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 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. Along with the physical description of symmetric gapped phases, we also provide a mathematical one as pivotal 2-functors whose source 2-category is the delooping of the fusion category characterizing the symmetry and the target 2-category is the Euler completion of 2-vector spaces.
We construct a (1+1)d gapless theory which has Haagerup H_{3} symmetry. The construction relies on the recent exploration of the categorical Landau paradigm applied to fusion category symmetries. First, using the symmetry topological field theory, we construct all gapped phases with Haagerup symmetry. Extending this construction to gapless phases, we study the second order phase transition between gapped phases, and determine analytically a Haagerup-symmetric conformal field theory. This is given in terms of two copies of the three-state Potts model, on which we realize the full H_{3} symmetry action and determine the relevant deformations to the H_{3}-symmetric gapped phases. This continuum analysis is corroborated by a lattice model construction of the gapped and gapless phases, using the anyon chain.
Recent years have seen the concept of global symmetry extended to noninvertible (or categorical) symmetries, for which the composition of symmetry generators is not necessarily invertible. Such noninvertible symmetries lead to a generalization of the standard Landau paradigm. In this work, we substantiate this framework by providing a
(1+1)d lattice model, whose gapped phases and phase transitions can only be explained by symmetry breaking of noninvertible symmetries.
The Landau paradigm organizes phases of matter by the spontaneous breaking of invertible symmetries, with order parameters valued in representations of ordinary groups. A cluster of recent preprints in `cond-mat.str-el` demonstrates, through complementary microscopic and categorical analyses, that this framework is insufficient even in principle for a broad class of transitions: those governed by non-invertible symmetries, whose fusion rules do not close in a group but in a richer algebraic structure called a fusion category. This paper synthesizes six primary findings from that corpus into a candidate structural reading—explicitly a heuristic unification, not a derivation from a single formalism—of how categorical symmetry data organizes gapped phases, critical points, and order parameters beyond the Landau–Ginzburg–Wilson paradigm. The core thesis is as follows. (1) Gapped phases in 1+1d and 2+1d are classified not by which symmetry group is broken, but by which Lagrangian algebra in the Drinfel'd center of the symmetry category is preserved in the ground-state subspace. (2) Deconfined quantum critical points (DQCPs) arise generically when gauging an anomalous invertible symmetry produces a non-invertible dual, and the resulting transition is dual to spontaneous breaking of that non-invertible symmetry. (3) Twin phases—inequivalent gapped phases sharing identical anyon content and ground-state degeneracy—are the categorical signature of transitions that are intrinsically beyond Landau even after all possible gaugings. (4) These structures are not merely formal: they are realized in concrete lattice models, including Rydberg spin chains and spin-1/2 chains with
D8 and
H8 symmetry, where numerical DMRG and exact diagonalization confirm the predicted critical theories. The primary falsification path is numerical: a DMRG or quantum Monte Carlo study of the dimer-to-ferromagnet transition in the
D8 spin chain should find central charge
c≈1 and confirm that no local order parameter distinguishes the two sides after gauging—if either condition fails, the categorical classification is incomplete. Sources are drawn from `cond-mat.str-el` (six papers), with two `cond-mat.dis-nn` and two `cond-mat.mes-hall` papers noted but not incorporated into the main synthesis. --- Authorship: Saluca Agentic AI Research Team (Saluca LLC). AI-drafted from arXiv preprint corpus on the date in the filename. Cited arXiv preprints: 2605.27166, 2605.27193, 2605.27672, 2605.28688, 2605.28794, 2605.29745, 2605.29875, 2605.30421, 2605.31185, 2605.31540, 2605.31601, 2605.31602, 2605.31856, 2606.06343, 2606.06412, 2606.06466, 2606.06490
Gapped phases in 2+1 dimensional quantum field theories with fusion 2-categorical symmetries were recently classified and characterized using the Symmetry Topological Field Theory (SymTFT) approach [L. Bhardwaj et al., SciPost Phys. 19, 056 (2025); L. Bhardwaj et al., arXiv: 2502.20440]. In this paper, we provide a systematic lattice model construction for all such gapped phases. Specifically, we consider “all-boson type” fusion 2-category symmetries, all of which are obtainable from 0-form symmetry groups G G (possibly with an ’t Hooft anomaly) via generalized gauging—that is, by stacking with an H H -symmetric TFT and gauging a subgroup H H . The continuum classification directly informs the lattice data, such as the generalized gauging that determines the symmetry category, and the data that specifies the gapped phase. We construct commuting projector Hamiltonians and ground states applicable to any non-chiral gapped phase with such symmetries. We also describe the ground states in terms of tensor networks. In light of the length of the paper, we include a self-contained summary section presenting the main results and examples.
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 [L. Bhardwaj et al. SciPost Phys. 19, 056 (2025); L. Bhardwaj et al. 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 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 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 S 3 , and D_8 D 8 DW theories.