Fock Parafermions and Self-Dual Representations of the Braid Group

TL;DR

This paper introduces Fock parafermions, a new class of anyons with specific exclusion and exchange statistics, and develops self-dual braid group representations relevant for topological quantum computing.

Contribution

It provides a second-quantization framework for Fock parafermions and constructs self-dual braid group representations suitable for quantum information processing.

Findings

Defined Fock parafermions with specific statistics

Developed a Fock algebra with well-defined normal-ordering

Constructed self-dual braid group representations

Abstract

We introduce and describe in second quantization a family of particle species with \(p=2,3,\dots\) exclusion and \(\theta=2\pi/p\) exchange statistics. We call these anyons Fock parafermions, because they are the particles naturally associated to the parafermionic zero-energy modes, potentially realizable in mesoscopic arrays of fractional topological insulators. Their second-quantization description entails the concept of Fock algebra, i.e., a Fock space endowed with a statistical multiplication that captures and logically correlates these anyons' exclusion and exchange statistics. As a consequence normal-ordering continues to be a well-defined operation. Because of its relevance to topological quantum information processing, we also derive families of self-dual representations of the braid group for any $p$, with the Gaussian representation being a special case. The self-dual representations can be realized in terms of local quadratic combinations of either parafermions or Fock parafermions, an important requisite for physical implementation of quantum logic gates.