with doublet Dirac fermions exhibit non-Abelian braid group representations.

The results include and generalize previous single-fermion vortex statistics.

Genuine non-Abelian statistics are found among identical vortices with the same number of Dirac fermions.

Abstract

We extend our previous analysis on the exchange statistics of vortices having a single Dirac fermion trapped in each core, to the case where vortices trap two Dirac fermions with U(2) symmetry. Such a system of vortices with non-Abelian Dirac fermions appears in color superconductors at extremely high densities, and in supersymmetric QCD. We show that the exchange of two vortices having doublet Dirac fermions in each core is expressed by non-Abelian representations of a braid group, which is explicitly verified in the matrix representation of the exchange operators when the number of vortices is up to four. We find that the result contains the matrices previously obtained for the vortices with a single Dirac fermion in each core as a special case. The whole braid group does not immediately imply non-Abelian statistics of identical particles because it also contains exchanges between vortices with different numbers of Dirac fermions. However, we find that it does contain, as its subgroup, a genuine non-Abelian statistics for the exchange of the identical particles, that is, vortices with the same number of Dirac fermions. This result is surprising compared with conventional understanding because all Dirac fermions are defined locally at each vortex, unlike the case of Majorana fermions for which Dirac fermions are defined non-locally by Majorana fermions located at two spatially separated vortices.