Fock Parafermions and Self-Dual Representations of the Braid Group

Abstract

We introduce and describe in second quantization a family of particle species with \(p=2,3,\dots\) exclusion and \(θ=2π/p\) 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.

35 pages. Submitted to Nuclear Physics B

References in corpus (9)

• Non-Abelian Anyons and Topological Quantum Computation
• Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices
• Evidence of Majorana fermions in an Al - InAs nanowire topological superconductor
• Exotic non-Abelian anyons from conventional fractional quantum Hall states
• Fractionalizing Majorana fermions: non-abelian statistics on the edges of abelian quantum Hall states
• Superconducting Proximity Effect on the Edge of Fractional Topological Insulators
• Fractional topological superconductors with fractionalized Majorana fermions
• Qudit surface codes and gauge theory with finite cyclic groups
• Arbitrary Dimensional Majorana Dualities and Network Architectures for Topological Matter

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