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.

该项目重点研究外部磁场下晶格材料结构（例如石墨烯）上的电子。重要的问题是：随着时间的推移，电子是否会逸出，从而表现出材料的类金属行为，或者电子是否会被限制在其原始位置附近，从而表现出类绝缘体的行为？此外，是否有数学方法来量化这些行为？了解包括多层石墨烯在内的不同材料的这些特征具有重要的应用，包括电力传输、室温超导体的设计和量子计算。该项目的教育部分包括领导本科生研究小组、指导研究生、开发材料科学和光谱理论研究生课程、为初级研究人员组织会议和讲习班以展示和交流想法，以及为中学生和高中生开发现代研究主题的在线讲座系列。该项目特别感兴趣的是磁拉普拉斯分析，它描述了电子在外部磁场下的运动，是量子领域的一个中心主题 机械师。对于无理量级的磁通量，方晶格上离散磁拉普拉斯的谱形成了一个美丽的自相似分形结构，称为霍夫施塔特蝴蝶。这种分形结构，特别是光谱间隙的存在，是量子霍尔效应和拓扑绝缘体理论首次推导的基石。计划的重点在于开发技术来研究各种离散晶格（包括双层石墨烯晶格）上的磁性拉普拉斯算子。研究主题包括魔角双层石墨烯、金属-绝缘体跃迁、光谱间隙和量子霍尔效应、安德森局域化和本征函数渐近以及量子动力学。为了解决这些问题，需要结合调和分析、动力系统、数论和其他领域的深入成果。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。