Topological aspects of the critical three-state Potts model

TL;DR

This paper characterizes topological defects in the critical three-state Potts model using lattice-based categorical methods, linking conformal field theory, tensor networks, and symmetries to understand boundary conditions and defects.

Contribution

It introduces a categorical lattice construction that lifts global symmetries to local MPO symmetries, enabling detailed analysis of topological defects and boundary conditions in the model.

Findings

Complete classification of topological defects on various surfaces.

Lifting of global symmetries to local MPO symmetries.

Extraction of conformal characters from partition functions.

Abstract

We explore the topological defects of the critical three-state Potts spin system on the torus, Klein bottle and cylinder. A complete characterization is obtained by breaking down the Fuchs-Runkel-Schweigert construction of 2d rational CFT to the lattice setting. This is done by applying the strange correlator prescription to the recently obtained tensor network descriptions of string-net ground states in terms of bimodule categories [Lootens, Fuchs, Haegeman, Schweigert, Verstraete, SciPost Phys. 10, 053 (2021)]. The symmetries are represented by matrix product operators (MPO), as well as intertwiners between the diagonal tetracritical Ising model and the non-diagonal three-state Potts model. Our categorical construction lifts the global transfer matrix symmetries and intertwiners, previously obtained by solving Yang-Baxter equations, to MPO symmetries and intertwiners that can be locally deformed, fused and split. This enables the extraction of conformal characters from partition functions and yields a comprehensive picture of all boundary conditions.