The authors show that one of the strongly correlated gapless states at the critical point of Landau symmetry breaking transition has an unbroken dual algebraic (n-1)-symmetry G^(n-1) in n-dimensional space, in addition to the usual unbroken symmetry G.
For a zero-temperature Landau symmetry breaking transition in n-dimensional space that completely breaks a finite symmetry G, the critical point at the transition has the symmetry G. In this paper, we show that the critical point also has a dual symmetry - a (n−1)-symmetry described by a higher group when G is Abelian or an algebraic (n−1)-symmetry beyond higher group when G is non-Abelian. In fact, any G-symmetric system can be viewed as a boundary of G-gauge theory in one higher dimension. The conservation of gauge charge and gauge flux in the bulk G-gauge theory gives rise to the symmetry and the dual symmetry respectively. So any G-symmetric system actually has a larger symmetry called categorical symmetry, which is a combination of the symmetry and the dual symmetry. However, part (and only part) of the categorical symmetry must be spontaneously broken in any gapped phase of the system, but there exists a gapless state where the categorical symmetry is not spontaneously broken. Such a gapless state corresponds to the usual critical point of Landau symmetry breaking transition. The above results remain valid even if we expand the notion of symmetry to include higher symmetries and algebraic higher symmetries. Thus our result also applies to critical points for transitions between topological phases of matter. In particular, we show that there can be several critical points for the transition from the 3+1D Z2 gauge theory to a trivial phase. The critical point from Higgs condensation has a categorical symmetry formed by a Z2 0-symmetry and its dual - a Z2 2-symmetry, while the critical point of the confinement transition has a categorical symmetry formed by a Z2 1-symmetry and its dual - another Z2 1-symmetry.
FINDING: Standard Landau paradigm equates all phase transitions with symmetry breaking; topological/dual-phase work (Warman "Zwillingsphasen", Xie Chen) shows transitions can occur *without* hidden symmetry breaking — order parameters are not universal. | MATH: Landau free energy F = F₀ + a(T−T_c)η² + bη⁴ + …; symmetry breaking ⟺ η ≠ 0 below T_c and G → H (coset G/H). Non-Landau transitions: no local order parameter; characterized instead by topological invariants (Chern number C ∈ ℤ, Z₂ index ν ∈ {0,1}), anyon condensation, or categorical symmetry (fusion category 𝒞, modular tensor category). Wigner–Eckart corrections (arXiv:2007.03539): ⟨α'j'm'|T^k_q|αjm⟩ = (jm;kq|j'm')⟨α'j'‖T^k‖αj⟩ + O(1/√V) in broken phase, with finite-size scaling V^(−1/2). | CONNECTION: Symmetry breaking cosets G/H are homogeneous spaces whose dimension and topology encode crystallographic point/space groups (230 space groups, 32 point groups) and root systems (A_n, B_n, C_n, D_n, E₆₇₈, F₄, G₂). Topological phase Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
The Landau paradigm—classifying phases by local order parameters and transitions by spontaneous symmetry breaking—has long been understood to be incomplete. Deconfined quantum critical points (DQCPs), topological orders, and symmetry-protected phases resist Landau characterization. A candidate structural pattern now emerging across the cond-mat.str-el corpus is that the failure of the Landau paradigm is not merely an exception but a systematic consequence of three interlocking mechanisms: (1) the presence of non-invertible symmetries whose breaking generates order parameters obeying generalized algebraic structures rather than group representations; (2) the role of anomaly type—specifically the position of the Lieb-Schultz-Mattis (LSM) anomaly class in the Lyndon-Hochschild-Serre spectral sequence—in determining whether a critical point is dual to ordinary symmetry breaking or to non-invertible symmetry breaking; and (3) excitation-density control as the decisive factor separating phases that merely enter the topological regime from those that develop genuine topological order. This synthesis draws on six papers from cond-mat.str-el and one from cond-mat.mes-hall (as a mechanistic comparator), all posted within the last 30 days. The thesis is presented as a *heuristic structural reading*, not a derivation from a shared formalism: the cited papers use distinct mathematical frameworks (fusion categories, spectral sequences, sequential circuits, tensor networks, quadratic fermion mappings), and we argue that these frameworks share a common organizational principle—the algebra of defects and condensable objects—rather than a common Lagrangian. The primary falsification path is explicit: if a direct DQCP transition between twin phases (as defined by [corpus:arxiv:2605.31601v1]) can be shown to be continuously connected to a hidden symmetry-breaking transition upon deformation of the Hamiltonian, the structural independence of beyond-Landau transitions claimed here would be overturned. Numerical DMRG or tensor-network studies on the specific models of [corpus:arxiv:2606.06343v1] and [corpus:arxiv:2605.27672v1] with explicitly broken categorical symmetry would constitute such a test. --- Authorship: Saluca Agentic AI Research Team (Saluca LLC). AI-drafted from arXiv preprint corpus on the date in the filename. Cited arXiv preprints: 2605.27166v1, 2605.27193v1, 2605.27672v1, 2605.31601v1, 2605.31602v1, 2606.05856v1, 2606.06343v1, 2606.11303v1, 2606.11527v1
A global symmetry (0-symmetry) in an n-dimensional space acts on the whole space.A higher symmetry acts on closed submanifolds (i.e., loops and membranes, etc.), and those transformations form a higher group.In this paper, we introduce the notion of algebraic higher symmetry, which generalizes higher symmetry and is beyond higher group.We show that an algebraic higher symmetry in a bosonic system in n-dimensional space is characterized and classified by a local fusion n-category.We find that, when restricted to its symmetric sub-Hilbert space, an algebraic higher symmetry can be fully characterized by a noninvertible gravitational anomaly (i.e., a topological order in one higher dimension).Thus, we also refer to a noninvertible gravitational anomaly as categorical symmetry to stress its connection to symmetry.This provides a holographic and entanglement view of symmetries.For a system with a categorical symmetry, its gapped state must spontaneously break part (not all) of the symmetry, and the state with the full symmetry must be gapless.Using such a holographic point of view, we obtain (1) the gauging of the algebraic higher symmetry; (2) the classification of anomalies for an algebraic higher symmetry; (3) the equivalence between classes of systems, with different (potentially anomalous) algebraic higher symmetries or different sets of low-energy excitations, as long as they have the same categorical symmetry; and (4) the classification of gapped liquid phases for bosonic and/or fermionic systems with a categorical symmetry, as gapped boundaries of a topological order in one higher dimension (that corresponds to the categorical symmetry).This classification includes symmetry protected trivial (SPT) orders and symmetry enriched topological (SET) orders with an algebraic higher symmetry.
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.
A structural pattern is emerging across several recent preprints in strongly correlated and topological quantum matter: the machinery of non-invertible symmetry — symmetries whose defects do not compose invertibly — provides a coherent organizational framework for phases and transitions that resist conventional Landau-Ginzburg-Wilson description. This synthesis draws on seven papers from the cond-mat.str-el and cond-mat.mes-hall categories, spanning lattice spin chains, topological field theory, sequential circuit invariants, and mesoscopic collider experiments. The thesis, offered as a heuristic reading rather than a derivation from a shared formalism, is as follows: non-invertible symmetry simultaneously (i) enriches the classification of topological orders beyond the invertible-symmetry-enriched (SET) framework, (ii) controls the universal categorical structure of deconfined quantum critical points (DQCPs) via LSM anomalies in spectral sequences, (iii) generates order parameters organized as Lagrangian algebras in Drinfeld centers, (iv) enables twin phases — inequivalent phases with isomorphic ground-state spaces that undergo transitions without any hidden symmetry breaking, and (v) produces measurable loop-excitation statistics in (3+1)D that are detectable through sequential-circuit invariants. A concrete falsification path runs through numerics: DMRG computation of the central charge and entanglement spectrum at the Rep(H₈)-type DQCP candidate in the spin-1/2 D₈ chain should yield c ≈ 1 with specific operator content; deviation from this prediction, or discovery of a first-order signal, would overturn the categorical Landau description reported for that transition. The synthesis is limited by abstract-only reading and the heuristic nature of the cross-paper unification. --- Authorship: Saluca Agentic AI Research Team (Saluca LLC). AI-drafted from arXiv preprint corpus on the date in the filename. Cited arXiv preprints: 2605.27193, 2605.27672, 2605.28688, 2605.28794, 2605.30421, 2605.31601, 2605.31602, 2606.04958, 2606.05856, 2606.06343, 2606.11147, 2606.11303, 2606.11527