Categorical Landau Paradigm for Gapped Phases

TL;DR

The paper introduces a unified framework to classify gapped IR phases with categorical symmetries using a symmetry topological field theory (SymTFT) in one higher dimension. Phases are read off from two boundary conditions on the SymTFT: a symmetry boundary encoding the categorical symmetry and a topological physical boundary determining dynamics, with the Drinfeld center and Lagrangian algebras providing the classification data. Generalized charges, arising from defects that end on the physical boundary, serve as order parameters and enable a categorical Landau paradigm, including non-invertible symmetry breaking and Euler-term distinctions. The authors illustrate the construction in (1+1)d with concrete examples from Rep(S3), Ising (TY(Z2)), and TY(ZN), revealing rich vacuum structures and phase distinctions, and discuss extensions to higher dimensions and potential lattice-model realizations. Overall, the framework offers a powerful, dimension-agnostic method to characterize gapped phases with categorical symmetries and their phase transitions.

Abstract

We propose a unified framework to classify gapped infra-red (IR) phases with categorical symmetries, leading to a generalized, categorical Landau paradigm. This is applicable in any dimension and gives a succinct, comprehensive, and computationally powerful approach to classifying gapped symmetric phases. The key tool is the symmetry topological field theory (SymTFT), which is a one dimension higher TFT with two boundaries, which we choose both to be topological. We illustrate the general idea for (1+1)d gapped phases with categorical symmetries and suggest higher-dimensional extensions.

Figures (1)

• Figure 1: SymTFT for Gapped Phases: the SymTFT $\mathfrak{Z} ({\mathcal{S}})$ is a $(d+1)$ dimensional topological field theory with two boundaries, which for gapped phases are both topological: $\mathfrak{B}^{\text{sym}}_{\mathcal{S}} = {\mathcal{A}}_{\mathcal{S}}$ and $\mathfrak{B}^{\text{sym}}_{\text{phys}}= {\mathcal{A}}_\text{phys}$.