Studies of Model Systems in Statistical Mechanic and QuantumField Theory

将对可解的统计力学和量子场理论模型进行研究。我们将特别强调手征波茨模型。手征Potts模型最近已被求解其本征值，并显示出许多新的效应。该模型显示了一个公度-非公度相、旋光性的磁性模拟和Pokrovsky-Talapov相变。这类精确的能级交叉跃迁可能会对强相互作用的费米子系统产生影响，我们将对此进行进一步的研究。将计算无公度相中的序参数和关联函数。这个可解水平交叉系统的结果将被用来研究更一般的不可积模型。我们将研究更一般的可积模型。本课程将探讨与单极子数学、代数几何和拓扑场论的联系。研究复杂的、相互作用的多粒子系统是凝聚态物理和材料物理的基础。由于这些系统的复杂性，为了获得解，经常需要进行径向近似。然而，有时一个物理模型，如伊辛模型，能够被精确求解。这些精确的解决方案为理解精确模型和相关模型的基本物理和数学结构提供了宝贵的指导。手征Potts模型最近得到了精确求解，本研究将研究该模型在各类物理Pemonena中的应用，并进一步探索解的性质。

Research will be conducted on solvable statistical mechanical and quantum field theoretical models. Particular emphasis will be placed on the chiral Potts model. The chiral Potts model has recently been solved for its eigenvalues and displays a number of novel effects. The model exhibits a commensurate-incommensurate phase, the magnetic analog of optical activity and a Pokrovsky- Talapov transition. Further studies will be undertaken of this class of exact level crossing transitions which may impact strongly interacting fermion systems. Order parameters and correlation functions in the incommensurate phase will be computed. The results of this solvable level crossing system will be used to study more general non-integrable models. More general integrable models will be examined. Connections with the mathematics of monopoles, algebraic geometry and topological field theory will be explored. %%% The study of complex, interacting, many-particle systems lies at the foundation of condensed matter physics and materials physics. Because of the complexity of these systems, radical approximations frequently need to be made in order to obtain solutions. However, occasionally a physical model, such as the Ising model, is able to be solved exactly. These exact solutions provide invaluable guidance to understanding the underlying physics and the mathematical structure of the exact model and related models. The chiral Potts model has recently been solved exactly and the proposed research will investigate the application of this model to various classes of physical phemonena and further explore the nature of the solutions.