Twisted semiconductors in the semiclassical regime

This project investigates the mathematical foundations of new phases of matter in condensed matter physics, such as stacked and twisted two-dimensional semiconductors/graphene, which are at the forefront of modern quantum science. Although experimental observations have been very fruitful, understanding their novel electronic and physical properties still requires new mathematical methods. This project develops new tools to analyze these materials in the semiclassical regime: the limit where quantum and classical physics meet. This project supports the design of future electronic and quantum devices and the prediction of new quantum phenomena, with potential applications in quantum information technology and materials science. This project also contributes to emerging technologies that rely on two dimensional materials such as superconductive or quantum computing devices. The project serves the national interest by advancing foundational science in applied mathematics, quantum science and materials science. The project also supports undergraduate and graduate education, as well as promotes interdisciplinary collaboration and dissemination of scientific knowledge via scientific workshops and seminars. The investigator studies the semiclassical and spectral analysis of matrix-valued Schrödinger operators arising in twisted two-dimensional semiconductors and twisted bilayer graphene. These systems exhibit rich spectral and topological properties, especially at small twisting angles, where moiré patterns produce new low-energy effective theories. While semiclassical analysis is well-developed for scalar operators, this project develops new methods for matrix-valued Hamiltonians, essential for describing multi-band and spin-orbit coupled systems in these materials. Another goal is to extend semiclassical methods to problems in quantum information theory and quantum computing, where understanding spectral gaps and asymptotic behavior of Hamiltonians is critical for error correction and algorithm design. The project also aims to study problems in the spectral theory of quasi-periodic operators, particularly continuous Schrödinger operators with incommensurable electromagnetic fields. Current understanding is largely based on discrete models such as the Almost Mathieu operator, while this research seeks to overcome these restrictions by directly analyzing the spectrum of continuous Schrödinger operators with such electromagnetic fields in the semiclassical limit. The project integrates techniques from partial differential equations, semiclassical analysis, representation theory, Bloch-Floquet theory, algebraic topology, and differential geometry. The outcomes will provide rigorous mathematical foundations for understanding band structure and band topology in moiré materials, enable interdisciplinary collaboration, and help guide experimental discoveries in physics and material science, such as magic angles or new phases of matter in twisted semiconductors and twisted multilayer graphene. 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.