Integrable and critical Haagerup spin chains

We construct the first integrable models based on the Haagerup fusion category $H_3$. We introduce a Haagerup version of the anyonic spin chain and use the boost operator formalism to identify two integrable Hamiltonians of PXP type on this chain. The first of these is an analogue of the golden chain; it has a topological symmetry based on $H_3$ and satisfies the Temperley-Lieb algebra with parameter $\delta=(3+\sqrt{13})/2$. We prove its integrability using a Lax formalism, and construct the corresponding solution to the Yang--Baxter equation. We present numerical evidence that this model is gapless with a dynamical critical exponent $z\neq 1$. The second integrable model we find breaks the topological symmetry. We present numerical evidence that this model reduces to a CFT in the large volume limit with central charge $c\sim3/2$.