Lattice Models for Phases and Transitions with Non-Invertible Symmetries

TL;DR

This paper develops lattice models based on SymTFT to realize and analyze phases and phase transitions with non-invertible symmetries, connecting topological and lattice descriptions.

Contribution

It provides a method to convert SymTFT data into UV lattice models, including quantum spin chains, for studying non-invertible symmetry phases and transitions.

Findings

Constructed lattice models realizing non-invertible symmetry phases.

Identified operators acting as order parameters for these phases.

Analyzed symmetry twisted sectors in lattice models.

Abstract

Non-invertible categorical symmetries have emerged as a powerful tool to uncover new beyond-Landau phases of matter, both gapped and gapless, along with second order phase transitions between them. The general theory of such phases in (1+1)d has been studied using the Symmetry Topological Field Theory (SymTFT), also known as topological holography. This has unearthed the infrared (IR) structure of these phases and transitions. In this paper, we describe how the SymTFT information can be converted into an ultraviolet (UV) anyonic chain lattice model realizing, in the IR limit, these phases and transitions. In many cases, the Hilbert space of the anyonic chain is tensor product decomposable and the model can be realized as a quantum spin-chain Hamiltonian. We also describe operators acting on the lattice models that are charged under non-invertible symmetries and act as order parameters for the phases and transitions. In order to fully describe the action of non-invertible symmetries, it is crucial to understand the symmetry twisted sectors of the lattice models, which we describe in detail. Throughout the paper, we illustrate the general concepts using the symmetry category $\mathsf{Rep}(S_3)$ formed by representations of the permutation group $S_3$, but our procedure can be applied to any fusion category symmetry.