Generalizations of Kitaev's honeycomb model from braided fusion categories

TL;DR

This paper extends Kitaev's honeycomb model using braided fusion categories to construct fusion surface models that exhibit rich topological phases, including chiral and parafermion orders, on the honeycomb lattice.

Contribution

It introduces fusion surface models based on braided fusion categories, generalizing Kitaev's model and exploring their topological phases and symmetries.

Findings

Models reduce to Levin-Wen string-nets in the anisotropic limit

Certain models realize chiral Ising and parafermion topological orders

Fibonacci honeycomb model features a non-invertible 1-form symmetry

Abstract

Fusion surface models, as introduced by Inamura and Ohmori, extend the concept of anyon chains to 2+1 dimensions, taking fusion 2-categories as their input. In this work, we construct and analyze fusion surface models on the honeycomb lattice built from braided fusion 1-categories. These models preserve mutually commuting plaquette operators and anomalous 1-form symmetries. Their Hamiltonian is chosen to mimic the structure of Kitaev's honeycomb model, which is unitarily equivalent to the Ising fusion surface model. In the anisotropic limit, where one coupling constant is dominant, the fusion surface models reduce to Levin-Wen string-nets. In the isotropic limit, they are described by weakly coupled anyon chains and are likely to realize chiral topological order. We focus on three specific examples: (i) Kitaev's honeycomb model with a perturbation breaking time-reversal symmetry that realizes chiral Ising topological order, (ii) a $\mathbb{Z}_N$ generalization proposed by Barkeshli et al., which potentially realizes chiral parafermion topological order, and (iii) a novel Fibonacci honeycomb model featuring a non-invertible 1-form symmetry.