Collaborative Research: Mathematical Studies of Short-Ranged Spin Glasses

New York University, New York NY

Investigators

Abstract

NSF Award Abstract - DMS-0102541 Mathematical Sciences: Collaborative Research: Mathematical Studies of Short-Ranged Spin Glasses Abstract DMS-0102541 Stein The work under this grant addresses the mathematical foundations of the theory of disordered magnets known as spin glasses. Although there exist proposed solutions of idealized (and unrealistic) spin glass models, the principal investigators' interests center on answering fundamental statistical mechanical questions that bear on the behavior of real laboratory spin glasses. These questions, many of which remain controversial despite two decades of intensive study, include the nature of ordering in the equilibrium spin glass phase, understanding anomalous behavior, such as slow relaxation and aging arising from nonequilibrium dynamics, and proving the presence or absence of a phase transition in finite dimensions. The methods used and concepts introduced should be relevant not only for spin glasses but also for other disordered systems, many of which remain poorly understood. Moreover, the generality of the principal investigators' approach to dynamics should yield progress in certain aspects of nonequilibrium dynamics in both homogeneous and disordered systems. Our deep physical and mathematical understanding of ordered systems in the solid and liquid state --- for example, crystals, ferromagnets, superconductors, liquid crystals, and many others --- has been both of fundamental scientific importance and has spurred profound technological change throughout the second half of the last century. However, there exist many systems, both familiar and unfamiliar, in which randomness or disorder plays a key role, and in which our mathematical and physical understanding remains comparatively primitive. One familiar example is ordinary window glass, where the atoms or molecules are "stuck" in random locations (as opposed to a regular crystalline array as would be found, for example, in ice). Spin glasses are disordered magnetic systems which are thought to be prototypes for this kind of macroscopic "frozen-in" disorder. Disordered systems in general present both fundamental scientific challenges and at the same time hold great promise for applications. The latter includes not only the possibility of new materials and devices but also the creation of new algorithms and applications to the biological and other sciences. Progress in understanding these systems is therefore greatly desirable. Spin glasses may be more amenable to mathematical analysis than other materials in this class. Nevertheless, little fundamental progress has been made even here. The principal investigators' work is aimed at resolving basic mathematical and physical issues concerning these materials and at providing a general theoretical approach for a wide variety of disordered systems.