What happens when exotic particles obey integrability rules?

Integrable systems—chains of quantum spins that can be solved exactly—normally work in unrestricted Hilbert spaces. This work extends that framework to anyonic chains, where constraints from fusion rules (like Rydberg blockade) eliminate certain states. The authors identify which fusion categories produce Temperley-Lieb algebras, discover new integrable models including spin-3/2 variants and Tambara-Yamagami chains, and compute properties for several exotic fusion categories.