Integrable and critical Haagerup spin chains

Abstract

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 $δ=(3+\sqrt{13})/2$. 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 $c\sim3/2$.

Figures (3)

• Figure 1: Adjacency rules for the Haagerup Hilbert space. Nodes represent simple objects in $H_3$. If there is no edge between a pair of objects then this is a disallowed pair in $V^L$.
• Figure 2: Energy gap up to $L=60$ on the periodic chain vs. an inverse shifted length $(L-\alpha)^{-1}$. For model 1 we take $\alpha=0$ and for model 2 we take $\alpha=5/2$. For the model 2 fit we used the last 10 points.
• Figure 3: Half-chain entanglement entropy up to $L=42$ on the periodic chain. For the fit we used the last 10 points.