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Strongly interacting models often possess ``dualities'' subtler than aone-to-one mapping of energy levels. The maps can be non-invertible, asapparent in the canonical example of Kramers and Wannier. We analyse analgebraic structure common to the XXZ spin chain and three other models:Rydberg-blockade bosons with one particle per square of a ladder, a three-stateantiferromagnet, and two Ising chains coupled in a zigzag fashion. Thestructure yields non-invertible maps between the four models while alsoguaranteeing all are integrable. We construct these maps explicitly utilisingtopological defects coming from fusion categories and the lattice version ofthe orbifold construction, and use them to give explicit conformal-field-theorypartition functions describing their critical regions. The Rydberg and Isingladders also possess interesting non-invertible symmetries, with thespontaneous breaking of one in the former resulting in an unusual ground-statedegeneracy.

From the XXZ chain to the integrable Rydberg-blockade ladder via non-invertible duality defects