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 [1], [2]. Here we define new web models corresponding to each of the rank-two spiders considered by Kuperberg [3]. These models are based on the A2, G2 and B2 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 B2 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 A2 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 G2 case we identify four regimes numerically. The B2 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.