Internal Levin-Wen models

TL;DR

This paper generalizes Levin-Wen models to lattice systems within arbitrary topological phases described by modular fusion categories, constructing new models and phases via orbifold data and characterizing their topological properties.

Contribution

It introduces a framework for internal Levin-Wen models using orbifold data in modular fusion categories, unifying and extending existing topological lattice models.

Findings

Constructed a state space and Hamiltonian $H_{\mathbb{A}}$ for internal Levin-Wen models.

Characterized the topological phase of ground states by a new modular fusion category $\mathcal{C}_{\mathbb{A}}$.

Recovered known models like Kitaev and Levin-Wen as special cases.

Abstract

Levin-Wen models are a class of two-dimensional lattice spin models with a Hamiltonian that is a sum of commuting projectors, which describe topological phases of matter related to Drinfeld centres. We generalise this construction to lattice systems internal to a topological phase described by an arbitrary modular fusion category $\mathcal{C}$. The lattice system is defined in terms of an orbifold datum $\mathbb{A}$ in $\mathcal{C}$, from which we construct a state space and a commuting-projector Hamiltonian $H_{\mathbb{A}}$ acting on it. The topological phase of the degenerate ground states of $H_{\mathbb{A}}$ is characterised by a modular fusion category $\mathcal{C}_{\mathbb{A}}$ defined directly in terms of $\mathbb{A}$. By choosing different $\mathbb{A}$'s for a fixed $\mathcal{C}$, one obtains precisely all phases which are Witt-equivalent to $\mathcal{C}$. As special cases we recover the Kitaev and the Levin-Wen lattice models from instances of orbifold data in the trivial modular fusion category of vector spaces, as well as phases obtained by anyon condensation in a given phase $\mathcal{C}$.