Anyon exclusions statistics on surfaces with gapped boundaries

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.

Figures (16)

• Figure 1: (a) A system of 5 real elementary anyons and 4 gapped boundaries. Each gapped boundary can be transformed to a small hole as in (b), without affecting the Hilbert space structure of the system. In this perspective, the Hilbert space consists of 5 elementary anyons and 4 composite anyons respectively identified as the 4 boundaries components with their topological charges.
• Figure 2: The extended Levin-Wen model defined on a hexagonal lattice embedded in a disk. Thick (thin) black lines are the bulk (boundary) edges of the lattice. The two red dots represent two $\tau\bar{\tau}$'s on the two plaquettes, for instance.
• Figure 3: (Color online.) A sketch of the basis states of multi-$\tau\bar{\tau}$ Hilbert space on a disk. The line $a$ in red represents either $1$ or $\tau\bar{\tau}$.
• Figure 4: Our convention for the basis of multi-$\tau\bar{\tau}$ Hilbert space on a disk. A $\tau\bar{\tau}$ in the bulk is denoted by a black dot. A place-holder for the pseudo-species is a box. (a) Abstract presentation of the disk's boundary. We always draw the boundary on the right. (b) A single $\tau\bar{\tau}$ in the bulk; it must be attached to the boundary by an edge. (c) Adding the second $\tau\bar{\tau}$, which introduces a place-holder on the root of the tree. We always add new $\tau\bar{\tau}$'s from the left to right. (d) Adding the third $\tau\bar{\tau}$, which introduces a place-holder on its left (the lower box) and one on its right (the upper box). (e) The basis with place-holders for multiple $\tau\bar{\tau}$'s. The root always has an upper box.
• Figure 5: Simplified basis of a multi-$\tau\bar{\tau}$ Hilbert space on a disk. the upper right box is the one above the root in Fig. \ref{['fig:basisDiskWithBox']}(e) and is accompanied with a dashed box underneath.