Invertible symmetries — those with an inverse operation — are the classical setting for symmetry-protected topological phases. Non-invertible symmetries, like Rep(G) (the representation category of a group), lack inverses: fusing two representations doesn't always decompose into a single representation. The category is richer and harder.
The paper classifying intrinsically mixed 1+1D non-invertible SPT phases (arXiv: 2603.24289) achieves a complete classification for the fusion-category symmetry H = Rep(G) × Vec_G.
“Intrinsically mixed” means the phase is trivial when restricted to either factor alone — it requires both Rep(G) and G symmetry simultaneously. The classification is surprisingly clean: such phases are parametrized by endomorphisms φ ∈ End(G). Each φ determines a condensable algebra in the bulk topological order (the Drinfeld double D_G²) and a lattice realization as a modified Kitaev quantum double with a domain wall.
The lattice models are explicit: contracting a 2D Kitaev model with a φ-dependent domain wall to a 1D chain produces a (possibly twisted) group-based cluster state. The ribbon-generated symmetry operators of the chain encode the same endomorphism φ.
The through-claim: non-invertible SPT phases are classified by the algebra of the symmetry's self-maps. The endomorphism group End(G) — how G maps to itself — parametrizes the phases. Each self-map creates a different way to weave the two symmetry factors together, and each weaving is a distinct topological phase. The classification lives in the symmetry's internal structure.
2603.24289. Condensed matter / non-invertible symmetry / SPT phases / fusion categories / Kitaev models.