Temperley-Lieb integrable models and fusion categories

TL;DR

This paper establishes a connection between fusion categories with self-dual objects and integrable anyonic chains governed by the Temperley-Lieb algebra, analyzing their spectral properties and gapped nature.

Contribution

It introduces a method to construct integrable anyonic chains from fusion categories and relates them to ADE lattice models, expanding understanding of their spectral characteristics.

Findings

Models are gapped when quantum dimension > 2

Finite size effects are significant near dimension 2

Spectrum analysis relates to Temperley-Lieb algebra structure

Abstract

We show that every fusion category containing a non-invertible, self-dual object $a$ gives rise to an integrable anyonic chain whose Hamiltonian density satisfies the Temperley-Lieb algebra. This spin chain arises by considering the projection onto the identity channel in the fusion process $a\otimes a$. We relate these models to Pasquier's construction of ADE lattice models. We then exploit the underlying Temperley-Lieb structure to discuss the spectrum of these models and argue that these models are gapped when the quantum dimension of $a$ is greater than 2. We show that for fusion categories where the dimension is close to 2, such as the Fib$\times$Fib and Haagerup fusion categories, the finite size effects are large and they can obscure the numerical analysis of the gap.