Quantum Statistical Mechanics of Lattice Systems

摘要：特纳盖勒将研究量子统计力学中的各种问题。在第一组问题中，他将研究磁性材料的量子自旋系统模型。这里主要关注的是畴壁和其他结构（如液滴）的平衡和非平衡性质。在Messager中，他为海森堡XXZ铁磁体中的111界面提出了一个新的量子固体对固体模型。对于该模型，将研究界面在严格正温度下的稳定性。本文将在一类一维模型中研究界面波动的普适性。另一个目标是证明零温度下量子自旋模型中液滴的存在，并为研究其动力学迈出第一步。在连续统量子统计力学中，snachtergaele将从相互作用费米子的薛定谔动力学推导出欧拉方程，并对各种类型的玻色凝聚进行表征。后者是与萨格勒布诺夫和布鲁合作的一个项目。第一项工作已经开始与HT Yau合作。量子统计力学是一个越来越受关注的领域，因为今天在凝聚态物理和材料科学中发现和研究的许多现象本质上都是量子力学：二次致密、超流动性、超导性、量子光学、量子计算，以及微观尺度上磁性的许多最有趣的方面。除了对材料科学的影响外，这项工作还将为量子统计力学的基本问题提供新的见解，并为解决这些问题提供分析方法。量子统计力学的数学模型虽然与偏微分方程、表示理论、算符的谱理论和概率论中的问题有关，但它们自己也提出了新的数学挑战。到目前为止，许多有趣的问题都无法用现有的分析和数值方法解决。这个项目的结果将有助于我们理解量子力学模型的基本数学结构，从长远来看，可能会导致量子力学模型的数值工作的创新，这在纯科学和应用科学中越来越需要。

ABSTRACTNachtergaele will work on a variety of problems in quantum statistical mechanics. In a first group of problems he will study quantum spin systemsas models for magnetic materials. The main focus here is on equilibriumand non-equilibrium properties of domain walls and other structuressuch as droplets. With Messager, he proposes a new quantum solid-on-solid model for the 111 interface in the Heisenberg XXZ ferromagnet. For this model the stability of the interface at strictly positive temperatures will be studied. The universality of fluctuations of interfaces will be investigated in a class of one-dimensional models. Another goal is to prove the existence of droplets in quantum spin models at zero temperature and to make first steps towards the study of their dynamics. In continuum quantum statistical mechanicsNachtergaele will work on the derivation of the Euler equations from theSchroedinger dynamics of interacting fermions, and on a characterization of the various types of Bose condensation. The latter is a project incollaboration with Zagrebnov and Bru. The first is work already begun with HT Yau. Quantum statistical mechanics is an area of growing interest because many of the phenomena that are being discovered and studied today in condensed matterphysics and material science are intrinsically quantum mechanical: Bosecondensation, superfluidity, superconductivity, quantum optics, quantumcomputation, and many of the most interesting aspects of magnetism at themicroscopic scale. In addition to the implications in material science, thiswork will provide new insights in fundamental problems of quantum statisticalmechanics and lead to analytical methods for their solution. The mathematicalmodels of quantum statistical mechanics, though related to problems in partialdifferential equations, representation theory, spectral theory of operators, and probability theory, pose new mathematical challenges of their own. Many fascinating questions have so far resisted existing analytical and numerical methods. The results of this project will contribute to ourunderstanding of the fundamental mathematical structures of quantum mechanicalmodels and, in the long run, may lead to innovations in numerical workon quantum mechanical models, which is increasingly needed in pure and applied science.