Mathematical Studies of Short-Ranged Spin Glasses

在这项资助下支持的工作涉及被称为自旋玻璃的无序磁铁理论的数学基础。虽然存在理想（和不现实）自旋玻璃模型的建议解决方案，但我们感兴趣的是回答与真实实验室自旋玻璃行为有关的基本统计力学问题。尽管经过二十年的深入研究，其中许多问题仍然存在争议，包括自旋玻璃相的有序性质，基态（和低温纯相）的数量，自旋玻璃亚稳态的性质（热力学相的集合），理解异常动力学行为，如慢弛豫和老化，并证明相变的存在或不存在。所使用的方法和引入的概念不仅适用于自旋玻璃，也适用于其他无序系统，其中许多仍然知之甚少。我们对固态和液态有序系统的深刻物理和数学理解——例如晶体、铁磁体、超导体、液晶等——在本世纪下半叶具有重要的科学和技术基础。然而，存在着许多系统，无论是日常使用中熟悉的还是不熟悉的，其中随机性或无序性起着关键作用，我们对这些系统的数学和物理理解仍然相对原始。一个熟悉的例子是普通的窗玻璃，其中的原子或分子“卡”在随机的位置（与在冰中发现的规则晶体阵列相反）。自旋玻璃是一种无序磁系统，被认为是这种宏观“冻结”无序的原型，它们可能比本类中的其他材料更适合数学分析。然而，即使在这方面也没有取得什么根本性的进展。这项工作旨在解决有关这些材料的基本数学和物理问题，并为各种各样的无序系统提供一般的理论方法。

The work that is supported 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, we are interested in answering fundamental statistical mechanical questions that bear on the behavior of real laboratory spin glasses. These questions, many of which remain controversial despitetwo decades of intensive study, include the nature of ordering in the spin glass phase, the number of ground states (and low-temperature pure phases), the nature of the spin glass metastate (an ensemble of thermodynamic phases), understanding anomalous dynamical behavior such as slow relaxation and aging, and proving the presence or absence of a phase transition. The methods used and concepts introduced should be relevant not only for spin glasses but should also be applicable to other disordered systems, many of which remain poorly understood.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 of fundamental scientific and technological importance throughout the second half of this century. However, there exist many systems, both familiar and unfamiliar in everyday use, in which randomness or disorder playsa 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 forthis kind of macroscopic "frozen-in" disorder, and they may be more amenable to mathematical analysis than other materials in this class. Nevertheless, little fundamental progress has been made even here. This work is aimed at resolving basic mathematical and physical issues concerningthese materials and at providing a general theoretical approach for a wide variety of disordered systems.