论文标题
循环和Potts模型的几何代数和代数几何形状
Geometric Algebra and Algebraic Geometry of Loop and Potts Models
论文作者
论文摘要
我们在可集成模型中发现了两个看似分开的主体之间的联系：仿射temperley-lieb代数的表示理论，以及解决方案的代数结构与xxz自旋链的贝尔斯方程的代数结构。我们通过计算代数几何形状分析地研究贝尔斯方程的解，并发现解决方案空间编码有关temperley-lieb代数代表理论的丰富信息。使用这些连接，我们通过两种完全不同的方法计算了完全包装的环路模型的分区和密切相关的随机群集Potts模型的分区函数。我们考虑无限长托里的部分热力学极限，并分析分区函数零的相应冷凝曲线。这些曲线的两个组成部分是在整个热力学极限下进行分析获得的。
We uncover a connection between two seemingly separate subjects in integrable models: the representation theory of the affine Temperley-Lieb algebra, and the algebraic structure of solutions to the Bethe equations of the XXZ spin chain. We study the solution of Bethe equations analytically by computational algebraic geometry, and find that the solution space encodes rich information about the representation theory of Temperley-Lieb algebra. Using these connections, we compute the partition function of the completely-packed loop model and of the closely related random-cluster Potts model, on medium-size lattices with toroidal boundary conditions, by two quite different methods. We consider the partial thermodynamic limit of infinitely long tori and analyze the corresponding condensation curves of the zeros of the partition functions. Two components of these curves are obtained analytically in the full thermodynamic limit.
