Equilibrium and Non-Equilibrium Quantum Statistical Mechanics

Bruno Nachtergaele将研究平衡和非平衡量子统计力学问题。重点是量子自旋模型，但也将研究其他一些量子多体系统。在这个项目中有四组问题。1)严格研究量子自旋模型中的陈-西蒙斯相，特别是弗里德曼模型。2)玻色化和量子中心极限定理，及其在XXZ海森堡链低位谱大自旋渐近分析中的应用。3)磁界面动力学的微观模型。这包括研究畴壁和液滴在杂质处的束缚态、畴壁的扩散、更高维度的畴以及具有畴壁的磁线中的电子散射。4)非平衡定态、熵产生与热力学流的微观描述。对于最后两个主题，Nachtergaele将研究简单模型的具体动力学问题，这些模型具有低能态和非平凡的几何结构，如界面。长期目标之一是在非平衡态统计力学理论方面取得进展。为了定量地了解纳米尺度下电子元件和测量设备的许多特性，需要能够研究同时考虑量子力学效应和界面现象的模型的分析方法。Nachtergaele将研究模型问题以开发此类方法。特别强调将放在磁畴和它们的动态之间的接口。这是理解磁存储设备（如计算机硬盘）动态特性的一个基本问题。同样的数学技术也适用于自旋电子学的新领域。方法是研究一些特定的模型，期望这些模型的足够详细的解决方案将为更一般的系统中类似问题的答案指明方向。该项目包含了一些有吸引力的研究问题，为学生从各种各样的背景，以及数学学生有兴趣的机会，成为熟悉数学以外的领域，其中数学是应用在一个高水平。现代材料科学，非常依赖于材料的计算建模和设计，就是一个例子。

Bruno Nachtergaele will work on problems in equilibrium and non-equilibrium quantum statistical mechanics. The main focus is on quantum spin models, but also some other quantum many-body systems will be studied. There are four groups of problems in the project. 1) Rigorous study of Chern-Simons phases in quantum spin models, in particular in the Freedman model. 2) Bosonization and quantum central limit theorems, with applications to the analysis of the asymptotics for large spin of the low-lying spectrum of the XXZ Heisenberg chain. 3) Microscopic models for magnetic interface dynamics. This includes the study of bound states of domain walls and droplets at impurities, diffusion of domain walls, domains in dimensions higher dimensions and electron scattering in a magnetic wire with domain walls. 4) Non-equilibrium stationary states, entropy production, and the microscopic description of thermodynamic currents. For the last two topics, Nachtergaele will study concrete dynamical questions of simple models that have low-energy states with non-trivial geometric structures such as interfaces. One of the long-term goals is to make progress in the theory of non-equilibrium statistical mechanics.In order to quantitatively understand many properties of electronic components and measuring devices at the nanometer scale, analytic methods are needed that allow us to study models that take into account both quantum mechanical effects and interface phenomena. Nachtergaele will work on model problems to develop such methods. Particular emphasis will be put on interfaces between magnetic domains and their dynamics. This is a fundamental problem in understanding the dynamics of magnetic memory devices, such as computer hard disks. The same mathematical techniques also apply to the new field of spintronics. The approach is to study a number of specific models with the expectation that a sufficiently detailed solution of these models will point the way to answers for similar questions in more general systems. The project contains a number of attractive research problems for students from a wide variety of backgrounds, as well as for mathematics students interested in opportunities to become familiar with areas outside of mathematics where mathematics is applied at a high level. Modern materials science, which very much relies on computational modeling and design of materials, is an example.