Anyon exclusions statistics on surfaces with gapped boundaries

Anyon exclusion statistics, proposed by Haldane, generalizes the Bose-Einstein and Fermi-Dirac statistics. When fusion of anyons is involved, certain ‘pseudo-species’ anyons appear in the exotic statistical weights of non-Abelian anyon systems, whose meaning and significance remains an open problem. The relevant past studies had considered only anyon systems without any physical boundary. 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. 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. The basis construction offers 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 that a pseudo-species has different ‘excitation’ modes corresponding 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.