Quantum Statistical Mechanics of Lattice Systems

Nachtergaele该项目的主要焦点是在凝聚态物理中出现的量子自旋模型。Nachtergaele将研究量子自旋（及相关）系统中的三类问题。首先是无限体积中基态完备集的表征。在这里，基态被定义为在任意局部扰动下的稳定性。对于具有唯一基态的模型，例如AKLT模型，最好处理这个问题。一个更难但也更重要的问题是描述完整的基态集合，当这个集合包括非周期状态时，比如界面状态，这是情况，例如，对于XXZ海森堡模型。第二个问题是研究零温度和非零温度下量子统计力学模型中界面态的涨落特性和低洼激发。我们的建议是在Falicov-Kimball模型中证明111界面在足够低的温度下的稳定性，并在第二阶段，在XXZ Heisenberg模型中研究111界面，在那里应该期待与连续光谱相关的新现象。第三条建议的研究路线是分析和使用密度矩阵重整化群方法的新实现来研究准一维系统中的各种现象。量子统计力学是一个越来越受关注的领域，因为今天发现和研究的许多现象本质上都是量子力学：玻色凝聚、超流动性、超导性、量子光学、量子计算，以及磁性的许多最有趣的方面。量子自旋模型是多种材料的磁性模型，其中许多是最近才合成的。磁性的重要性不仅在于它使我们能够理解磁铁的行为。通常材料的磁性（自旋）结构与电子和几何结构之间存在相互作用。在过去几年中备受关注的新型自旋-佩尔斯材料就是这种相互作用的一个很好的例子。人们早就知道反铁磁相关是高温超导体的一个重要方面。另一个磁性和输运性质相互影响的例子是巨磁阻材料。这个项目的方法是数学的，是对化学家和物理学家对相同材料的实验和理论研究的补充。这个提议解决了关于量子晶格模型的数学结构的基本问题。Nachtergaele还将致力于将新的数学见解应用到更有效的计算方法中。这项研究将为研究生和本科生提供动手学习的机会，他们不仅想精通数学，而且对具体应用感兴趣。在这个过程中，这些学生将熟悉从事相同或相关问题的物理学家和化学家的语言。这将为他们未来在跨学科环境中的职业生涯做好准备。

9706599 Nachtergaele The main focus of the project is on quantum spin models that arise in condensed matter physics. Nachtergaele will uork on three classes of problems in quantum spin (and related) systems. The first is the characterization of the complete set of ground states in infinite volume. Here, ground states are defined in the sense of stability under arbitrary local perturbations. This problem is best tractable for models with a unique ground state, such as the AKLT model. A harder but also more important problem is to characterize the complete set of ground states when this set includes non-periodic states, such as interface states, which is the case, e.g., for the XXZ Heisenberg model. A second problem is to study the fluctuation properties and low-lying excitations of interface states in quantum statistical mechanics models both at zero and non-zero temperature. The proposal is to prove stability of the 111 interface at sufficiently low temperatures in the Falicov-Kimball model, and, in a second phase, to study 111 interfaces in the XXZ Heisenberg model, where new phenomena related to continuous spectrum should be expected. The third proposed line of research is to analyze and use a new implementation of the Density Matrix Renormalization Group method to study various phenomena in quasi-one-dimensional systems. Quantum statistical mechanics is an area of growing interest because many of the phenomena that are being discovered and studied today are intrinsically quantum mechanical: Bose condensation, superfluidity, superconductivity, quantum optics, quantum computation, and many of the most interesting aspects of magnetism. Quantum spin models are models for the magnetic properties of a large variety of materials, many of which have only been recently synthesized. The importance of magnetic properties is not just that they allow us to understand the behavior of magnets. Often there is an interaction between magnetic (spin) structure and elect ronic and geometric structure of the material. The novel spin-Peierls materials that in the last couple of years received so much attention are a good example of this interaction. It has been known for some time that antiferromagnetic correlations are an important aspect of high-temperature superconductors. Another example where magnetic and transport properties influence each other are the Giant Magneto Resistance materials. The approach in this project is mathematical, and is complementary to the experimental and theoretical research on the same materials by chemists and physicists. This proposal addresses fundamental questions about the mathematical structure of quantum lattice models. Nachtergaele will also work on the implementation of new mathematical insights into more efficient computational methods. This research will provide hands-on learning opportunities for graduate and undergraduate students who not only want to become well-versed in mathematics but also have an interest in concrete applications. In the process these students will become familiar with the language of physicists and chemists working on the same or related problems. This will prepare them better for future careers in an interdisciplinary environment.