Integrability of rank-two web models

We continue our work on lattice models of webs, which generalise the well-known loop models to allow for various kinds of bifurcations [arXiv:2101.00282, arXiv:2107.10106]. Here we define new web models corresponding to each of the rank-two spiders considered by Kuperberg [arXiv:q-alg/9712003]. These models are based on the $A_2$, $G_2$ and $B_2$ Lie algebras, and their local vertex configurations are intertwiners of the corresponding $q$-deformed quantum algebras. In all three cases we define a corresponding model on the hexagonal lattice, and in the case of $B_2$ also on the square lattice. For specific root-of-unity choices of $q$, we show the equivalence to a number of three- and four-state spin models on the dual lattice. The main result of this paper is to exhibit integrable manifolds in the parameter spaces of each web model. For $q$ on the unit circle, these models are critical and we characterise the corresponding conformal field theories via numerical diagonalisation of the transfer matrix. In the $A_2$ case we find two integrable regimes. The first one contains a dense and a dilute phase, for which we have analytic control via a Coulomb gas construction, while the second one is more elusive and likely conceals non-compact physics. Three particular points correspond to a three-state spin model with plaquette interactions, of which the one in the second regime appears to present a new universality class. In the $G_2$ case we identify four regimes numerically. The $B_2$ case is too unwieldy to be studied numerically in the general case, but it found analytically to contain a simpler sub-model based on generators of the dilute Birman-Murakami-Wenzl algebra.