CAREER: Quantum Systems with Deterministic Disorder

职业：具有确定性无序的量子系统

基本信息

批准号：
1846114
负责人：
Ilya Kachkovskiy
金额：
$ 40万
依托单位：
Michigan State University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-07-01 至 2025-06-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1846114&HistoricalAwards=false
关键词：
CAREER Quantum Systems Deterministic Disorder

项目摘要

Disordered quantum systems are fundamental objects in modern mathematical physics, since, due to presence of heat, any real-life system has some level of noise in it. Systems with noise/disorder are often studied by probabilistic methods and assume that the noise is random. The main scope of the project is to study systems where the noise has additional structure and has deterministic (non-random) nature. In physics, strong random disorder often implies that the system becomes an insulator and prevents electrons in it from moving freely. Would it also be true for some systems with non-random disorder? If yes, which properties make non-random systems behave like random and can it be measured? Are there any additional effects if the disorder is not strong? The main goal of the project is to address these questions on many levels, which will include work with undergraduate and graduate students, development of graduate courses on dynamics and spectral theory, and developing a course for high school students that would illustrate connections between basic linear algebra and physics, providing them skills and motivation for possible further education in STEM.This project incorporates teaching and research activities on the analysis of a class of models of mathematical quantum physics, including developing abstract techniques of operator theory and establishing rigorous results on more concrete systems. All proposed models involve disorder, however, unlike usual probabilistic view on disordered systems, the main goal will be studying the disorder in a completely deterministic setting, or with a very small number of random/ergodic parameters. An example of such system would be a Schrodinger operator with dynamically-defined potential, where the underlying dynamical system has small dimension and low degree of mixing (for example, irrational rotation). Typically, quantum systems with large random disorder tend to prevent electrons from moving freely (Anderson localization). To answer even basic questions about electron transport in deterministic disordered systems, one must replace usual probabilistic methods by methods of number theory, ergodic theory, semi-algebraic geometry, and other deep areas. The main directions of the project involve analysis of localization/delocalization for systems of interacting quasiperiodic particles and the effect of interaction, perturbative methods for single-particle operators with rough potentials, perturbation properties for spectral bands of periodic operators, and abstract methods of operator theory applied to almost commuting operators and matrices, with applications to quantum 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.

无序量子系统是现代数学物理学中的基本对象，因为由于存在热量，任何现实生活中的系统都具有一定程度的噪声。经常通过概率方法研究具有噪声/混乱的系统，并假设噪声是随机的。该项目的主要范围是研究噪声具有额外结构并具有确定性（非随机）性质的系统。在物理学中，强大的随机疾病通常意味着系统成为绝缘体，并防止其中的电子自由移动。对于某些非随机疾病的系统也是如此吗？如果是，哪些属性会使非随机系统的行为像随机，并且可以测量吗？如果该疾病不强，是否会有其他影响吗？该项目的主要目标是在许多层面上解决这些问题，其中包括与本科生和研究生的合作，开发研究生课程，有关动态和光谱理论的研究生课程，并为高中生开发一门课程，以说明基本线性代数和物理学之间的联系，从操作者理论并在更具体的系统上建立严格的结果。但是，所有提出的模型都涉及混乱，但是，与无序系统上的常见概率观点不同，主要目标是在完全确定性的环境中研究该疾病，或者使用少量的随机/嗜酸群参数。这样的系统的一个例子是具有动态定义电势的Schrodinger操作员，其中基础动力学系统具有较小的尺寸和较低的混合度（例如，非理性旋转）。通常，具有较大随机疾病的量子系统倾向于防止电子自由移动（安德森本地化）。为了回答有关确定性混乱系统中电子传输的基本问题，必须通过数字理论，千古理论，半代数几何形状和其他深层区域替代通常的概率方法。该项目的主要方向涉及分析相互作用的准膜颗粒系统的定位/定位化以及相互作用的效果，具有粗糙电位的单粒子操作员的扰动方法，对周期性操作员的光谱带的扰动特性，对经期间操作员的光谱带的扰动特性，以及对操作员的摘要和统计型统计型和统计的统计范围，并适用于运营商的应用程序，并适用于统计的统计信息，并适用于通勤的系统，并适用于系统的应用。使用基金会的知识分子优点和更广泛的审查标准，通过评估被认为值得支持。