Abstract In a physical system undergoing a continuous quantum phase transition, spontaneous symmetry breaking occurs when certain symmetries of the Hamiltonian fail to be preserved in the ground state. In the traditional Landau theory, a symmetry group can break down to any subgroup. However, this no longer holds across a continuous phase transition driven by anyon condensation in symmetry enriched topological orders (SETOs). For a SETO described by aG-crossed braided extension $$ \mathcal{C}\subseteq {\mathcal{C}}_G^{\times } $$ C⊆CG× , we show that physical considerations require that a connected étale algebraA∈ $$ \mathcal{C} $$ C admit aG-equivariant algebra structure for symmetry to be preserved under condensation ofA. Given any categorical actionG→EqBr( $$ \mathcal{C} $$ C ) such thatg(A) ≅Afor allg∈G, we show there is a short exact sequence whose splittings correspond toG-equivariant algebra structures. The non-splitting of this sequence forces spontaneous symmetry breaking under condensation ofA, while inequivalent splittings of the sequence correspond to different SETOs resulting from the anyon-condensation transition. Furthermore, we show that if symmetry is preserved, there is a canonically associated SETO of $$ {\mathcal{C}}_A^{\mathrm{loc}} $$ CAloc , and gauging this symmetry commutes with anyon condensation.
Abstract In a physical system undergoing a continuous quantum phase transition, spontaneous symmetry breaking occurs when certain symmetries of the Hamiltonian fail to be preserved in the ground state. In the traditional Landau theory, a symmetry group can break down to any subgroup. However, this no longer holds across a continuous phase transition driven by anyon condensation in symmetry enriched topological orders (SETOs). For a SETO described by aG-crossed braided extension C⊆CG× C⊆CG× , we show that physical considerations require that a connected étale algebraA∈ C C admit aG-equivariant algebra structure for symmetry to be preserved under condensation ofA. Given any categorical actionG→EqBr( C C ) such thatg(A) ≅Afor allg∈G, we show there is a short exact sequence whose splittings correspond toG-equivariant algebra structures. The non-splitting of this sequence forces spontaneous symmetry breaking under condensation ofA, while inequivalent splittings of the sequence correspond to different SETOs resulting from the anyon-condensation transition. Furthermore, we show that if symmetry is preserved, there is a canonically associated SETO of CAloc CAloc , and gauging this symmetry commutes with anyon condensation.
Condensable algebras in 2+1d non-chiral topological orders characterize gapped boundary conditions and interfaces. Applied to the Symmetry Topological Field Theory, they allow classification of symmetric gapped phases and impose sharp constraints on possible phase transitions. A condensable algebra is specified not only by its underlying set of anyons, which end on the boundary or interface, but also by its algebra structure. We introduce the concept of twin condensable algebras, which have the same anyon decomposition, but inequivalent algebra structure. We revisit the classification of condensable algebras in Z(VecGω), i.e. in group-theoretical topological orders for finite groups G with anomaly ω. In this context we are able to identify twin algebras that arise from different mechanisms, such as subgroup data, SPT cocycles, and symmetry actions. In particular, we construct infinite families of examples of twins from so-called Gassmann triples, and exhibit cases in which the reduced topological orders are inequivalent despite having identical anyon content. Physically, twin algebras describe distinct symmetric phases that have isomorphic spaces of ground states, but inequivalent order parameters. Such twin phases never exhibit relative spontaneous symmetry breaking, and can be used to construct phase transitions without hidden symmetry breaking, which are intrinsically beyond Landau transitions.
Abstract Although the mathematics of anyon condensation in topological phases has been studied intensively in recent years, a proof of its physical existence is tantamount to constructing an effective Hamiltonian theory. In this paper, we concretely establish the physical foundation of anyon condensation by building the effective Hamiltonian and the Hilbert space, in which we explicitly construct the vacuum of the condensed phase as the coherent states that are the eigenstates of the creation operators creating the condensate anyons. Along with this construction, which is analogous to Laughlin’s construction of wavefunctions of fractional quantum hall states, we generalize the Goldstone theorem in the usual spontaneous symmetry breaking paradigm to the case of anyon condensation. We then prove that the condensed phase is a symmetry enriched (protected) topological phase by directly constructing the corresponding symmetry transformations, which can be considered as a generalization of the Bogoliubov transformation.
A candidate reading of seven recent preprints from cond-mat.str-el and physics.atom-ph suggests that the classical toolkit for classifying gapped phases and their transitions—built on Landau symmetry breaking, local order parameters, and invertible group symmetries—requires systematic extension at the level of microscopic algebraic structure, not merely at the phenomenological level. The emerging picture, drawn from the corpus, is that *generalized symmetries* (non-invertible, categorical, and higher-form) impose precise algebraic constraints on what constitutes a valid order parameter, what boundary conditions are permissible, and which phase transitions can occur without any form of spontaneous symmetry breaking, even after arbitrary gauging. Concretely: (i) condensable algebras in 2+1d topological orders classify gapped boundaries, and twin algebras sharing identical anyon content but inequivalent algebra structure describe physically distinct phases with isomorphic ground-state degeneracy but inequivalent order parameters [corpus:arxiv:2605.31602]; (ii) these twin phases can undergo direct transitions that are provably beyond-Landau even after partial gauging [corpus:arxiv:2605.31601]; (iii) string order parameters in 1d systems organize into Lagrangian algebras in the Drinfel'd center of the symmetry category, making the fusion rule of the symmetry category—not just its representation theory—the operative diagnostic [corpus:arxiv:2605.27193]; (iv) non-invertible symmetry-breaking transitions can be mapped via generalized gauging to deconfined quantum critical points, and vice versa, establishing a precise duality [corpus:arxiv:2605.27672]; and (v) non-invertible symmetry enriched topological orders (NI-SETOs) can be constructed from string-net models, extending symmetry enrichment beyond the group-theoretic setting [corpus:arxiv:2605.28794]. Two additional corpus entries—a Rydberg spin-chain study [corpus:arxiv:2605.27166] and a topological lattice gauge theory paper [corpus:arxiv:2605.28688]—provide microscopic anchoring and structural context, respectively. The central falsification path is explicit: if a twin-phase transition can be shown to spontaneously break some hidden symmetry after arbitrary gauging, the framework collapses. Concrete tensor-network (DMRG/MPS) constructions of twin-phase order parameters in 1+1d would constitute a direct test. --- 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
A central dogma of twentieth-century condensed matter physics holds that every phase transition either breaks a symmetry spontaneously or restores one—Landau's paradigm. A cluster of recent preprints, drawn primarily from `cond-mat.str-el` and `physics.atom-ph`, collectively challenges this dogma at the level of microscopic mechanism rather than abstract categorical assertion. This synthesis identifies and cross-connects five specific findings: (1) the existence of "twin phases" and "twin condensable algebras" that are inequivalent gapped phases sharing identical anyon content and isomorphic ground-state degeneracy, yet connected by transitions that cannot be cast as any form of symmetry breaking even after gauging [corpus:arxiv:2605.31601v1][corpus:arxiv:2605.31602v1]; (2) the demonstration that spontaneous breaking of non-invertible symmetries can be dual, via generalized gauging, to deconfined quantum critical points of invertible symmetries, establishing a precise duality dictionary rather than a mere analogy [corpus:arxiv:2605.27672v1]; (3) the characterization of order parameters in one-dimensional systems as Lagrangian algebras in the Drinfeld center, providing a tensor-network-computable diagnostic that generalizes the string-order concept beyond the invertible setting [corpus:arxiv:2605.27193v1]; (4) the discovery of "diabolical textures" as a new topological invariant in inhomogeneous systems, producing "unnecessary critical" surfaces stable to interactions [corpus:arxiv:2605.30421v1]; and (5) the identification of an exact factorization line embedded within an antiferromagnetically ordered phase of a Rydberg spin chain, providing an experimentally accessible zero-entanglement reference point for probing these structures [corpus:arxiv:2605.27166v1]. The thesis is that beyond-Landau transitions are not exotic edge cases but arise from a systematic mismatch between the algebraic structure of condensable algebras and the anyon content that naively classifies phases—a mismatch that is detectable, in principle, by tensor-network diagnostics and Rydberg-atom experiments. The primary falsification path is the construction of a gauging sequence that maps any twin-phase transition to a conventional symmetry-breaking transition; no such construction has been exhibited in the corpus. --- Authorship: Saluca Agentic AI Research Team (Saluca LLC). AI-drafted from arXiv preprint corpus on the date in the filename. Cited arXiv preprints: 2605.27166v1, 2605.27193v1, 2605.27672v1, 2605.28688v1, 2605.28794v1, 2605.29745v1, 2605.29875v1, 2605.30421v1, 2605.31540v1, 2605.31601v1, 2605.31602v1
Anyon Condensation In Symmetry-Enriched Topological Phases: G-Grading of Multifusion Categories
Although anyon condensation is a standard mechanism for relating topological orders, anyon condensation in symmetry-enriched topological (SET) phases is more intricate because the condensate must also be compatible with the global symmetry. We study symmetry-preserving anyon condensation in SET phases described by the enlarged Hu-Geer-Wu (HGW) string-net model with multifusion-category input data. We show that a G-preserving condensation is characterized by a compatible grading of the input multifusion category, and that this grading constructs the multifusion-category input of the child SET phase. To make this construction concrete, we consider the case where the relevant data come from a finite group extension E of G by N: an E-graded fusion category induces a G-SET input by passing to the quotient symmetry G, and the resulting input naturally carries a compatible N-grading that implements the further condensation inside the SET phase while preserving G. We illustrate the construction using three quantum-double examples: the trivial extension Z2×Z2, the non-Abelian semidirect product S3, and the nontrivial central extension Z4. The Z4 example further shows that symmetry fractionalized anyons do not obstruct symmetry-preserving condensation once the condensate is treated as a physical coherent state.
We analyze the effect of perturbations on a toric code-like Hamiltonian enriched with global symmetries. The ground states of this Hamiltonian are superpositions of loops decorated with cluster states, resulting in anyonic excitations that fractionalize a global on-site Z2×Z2, inversion and time reversal symmetry. We use a modified version of the string order parameters proposed in [New Journal of Physics 21, 113016 (2019)], adapted to work beyond the framework of tensor networks, to capture the symmetry fractionalization pattern of the anyons. We show that this order parameter accurately captures the symmetry fractionalization in a direct Hamiltonian interpolation between the decorated and trivial toric codes, and in the presence of symmetry-preserving local magnetic fields and Ising interactions. We also observe how the condensation of an anyon that fractionalizes the symmetry forces the symmetry to spontaneously break at the phase transition. This notably changes the phase diagram of this Hamiltonian under parallel magnetic fields compared to the regular toric code since the condensation of charge and flux anyons drives the system to two distinct trivial phases, a symmetry broken phase and symmetric phase. Moreover, we provide a proof of spontaneous symmetry breaking from anyon condensation in the framework of projected entangled pair states.