Gapped Phases with Non-Invertible Symmetries: (1+1)d

TL;DR

The paper develops a systematic framework to classify (1+1)d gapped phases with non-invertible (categorical) symmetries using Symmetry Topological Field Theory (SymTFT). By encoding the symmetry as a boundary condition and selecting a second gapped boundary, the interval compactification yields the IR theory while the SymTFT supplies a unified description of vacua, symmetry action, and order parameters, including mixtures of untwisted and twisted sector operators. The construction is applied to explicit cases: Rep$(S_3)$ (group-theoretical) and Tambara-Yamagami categories TY$(\mathbb Z_N)$ (intrinsic non-invertible), deriving the full set of gapped phases, generalized charges, vacua counts, relative Euler terms, and order parameters. A generalized, categorical Landau paradigm emerges, where phases are distinguished by the spectrum of generalized charges and by relative Euler data, with spontaneous breaking of non-invertible symmetries potentially producing physically distinguishable vacua. Overall, the work provides a computational, boundary-data–driven route to classify and characterize symmetric gapped phases across a broad class of non-invertible symmetries, with concrete results for Ising-like TY theories and Rep$(S_3)$.

Abstract

We propose a general framework to characterize gapped infra-red (IR) phases of theories with non-invertible (or categorical) symmetries. In this paper we focus on (1+1)d gapped phases with fusion category symmetries. The approach that we propose uses the Symmetry Topological Field Theory (SymTFT) as a key input: associated to a field theory in d spacetime dimensions, the SymTFT lives in one dimension higher and admits a gapped boundary, which realizes the categorical symmetries. It also admits a second, physical, boundary, which is generically not gapped. Upon interval compactification of the SymTFT by colliding the gapped and physical boundaries, we regain the original theory. In this paper, we realize gapped symmetric phases by choosing the physical boundary to be a gapped boundary condition as well. This set-up provides computational power to determine the number of vacua, the symmetry breaking pattern, and the action of the symmetry on the vacua. The SymTFT also manifestly encodes the order parameters for these gapped phases, thus providing a generalized, categorical Landau paradigm for (1+1)d gapped phases. We find that for non-invertible symmetries the order parameters involve multiplets containing both untwisted and twisted sector local operators, and hence can be interpreted as mixtures of conventional and string order parameters. We also observe that spontaneous breaking of non-invertible symmetries can lead to vacua that are physically distinguishable: unlike the standard symmetries described by groups, non-invertible symmetries can have different actions on different vacua of an irreducible gapped phase. This leads to the presence of relative Euler terms between physically distinct vacua. We also provide a mathematical description of symmetric gapped phases as 2-functors from delooping of fusion category characterizing the symmetry to Euler completion of 2-vector spaces.

Figures (3)

• Figure 1: The basic SymTFT sandwich: (1) LHS: The $d$-dimensional theory $\mathfrak{T}$ on the RHS is constructed as the interval compactification of $d+1$-dimensional SymTFT $\mathfrak{Z}(\mathcal{S})$ on the LHS, with two boundary conditions. The gapped, i.e. topological, boundary $\mathfrak{B}^{\text{sym}}_{\mathcal{S}}$ is on the left and the physical, possibly non-topological, boundary $\mathfrak{B}^{\text{phys}}_{\mathfrak{T}}$ is on the right. (2) RHS: In this paper, we will focus on sandwich constructions for $\mathcal{S}$-symmetric TQFTs (denoted TQFT${}_{\mathcal{S}}$) in which case the physical boundary $\mathfrak{B}^{\text{phys}}_\mathfrak{T}$ is also topological, $\mathfrak{B}^\text{phys}_{\text{top}}$ (though not necessarily the same as the symmetry boundary on the left).
• Figure 2: Rectangle identity: the square needs to commute for for all values of $\{a,b,c,\mu,\nu\}$. The arrows in the rectangle diagram denote the coefficients arising in each move. As is clear from the diagram, three of the moves involve $Q$-symbols, while one of them involves $F$-symbols. We have omitted the sum over internal topological lines, which can be reinstated straightforwardly using the definitions of $Q$ and $F$.
• Figure 3: The sandwich construction of a multiplet $\mathcal{M}$ of local operators carrying generalized charge $\bm{Q}_A$ in an $\mathcal{S}$-symmetric 2d theory $\mathfrak{T}$ involves interval compactification of the corresponding bulk topological line $\bm{Q}_A$ ending along the physical boundary $\mathfrak{B}^{\text{phys}}_\mathfrak{T}$ along a local operator $\mathcal{M}$. Along the symmetry boundary $\mathfrak{B}^{\text{sym}}_\mathcal{S}$, the bulk line $\bm{Q}_A$ can be attached to a boundary line $a$ along local operators $\mathcal{E}_\mu^{(a)}$. After the interval compactification, these local operators $\mathcal{E}_\mu^{(a)}$ become $a$-twisted sector local operators $\mathcal{O}_\mu^{(a)}$ in the multiplet $\mathcal{M}$, living at the end of topological line $D_1^{(a)}$ implementing $a$ symmetry in $\mathfrak{T}$.