Fock parafermions and selfdual representations of the braid group
Abstract
Because of potential relevance to topological quantum information processing, we introduce and study the selfdual family of representations of the braid group. Selfdual representations are physically well motivated and provide a natural generalization of the Majorana and Gaussian representations, which appear as particular instances. To show that selfdual representations admit a particle interpretation, we introduce and describe in second quantization a family of particle species with p =2,3,⋯ exclusion and θ =2π/p exchange statistics. We call these anyons Fock parafermions, because they are the particles naturally associated to the parafermionic zeroenergy modes, potentially realizable in mesoscopic arrays of fractional topological insulators. Josephson junctions modeled with Fock parafermions may display a 2πp periodic relation between the Josephson current and the phase difference across the junction. Selfdual 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. The secondquantization description of Fock parafermions 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 welldefined operation.
 Publication:

Physical Review A
 Pub Date:
 January 2014
 DOI:
 10.1103/PhysRevA.89.012328
 arXiv:
 arXiv:1307.6214
 Bibcode:
 2014PhRvA..89a2328C
 Keywords:

 03.67.Pp;
 05.30.Pr;
 11.15.q;
 Quantum error correction and other methods for protection against decoherence;
 Fractional statistics systems;
 Gauge field theories;
 Condensed Matter  Statistical Mechanics;
 Mathematical Physics;
 Quantum Physics
 EPrint:
 35 pages. Submitted to Nuclear Physics B