Quantum Cluster State Model with Haagerup Fusion Category Symmetry

TL;DR

This paper introduces a novel (1+1)D lattice model that realizes Haagerup fusion category symmetry using a weak Hopf algebra framework, extending the cluster state concept to incorporate complex topological symmetries.

Contribution

It constructs a new lattice model based on Haagerup fusion categories, utilizing weak Hopf algebras and SymTFT, and provides an explicit ground state solution as a weak Hopf matrix product state.

Findings

Supports Haagerup fusion category symmetry in a lattice model

Constructs a weak Hopf algebra from the Haagerup fusion category

Provides an explicit ground state as a weak Hopf matrix product state

Abstract

We propose a (1+1)D lattice model, inspired by a weak Hopf algebra generalization of the cluster state model, which realizes Haagerup fusion category symmetry and features a tensor product Hilbert space. The construction begins with a reconstruction of the Haagerup weak Hopf algebra $H_3$ from the Haagerup fusion category, ensuring that the representation category of $H_3$ is equivalent to Haagerup fusion category. Utilizing the framework of symmetry topological field theory (SymTFT), we develop an ultra-thin weak Hopf quantum double model, characterized by a smooth topological boundary condition. We show that this model supports Haagerup fusion category symmetry. Finally, we solve the ground state of the model in terms of a weak Hopf matrix product state, which serves as a natural generalization of the cluster state, embodying Haagerup fusion category symmetry.