Theory and application of Berry phase methods in solids

NONTECHNICAL SUMMARY An improved understanding of the electronic properties of materials is fundamental to the development of many modern technologies, especially those relying on electronic, magnetic, and optical materials. In recent years, a class of mathematical methods based on so-called "Berry phases" and related topological concepts have begun having a profound impact on our understanding of the electronic structure of materials and on our ability to compute their properties. This research project is focused on the translation of these mathematical concepts into practical tools for the computation of important properties of materials, including materials whose electronic configuration shows unconventional topological behavior. Part of the project focuses on a better understanding of the underlying theory, together with the development of new or improved computer codes that embody these mathematical methods. Another part of the work involves the application of these mathematical and computational tools to improve our understanding of known materials and to assist in the search for new materials with improved properties. The research is expected to advance the availability of materials with useful electronic, magnetic, and optical properties. The project involves training and mentorship of graduate students that will contribute to their career advancement and to the development of scientific workforce, while algorithms and computer codes developed by the project will be made available in open-source form for the benefit of the wider scientific research community. The project also holds out promise for the identification and evaluation of electronic materials that may ultimately find commercial applications. TECHNICAL SUMMARY This project is focused on theoretical research on the electronic properties of materials, with a special emphasis on physical properties whose underlying mathematical description involves geometric quantities based on Berry phases and curvatures. These are typically properties for which macroscopic orbital currents play an important role, and include electric polarization, orbital magnetization, anomalous Hall conductivity, circular and gyrotropic optical effects, and the subtle role of magnetism on lattice dynamics. The proper mathematical description of these properties underlies much recent progress in the theory of topological insulator and semimetal phases and of moire-scale multilayer systems that have been the focus of recent attention. The objectives of this project are: i) to further develop the formal theory of the aspects of electronic structure that depend upon a description in terms of geometric quantities; ii) to invent and disseminate accurate and efficient computational methods for computing materials properties related to these mathematical concepts; and iii) to use computational methods to identify promising new materials or structures in which these properties can manifest themselves, potentially leading to technological applications. The need to understand the relations between bulk and surface properties requires further progress in our ability to describe geometrical properties locally, not just globally, motivating further progress in the theory of "local markers" and related approaches. It is timely to extend theories developed for static systems to include frequency dependence, especially regarding ordinary and spatial-dispersive optical effects that occur in low-symmetry (e.g., chiral magnetic) materials. The influence of electronic Berry phases and curvatures on the dynamics of phonons in spin-orbit coupled magnetic materials will also be explored. As a cross-cutting theme, first-principles calculations will be carried out to quantify the physical properties that are predicted on the basis of the newly developed formal descriptions, with an eye towards identifying novel materials showing unusual or enhanced properties. The project is expected to result in the development of algorithms that will ultimately be implemented in open-source code packages and made available to the wider electronic-structure community. Training and mentorship of the graduate students will contribute to scientific workforce development. The project also holds out promise for the identification and evaluation of electronic materials that may ultimately find commercial applications. 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.