Mathematical Sciences: Classical and Quantum Mechanical Lattice Spin Systems

这个项目涉及晶格上的经典自旋系统和量子力学自旋系统。这项研究将从开发严格研究处于唯一吉布斯状态和关联指数衰减的阶段的系统的技术开始。这些技术将被应用于各种问题，这些问题可以以显示给定系统处于这样一个阶段的形式表现出来。这些问题包括多数规则重整化群变换，Haldane对自旋1海森堡链的预言，以及一些经典的统计力学模型，它们被认为在所有温度下都具有唯一的Gibbs态。其他需要研究的问题包括二维自旋1/2海斯堡反铁磁体中反铁磁序的存在，量子自旋系统的Rayleigh-Schrodinger微扰理论，以及Dobrushin和Schlosman关于经典系统中Gibbs态唯一性的工作在量子系统中的推广。

This project is concerned with both classical and quantum mechanical spin systems on a lattice. The research will begin by developing techniques for rigorously studying systems which are in a phase in which there is a unique Gibbs state and exponential decay of the correlations. These techniques will be applied to various problems which can be cast in the form of showing that a given system is in such a phase. These problems include the majority rule renormalization group transformation, the predictions of Haldane for the spin 1 Heisenberg chain, and some classical statistical mechanics models that are believed to have a unique Gibbs state at all temperatures. Other problems to be studied include the existence of antiferromagnetic order in the two-dimensional spin 1/2 Heisberg antiferromagnet, Rayleigh- Schrodinger perturbation theory for quantum spin systems, and the generalization of Dobrushin and Schlosman's work on the uniqueness of the Gibbs state in classical systems to quantum systems.