标题： 二阶网模型的可积性 显示英文标题

标题： Integrability of rank-two web models

摘要： 我们继续研究网络的格点模型，这些模型推广了众所周知的环模型，以允许各种类型的分叉 [arXiv:2101.00282, arXiv:2107.10106]。在这里，我们定义了与Kuperberg所考虑的每个二阶蜘蛛相对应的新网络模型 [arXiv:q-alg/9712003]。这些模型基于$A_2$、$G_2$和$B_2$李代数，其局部顶点配置是相应$q$变形量子代数的互变子。在所有三种情况下，我们在六边形格点上定义了相应的模型，在$B_2$的情况下还在正方形格点上定义了模型。对于$q$的特定单位根选择，我们证明了与对偶格点上若干三态和四态自旋模型的等价性。本文的主要结果是展示每个网络模型参数空间中的可积流形。对于单位圆上的$q$，这些模型是临界的，我们通过转移矩阵的数值对角化来表征相应的共形场论。 在$A_2$的情况下，我们发现两个可积的区域。 第一个区域包含一个密集相和一个稀疏相，我们通过库仑气体构造对其有解析控制，而第二个区域则更难以捉摸，可能隐藏着非紧致物理。 三个特定点对应于具有面元相互作用的三态自旋模型，其中第二个区域中的一个似乎表现出一个新的普适性类。 在$G_2$的情况下，我们通过数值方法识别了四个区域。 $B_2$的情况在一般情况下过于复杂，无法进行数值研究，但它被解析地发现包含一个基于稀疏 Birman-Murakami-Wenzl 代数生成器的更简单的子模型。 显示英文摘要

摘要： 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.