Collaborative Research: Semiclassical Methods for the Study of Spin Systems

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。研究将使用自旋相干态路径积分和其他数学和理论物理工具进行自旋系统的半经典行为。这项研究的一个重要部分将在理解各种实验可实现的物理系统的背景下进行。相干态路径积分，特别是自旋的路径积分，一直被认为是数学上难以处理的。然而，在过去的几年中，已经有一些进步的理解和使用这些路径积分开辟了许多新的研究领域。要追求的基本问题包括：（i）算子排序，和Bohr-Sommerfeld规则，可能高于半经典极限中的前导阶，（ii）基于最近对多自旋系统的半经典传播子的评估，发展Gutzwiller样迹公式，（iii）通过路径积分增加角动量的群论分解规则，（iv）这些规则的半经典行为，（v）从整体反常的观点研究多自旋传播子。物理系统的基本理论的发展将适用于包括（i）小磁性分子在分子固体，特别是通过跟踪公式，（ii）所谓的玻色哈伯德模型的原子凝聚在光学晶格，（iii）反铁磁自旋链，强调各向异性的作用，和（iv）晶格自旋模型。在自旋情况下涉及的许多概念是共享的粒子相干态路径积分，这也将被用来理解未解决的问题，在亚稳态系统的热统计逃逸。计算将主要是分析性的，但将根据需要采用数值方法。将路径积分方法扩展到自旋系统，并研究它们与其他研究半经典极限的方法的关系，是理论物理学家武器库中一个理想的补充，因为路径积分在统计和量子力学中已经被证明是一个重要的工具。更广泛的影响：该项目将通过对本科生和研究生的培训以及对博士后同事的指导，为未来的研究人员做好准备。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).Research will be carried out into the semiclassical behavior of spin systems, using spin-coherent-state path integrals and other mathematical and theoretical physics tools. A significant part of this research will be done in the context of understanding a variety of experimentally realizable physical systems. Coherent-state path integrals, especially those for spin, have long been regarded as mathematically difficult to work with. In the last few years, however, there have been several advances in the understanding and uses of these path integrals which have opened up many new areas for study. The basic problems to be pursued include (i) operator ordering, and Bohr-Sommerfeld rules, possibly to higher than leading order in the semiclassical limit, (ii) development of Gutzwiller-like trace formulas, based on the recent evaluation of the semiclassical propagator for multispin systems, (iii) group theoretic decomposition rules for the addition of angular momenta via path integrals, (iv) semiclassical behaviour of such rules, and (v) study of the multispin propagator from the viewpoint of the global anomaly. The physical systems to which the fundamental theoretical developments will be applied include (i) small magnetic molecules in molecular solids, especially via the trace formula, (ii) so-called Bose-Hubbard models for atomic condensates in optical lattices, (iii)antiferromagnetic spin chains, with emphasis on the role of anisotropy, and (iv) lattice spin models. Many of the concepts involved in the spin case are shared by particle coherent-state path integrals, which will also be employed to understand unresolved issues in the thermostatistical escape of metastable systems. The calculations will be largely analytic, but numerical approaches will be employed as needed. The extension of path-integral methods to spin systems, and study of their relationship to other methods for studying the semiclassical limit, is a desirable addition to theoretical physicists' arsenal, since path integrals have a proven record as an essential tool in statistical and quantum mechanics. Broader Impacts: The project will prepare future researchers through the training of undergraduate and graduate students, and the mentoring of post-doctoral associates.