Sector Theory of Levin-Wen Models I : Classification of Anyon Sectors

TL;DR

This paper classifies the fundamental anyon sectors in Levin-Wen models using the Drinfeld center of the underlying fusion category, linking topological quantum field theory with lattice models.

Contribution

It provides a complete classification of anyon sectors in Levin-Wen models via the Drinfeld center, with explicit constructions of operators and state spaces.

Findings

Classifies irreducible anyon sectors using Drinfeld center

Constructs Drinfeld insertion and string operators for anyons

Links Levin-Wen models to Turaev-Viro TQFT

Abstract

We classify the irreducible anyon sectors of Levin-Wen models over an arbitrary unitary fusion category $\mathcal{C}$, showing that they are in one-to-one correspondence with equivalence classes of simple objects of the Drinfeld center $Z(\mathcal{C})$. We achieve this by making explicit how the Levin-Wen Hamiltonian stabilizes subspaces isomorphic to state spaces of the corresponding Turaev-Viro TQFT, and developing a detailed understanding of these state spaces on punctured disks. In particular, we construct Drinfeld insertion operators on such spaces which can move anyons between the punctures, and can change their fusion channels. Using these Drinfeld insertions, we construct explicit string operators that excite anyons above the ground state. The fusion and braiding properties of these anyons will be analysed in a companion paper.