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。研究将进行到自旋系统的半经典行为，使用自旋相干态路径积分和其他数学和理论物理工具。本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完成。相干态路径积分，尤其是自旋的路径积分，一直被认为在数学上难以处理。然而，在过去的几年里，在理解和使用这些路径积分方面取得了一些进展，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，可能在半经典极限下高于导阶，（ii）基于最近对多自旋系统的半经典传播子的评估，开发类gutzwiller示踪公式，（iii）通过路径积分添加角动量的群理论分解规则，（iv）这些规则的半经典行为，(5)从全球异常的角度研究多自旋传播子。基础理论发展将应用的物理系统包括(i)分子固体中的小磁性分子，特别是通过痕量公式；（ii）光学晶格中原子凝聚物的所谓玻色-哈伯德模型；（iii）反铁磁自旋链，强调各向异性的作用；（iv）晶格自旋模型。自旋情况中涉及的许多概念是由粒子相干态路径积分共享的，这些积分也将用于理解亚稳态系统热统计逸出中尚未解决的问题。计算将主要是分析性的，但必要时也将采用数值方法。将路径积分方法扩展到自旋系统，并研究它们与研究半经典极限的其他方法的关系，是理论物理学家的理想补充，因为路径积分在统计和量子力学中已经被证明是一种重要的工具。更广泛的影响：该项目将通过培养本科生和研究生，以及博士后助理的指导来培养未来的研究人员。

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.