Lattice Models for Phases and Transitions with Non-Invertible Symmetries

TL;DR

The paper develops a concrete UV lattice realization of (1+1)d phases with non-invertible, fusion-category symmetries by translating SymTFT data into anyon-chain models. It provides a systematic procedure to construct untwisted and symmetry-twisted sectors, define symmetry actions, and identify generalized charges and local operators transforming under the non-invertible symmetry. The authors work in depth through explicit examples, notably Rep(S3) and abelian G, detailing gapped phases (via Frobenius-algebra/gauging data), gapless phases (via club-sandwich condensation), and phase transitions with corresponding order parameters and ground-state structures. The framework unifies continuum symmetry-categorical insights with microscopic lattice realizations, enabling precise diagnostics and potential numerical explorations of beyond-Landau phenomena in 1+1d and offering pathways to higher dimensions. Overall, the work demonstrates how to engineer and analyze lattice models that realize rich IR physics protected by non-invertible symmetries, using a consistent SymTFT-guided methodology applicable to general fusion-category symmetries.

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.

Figures (3)

• Figure 1: Three-dimensional sketch of the SymTFT picture: the (2+1)d TQFT $\mathfrak{Z}(\mathcal{C})$ has two boundaries, $\mathfrak{B}^{\text{sym}}_\mathcal{S}$ and $\mathfrak{B}^{\text{inp}}_\mathcal{C}$. The interface between these is the module category $\mathcal{M}$ (blue). The topological lines on $\mathfrak{B}^{\text{inp}}_\mathcal{C}$ form the category $\mathcal{C}$, whereas the ones on $\mathfrak{B}^{\text{sym}}_\mathcal{S}$ form $\mathcal{S}= \mathcal{C}^*_\mathcal{M}$. The latter is the symmetry of the spin-chain. We can think of the spin-chain as located along the interface specified by the module category, with $\rho\in \mathcal{C}$ extending into $\mathfrak{B}^{\text{inp}}_\mathcal{C}$. The symmetry acts from the left (i.e. taking a topological defect (green) of $\mathfrak{B}^{\text{sym}}_\mathcal{S}$ and pushing it parallel to $\mathcal{M}$).
• Figure 2: Lattice SymTFT description of local operators. The bulk topological line $\bm{Q}\in\mathcal{Z}(\mathcal{S})$ (teal) ends on both boundaries $\mathfrak{B}^{\text{sym}}_\mathcal{S}$ and $\mathfrak{B}^{\text{inp}}_\mathcal{C}$. The former extends along $\mathfrak{B}^{\text{sym}}_\mathcal{S}$ and is converted to $s$ by $\bm{Q}^s_\nu$, while the latter extends along $\mathfrak{B}^{\text{inp}}_\mathcal{C}$ and is absorbed into $\rho$ by $\bm{Q}_\mu^\rho$.
• Figure 3: SymTFT picture for gapless phases, aka the club-sandwich. The interface $\mathcal{I}_{\text{phys}}$ defined by the condensable algebra reduces the topological order $\mathfrak{Z}(\mathcal{S})$ to $\mathfrak{Z}'$. The symmetry boundary carries the symmetry $\mathcal{S}$. The physical boundary is given by $\mathfrak{B}_{\mathcal{C}'}$. We compactify the interval occupied by $\mathfrak{Z}'$, which results in the right hand side picture: a topological boundary $\mathfrak{B}_{\mathcal{C}}^{\text{inp}}$ (carrying $\mathcal{C}$ topological defects) for $\mathfrak{Z}(\mathcal{S})= \mathfrak{Z}(\mathcal{C})$, and a module category at the intersection of the symmetry boundary $\mathfrak{B}^{\text{sym}}_\mathcal{S}$ and $\mathfrak{B}_{\mathcal{C}}^{\text{inp}}$.