Localization and Diffusion in Open and Many Body Quantum Systems

这个项目的主要重点是研究无序对量子系统的影响。在更广泛的背景下，这项工作从一个基本的科学问题中获得灵感：“无序的影响是什么？”这是一个与任何科学模型相关的基本问题，即使其中没有明确包括无序。毕竟，任何真实世界的系统都受到少量噪声的影响，经验表明，即使是微弱的无序也可能对系统的行为产生深远的影响。本项目研究的方程起源于无序物质理论，但由于波动和无序的基本性质而引起普遍的兴趣。理解这些方程的解的进展将提高对理论物理和应用数学模型的基本理解。此外，这项研究的一个中心目标是教学：向学生介绍一门基础学科，并向他们传达数学和理论物理是充满活力、不断发展的领域。该项目将通过一个关于物理模型中无序影响的研究项目进行。研究的主要重点是理解受内在无序影响的量子系统的演化。当无序性较弱时，预期的结果是波函数的扩散演化。在更强的无序状态下，已知波函数的局域化可以发生，尽管在许多体和开放量子系统中局域化的确切性质尚不清楚。两个关键目标是1)分析波在弱无序介质中任意长时间的扩散，2)阐明许多物体模型中的局部化性质。近年来，PI和各种博士后和学生合作者研究了时变随机介质中的波扩散问题，其时间依赖性由马尔可夫过程产生。对于这样的模型，例如紧结合薛定谔方程，可以通过谱分析来建立扩散传播。本项目的目的之一是将时间无关方程作为这些时间相关方程的扰动来处理，这些时间相关方程具有共享预期定性行为的优点。同时，我们将研究相互作用系统的极化子型模型中无序诱导的局域化。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The main focus of this project is the study of the effects of disorder on quantum systems. In a broader context, the work draws inspiration from the basic scientific question: "What are the effects of disorder?" This is a fundamental question relevant to any scientific model, even one in which disorder is not explicitly included. After all, any real world system is subject to a small amount of noise, and experience shows that even weak disorder may have a profound effect on the behavior of the system. The equations studied in this project arise in the theory of disordered materials, but are of general interest because of the fundamental nature of both wave motion and disorder. Progress in understanding the solutions to these equations will improve basic understanding of models of theoretical physics and applied mathematics. In addition, a central goal of the research is pedagogical: to introduce students to a fundamental subject and convey to them that mathematical and theoretical physics are vibrant, growing fields. The project will proceed through a program of research on the effects of disorder in physical models. The main focus of the research is in understanding the evolution of quantum systems subject to intrinsic disorder. When disorder is weak, the expectation is that a diffusive evolution of the wave function results. At stronger disorder, it is known that localization of the wave function can occur, although the exact nature of localization in many body and open quantum systems is poorly understood. Two key goals are 1) to analyze the diffusion of waves in a weakly disordered medium over arbitrarily long times, and 2) to clarify the nature of localization in many body models. In recent years the PI and various post-doc and student collaborators have considered the problem of wave diffusion in time-dependent random media, with the time dependence generated by a Markov process. For such models the diffusive propagation, e.g., for the tight binding Schrodinger equation, can be established by spectral analysis. One aim of the present project is to approach the time independent equation as a perturbation of these time dependent equations, which have the virtue of sharing the expected qualitative behavior. In parallel, we will study disorder induced localization in polaron type models of a interacting systems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.