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.
