CAREER: Schrödinger Operators on Lattices

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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.