三维Ising模型的数学结构与精确解探索

MATHEMATICAL STRUCTURE AND THE CONJECTURED EXACT SOLUTION OF THREEDIMENSIONAL(3D)ISING MODEL

本文首先回顾Ising模型的研究历史,包括Ising模型简介、二维和三维Ising模型的研究进展,特别是二维Ising模型的精确解.然后介绍作者提出的有关三维Ising模型的2个猜想以及推定的精确解.从拓扑、代数和几何的角度对三维Ising模型的数学结构进行了评述.分析三维Ising模型的转移矩阵、拓扑理论中的纽结变换、Yang-Baxter方程和四面体方程之间的关系,还介绍了三维Ising模型中存在的非局域效应、与量子场论和规范理论的关系、权重因子的物理意义、无限大温度及附近的奇异性和拓扑相变.指出一些近似计算方法(例如,低温展开、高温展开、重整化群和Monte Carlo模拟等)在研究三维Ising模型时的局限性.

In this article, the history of study on Ising model was first reviewed briefly, including a brief introduction of Ising model, the advances in the study of two- dimensional（2D） and three- dimensional（3D） Ising models, with a special interest in the exact solution of the 2D Ising model. Then two conjectures and putative exact solution of the 3D Ising model were introduced, and the mathematical structure of the 3D Ising model was investigated from the aspects of topology, algebra and geometry. The transfer matrices of the 3D Ising model, the knot theory in the topology, the relations between the Yang- Baxter equations and the tetrahedron equations were analysized. The non-local effect in the 3D Ising model, the relation between quantum field theory and gauge theory, the physical significance of weight factors, the singularity and the topological phase transition at/near infinite temperature in the 3D Ising model were also discussed. Finally, it was pointed out that some approximation techniques（for examples, low-temperature expansions, high-temperature expansions, renormalization group and Monte Carlo simulations） have disadvantages for studying the 3D Ising model.