Gapless Phases in (2+1)d with Non-Invertible Symmetries

TL;DR

The paper develops a comprehensive framework to study gapless phases in (2+1)d with fusion 2-category (non-invertible) symmetries using Symmetry Topological Field Theory (SymTFT). It shows how to generate gapless phase transitions by applying Kennedy–Tasaki (KT) style transformations across interfaces (club sandwiches) between topological orders, starting from a known input phase with a smaller symmetry and producing a theory with a larger categorical symmetry. Interfaces are classified in terms of condensable algebras and module 2-categories, with gapped and gapless phases organized via Hasse diagrams; the reduced topological orders are typically DW theories for finite groups with twists, and multiple concrete examples (Z_4, S_3, D_8) illustrate intrinsically gapless SPTs (igSPTs) and spontaneously broken (igSSB) gapless phases. The work provides both a SymTFT sandwich and a module-category perspective that agree on the classification of gapless phases and their transitions, and it connects to étale algebras in mathematics, yielding a physically natural interpretation for condensable algebras. Overall, the results give systematic tools to construct and analyze second-order phase transitions and gapless critical theories in (2+1)d with categorical symmetries, with potential broad applications to generalized symmetries in quantum field theory and beyond.

Abstract

The study of gapless phases with categorical (or so-called non-invertible) symmetries is a formidable task, in particular in higher than two space-time dimensions. In this paper we build on previous works arXiv:2408.05266 and arXiv:2502.20440 on gapped phases in (2+1)d and provide a systematic framework to study phase transitions with categorical symmetries. The Symmetry Topological Field Theory (SymTFT) is, as often in these matters, the central tool. Applied to gapless theories, we need to consider the extension of the SymTFT to interfaces between topological orders, so-called ``club sandwiches", which realize generalizations of so-called Kennedy-Tasaki (KT) transformations. This requires an input phase transition for a smaller symmetry, such as the Ising transition for $\mathbb{Z}_2$, and the SymTFT constructs a transformation to a gapless phase with a larger categorical symmetry. We carry this out for categorical symmetries whose SymTFT is a (3+1)d Dijkgraaf-Witten (DW) theory for a finite group $G$ with twist -- so-called all bosonic fusion 2-categories. We classify such interfaces using a physically motivated picture of generalized gauging, as well as with a complementary analysis using (bi-)module 2-categories.This is exemplified in numerous abelian and non-abelian DW theories, giving rise to interesting gapless phases such as intrinsically gapless symmetry protected phases (igSPTs) and spontaneous symmetry breaking phases (igSSBs) from abelian, $S_3$, and $D_8$ DW theories.

Figures (7)

• Figure 1: Club Sandwich Picture that generalizes the KT transitions to categorical symmetries. This is applicable in any dimension. Here we have drawn the concrete case relevant for (2+1)d fusion 2-category symmetries. ${\bm{Q}}_d$ are the topological defects of the SymTFT, which will provide order parameters and symmetry generators.
• Figure 2: An $\mathcal{S}$-symmetric phase $\mathcal{P}$ is characterized by the set of condensed, confined and deconfined symmetry charges. Within the SymTFT, such a phase is modeled via a condensable algebra that defines a topological interface $\mathcal{I}_\mathcal{P}$ to a reduced topological order (TO). When $\mathcal{P}$ is a gapped phase, the reduced TO is trivial and $\mathcal{I}_{\mathcal{P}}$ is a gapped boundary.
• Figure 3: Schematic phase diagram for phases with ${\mathbb Z}_2$ symmetry. The ${\mathbb Z}_{2}$ SSB transition is described by the 3d Ising CFT, which is denoted as Ising. The phase transition between the ${\mathbb Z}_{2}$ SPT and the ${\mathbb Z}_{2}$ SSB phase is given by the stacking of the 3d Ising CFT with the ${\mathbb Z}_{2}$ SPT state, which we denote as $\text{Ising} \boxtimes {\mathbb Z}_{2}\text{-SPT}$. The transition between the ${\mathbb Z}_{2}$ trivial and the ${\mathbb Z}_{2}$ SPT phases was found to be described by a first order transition.
• Figure 4: Schematic phase diagram for phases with ${\mathbb Z}_{2}^{(1)}$ 1-form symmetry. We denote the trivial gapped phase with ${\mathbb Z}_{2}^{(1)}$ symmetry as ${\mathbb Z}_{2}^{(1)}$-Triv. DW$({{\mathbb Z}_{2}})_{0}$ denotes the DW theory without twist, or equivalently the Toric Code topological order. DW$({{\mathbb Z}_{2}})_{1}$ denotes the DW theory with the non-trivail twist, or equivalently the double semion topological order. The transition between the ${\mathbb Z}_{2}^{(1)}$ trivial and the DW$({{\mathbb Z}_{2}})_{0}$ phases is described by the gauged Ising CFT, which is denoted as $\text{Ising}/{\mathbb Z}_{2}$. The transition between the ${\mathbb Z}_{2}^{(1)}$ trivial and the DW$({{\mathbb Z}_{2}})_{0}$ phases is described by the CFT obtained from gauging the ${\mathbb Z}_{2}$ symmetry in $\text{Ising} \boxtimes {\mathbb Z}_{2}\text{-SPT}$, which we denote as $(\text{Ising} \boxtimes {\mathbb Z}_{2}\text{-SPT})/{\mathbb Z}_{2}$. The phase transition between $\text{DW}({{\mathbb Z}_{2}})_{0}$ and $\text{DW}({{\mathbb Z}_{2}})_{1}$ is expected to be a gauged first order transition.
• Figure 5: Hasse diagram for condensable algebras in $\mathcal{Z} (2\mathsf{Vec}_{{\mathbb Z}_4})$. Each box contains a condensable algebra, for which we provide generating objects and the group-theoretical data. We list $(H,N, \omega)$, and for the maximal algebras (bottom row) that correspond to gapped BCs, $H=N$. There can be a possible $N$-SPT: $\omega$ is the generator of $H^3({\mathbb Z}_4,U(1))={\mathbb Z}_4$, whose restriction to ${\mathbb Z}_2$ is the generator of $H^3({\mathbb Z}_2,U(1))={\mathbb Z}_2$.