CAREER: Schrödinger Operators on Lattices

职业：格子上的薛定谔算子

• 批准号: 2143369
• 负责人: Rui Han
• 金额: $ 46.48万
• 依托单位: Louisiana State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-09-01 至 2027-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2143369&HistoricalAwards=false
• 关键词: CAREER; Schr; dinger; Operators; Lattices

This project focuses on the study of electrons on a lattice material structure, for example, graphene, under external magnetic fields. Important questions are: As time evolves, will electrons escape, showing metal-like behavior of the material, or will electrons stay confined near their original positions, showing insulator-like behavior? Furthermore, are there mathematical ways to quantify these behaviors? Understanding these features in different materials, including multi-layer graphene, has important applications, including electricity transmission, design of room-temperature superconductor, and quantum computing. The educational part of this project includes leading undergraduate research groups, mentoring graduate students, developing graduate courses on material sciences and spectral theory, organizing conferences and workshops for junior researchers to present and exchange ideas, and developing an online lecture series on modern research topics for middle and high school students.Of specific interest in this project is the analysis of the magnetic Laplacian, which characterizes the motion of electrons under external magnetic fields and is a central topic in quantum mechanics. For magnetic fluxes with irrational magnitude, the spectra of the discrete magnetic Laplacians on the square lattice form a beautiful self-similar fractal structure called Hofstadter’s butterfly. This fractal structure, in particular the existence of spectral gaps, was a cornerstone of the first derivation of the quantum Hall effect and the theory of topological insulators. The planned focus lies on developing techniques to study magnetic Laplacians on various discrete lattices, including the bilayer graphene lattice. The topics of investigation include the magic-angle bilayer graphene, metal-insulator transitions, spectral gaps and quantum Hall effect, Anderson localization and eigenfunction asymptotics, and quantum dynamics. To tackle these questions, deep results of harmonic analysis, dynamical system, number theory, and other areas are to be combined.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.

该项目的重点是对晶格材料结构的电子研究，例如石墨烯，在外部磁场下。重要的问题是：随着时间的发展，电子会逃脱，显示材料的金属样行为，还是电子会限制在其原始位置附近，显示出类似绝缘体的行为？此外，是否有数学方法可以量化这些行为？在包括多层石墨烯在内的不同材料中了解这些特征，具有重要的应用，包括电动传输，室温超导体的设计和量子计算。该项目的教育部分包括领先的本科研究小组，心理研究生，开发有关物质科学和光谱理论的研究生课程，组织会议和研讨会，供初级研究人员提出和交流思想，并开发有关中学学生的现代研究主题的现代研究主题的在线讲座系列。该项目的特定磁力是磁性磁力的分析，该运动的分析是对磁性运动的分析，其特征是该运动的分析，其特征是该动作的特征，该动作构成了该项目的特征，该运动的特征是该项目的特征。力学。对于具有非理性幅度的磁通量，方格上离散的磁性拉板式的光谱形成了一种美丽的自相似分形结构，称为Hofstadter的蝴蝶。这种分形结构，尤其是光谱间隙的存在，是量子霍尔效应的第一个推导和拓扑绝缘体理论的基石。计划的重点在于开发在包括双层石墨烯晶格在内的各种离散晶格上研究磁性laplacian的技术。调查的主题包括魔法双层石墨烯，金属 - 绝缘体过渡，光谱间隙和量子厅效应，安德森定位和特征函数渐近学以及量子动力学。为了解决这些问题，将结合谐波分析，动态系统，数字理论和其他领域的深刻结果。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响来评估，被视为珍贵的支持。