Collaborative Research: Semiclassical Methods for the Study of Spin Systems

该奖项由2009年美国复苏和再投资法案(公法111-5)资助。将使用自旋相干态路径积分和其他数学和理论物理工具对自旋系统的半经典行为进行研究。这项研究的很大一部分将在理解各种可实验实现的物理系统的背景下进行。相干态路径积分，特别是自旋路径积分，长期以来一直被认为在数学上很难处理。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，这为研究开辟了许多新的领域。要解决的基本问题包括：(I)算符有序和Bohr-Sommerfeld规则，可能高于半经典极限中的主阶；(Ii)基于最近对多自旋系统半经典传播子的评估，发展类似Gutzwill er的迹公式；(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；以及(V)从整体反常的观点研究多自旋传播子。将应用基本理论发展的物理系统包括(I)分子固体中的小磁性分子，特别是通过迹公式，(Ii)光学晶格中原子凝聚的所谓玻色-哈伯德模型，(Iii)反铁磁自旋链，重点是各向异性的作用，以及(Iv)晶格自旋模型。自旋情形中涉及的许多概念都被粒子相干态路径积分所共享，这也将被用来理解亚稳态系统热统计逃逸中的未解决问题。计算将在很大程度上是解析的，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，并研究它们与研究半经典极限的其他方法的关系，是理论物理学家们理想的补充，因为路径积分作为统计和量子力学中的基本工具已经得到了证实。更广泛的影响：该项目将通过培训本科生和研究生，以及指导博士后助理，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.