CAREER: Schrödinger Operators on Lattices

该项目主要研究外部磁场作用下晶格材料结构(例如石墨烯)上的电子。重要的问题是：随着时间的推移，电子会逃逸，显示出材料的类似金属的行为，还是会被限制在其原始位置附近，表现出绝缘体的行为？此外，有没有数学方法来量化这些行为？了解不同材料中的这些特征，包括多层石墨烯，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。这个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的研究生课程，组织初级研究人员展示和交流想法的会议和研讨会，以及为初中生开发关于现代研究主题的在线讲座系列。对这个项目特别感兴趣的是对磁拉普拉斯的分析，它描述了电子在外部磁场下的运动，也是量子力学的中心主题。对于无理大小的磁通量，正方形晶格上离散的磁拉普拉斯算符的谱形成了一个美丽的自相似分形结构，称为霍夫斯塔特蝴蝶。这种分形结构，特别是光谱带隙的存在，是量子霍尔效应和拓扑绝缘体理论第一次推导的基石。计划的重点是开发技术来研究各种离散晶格上的磁性拉普拉斯效应，包括双层石墨烯晶格。研究的主题包括魔角双层石墨烯，金属-绝缘体跃迁，光谱禁带和量子霍尔效应，安德森定域和本征函数渐近性，以及量子动力学。为了解决这些问题，调和分析、动力系统、数论和其他领域的深刻结果将被结合起来。这个奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.