Fock Parafermions and Self-Dual Representations of the Braid Group

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.