Anyon exclusions statistics on surfaces with gapped boundaries

TL;DR

This paper extends anyon exclusion statistics to surfaces with gapped boundaries, introducing mutual exclusion parameters and clarifying the role of pseudo-species in non-Abelian anyon systems, relevant for topological quantum computing.

Contribution

It proposes a new framework for anyon exclusion statistics on bounded surfaces, including a systematic basis construction and a formula for statistical weights that incorporates boundaries and pseudo-species.

Findings

Developed a formula for statistical weights with boundaries

Provided a basis construction for non-Abelian anyons on surfaces

Clarified the meaning of pseudo-species as excitation modes

Abstract

An anyon exclusion statistics, which generalizes the Bose-Einstein and Fermi-Dirac statistics of bosons and fermions, was proposed by Haldane[1]. The relevant past studies had considered only anyon systems without any physical boundary but boundaries often appear in real-life materials. When fusion of anyons is involved, certain `pseudo-species' anyons appear in the exotic statistical weights of non-Abelian anyon systems; however, the meaning and significance of pseudo-species remains an open problem. In this paper, we propose an extended anyon exclusion statistics on surfaces with gapped boundaries, introducing mutual exclusion statistics between anyons as well as the boundary components. Motivated by Refs. [2, 3], we present a formula for the statistical weight of many-anyon states obeying the proposed statistics. We develop a systematic basis construction for non-Abelian anyons on any Riemann surfaces with gapped boundaries. From the basis construction, we have a standard way to read off a canonical set of statistics parameters and hence write down the extended statistical weight of the anyon system being studied. The basis construction reveals the meaning of pseudo-species. A pseudo-species has different `excitation' modes, each corresponding to an anyon species. The `excitation' modes of pseudo-species corresponds to good quantum numbers of subsystems of a non-Abelian anyon system. This is important because often (e.g., in topological quantum computing) we may be concerned about only the entanglement between such subsystems.