A Defect Verlinde Formula

TL;DR

This paper derives a Verlinde-like formula for boundary excitations in 2+1D topological orders, relating fusion rules and half-linking numbers, with explicit computations for Abelian and non-Abelian cases, aiding quantum computing platform design.

Contribution

It introduces a novel Verlinde-like formula connecting boundary fusion rules and half-linking numbers in topological orders, with practical computation methods for various anyon models.

Findings

Derived a boundary Verlinde formula for topological orders.

Explicit calculations of half-linking numbers in Abelian and non-Abelian cases.

Potential applications in designing quantum computing platforms.

Abstract

We revisit the problem of boundary excitations at a topological boundary or junction defects between topological boundaries in non-chiral bosonic topological orders in 2+1 dimensions. Based on physical considerations, we derive a formula that relates the fusion rules of the boundary excitations, and the "half-linking" number between condensed anyons and confined boundary excitations. This formula is a direct analogue of the Verlinde formula. We also demonstrate how these half-linking numbers can be computed in explicit Abelian and non-Abelian examples. As a fundamental property of topological orders and their allowed boundaries, this should also find applications in finding suitable platforms realizing quantum computing devices.