Symmetry Without Inversion: How Non-Invertible Symmetries, Twin Phases, and Generalized Order Parameters Redefine the Boundaries of Landau Theory

Version 2 — revised in response to an external structural review and an automated critique pass. See "Response to Review" appendix in the PDF for the change log.

Landau's framework for classifying phases and phase transitions rests on three interlocking pillars: a symmetry group, its spontaneous breaking, and a local order parameter that distinguishes phases. A cluster of recent preprints in `cond-mat.str-el` and `physics.atom-ph` collectively stress-tests all three pillars simultaneously. This synthesis argues—as a heuristic reading, not a formal derivation—that the emerging picture is best understood through a single organizing question: *what is the minimal algebraic structure needed to characterize a gapped phase and the transition out of it?* We draw on six primary sources spanning (i) twin condensable algebras and twin phases that are inequivalent yet share identical order-parameter spaces [corpus:arxiv:2605.31602v1][corpus:arxiv:2605.31601v1], (ii) non-invertible symmetry-enriched topological orders and their lattice gauge realizations [corpus:arxiv:2605.28794v1][corpus:arxiv:2605.28688v1], (iii) the algebraic organization of string order parameters in 1D gapped phases [corpus:arxiv:2605.27193v1], and (iv) spontaneous breaking of non-invertible symmetries and duality to beyond-Landau transitions [corpus:arxiv:2605.27672v1]. Two further sources are incorporated with explicit caveats: a concrete Rydberg-atom spin chain [corpus:arxiv:2605.27166v1] whose connection to the beyond-Landau theme is a heuristic extrapolation rather than a direct result of the abstract, and a diabolical-textures paper [corpus:arxiv:2605.30421v1] whose classification framework rests on an unproven conjecture. Two additional corpus papers—on Localization Landscape Theory on the Bethe lattice [corpus:arxiv:2605.29745v1] and quantum annealing in spin glasses [corpus:arxiv:2605.29875v1]—are noted but placed in a weakly-connected addendum, as their connection to the thesis is analogical rather than mechanistic. The central falsification path is explicit: if a tensor-network computation can find a local order parameter that distinguishes any pair of identified twin phases, the twin-phase construction fails and the Landau framework is restored for that case. ---

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