Theory and Application of Berry Phase Methods in Solids

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NONTECHNICAL SUMMARYAn improved understanding of the electronic properties of materials is fundamental to the development of many modern technologies, especially in regard to electronic, magnetic, and optical materials. In recent years, a class of mathematical methods based on so-called "geometric phases" or "Berry phases" have begun to have a profound impact on our understanding of the electronic structure of materials and our ability to compute their properties. This research program is focused on further development of these methods, both at the level of formal theory and in terms of computational implementation. Much of the work is targeted to topological materials, i.e., those in which the quantum-mechanical electronic wave functions are twisted in a certain sense. Such materials are often characterized by certain bulk properties (i.e., those that depend only on the interior of the crystal sample) that nevertheless have specific manifestations on their surfaces. An important theme of this project is to obtain a better understanding of this kind of "bulk-boundary correspondence," especially as it applies to magnetic and magnetoelectric properties. This award also supports the training and mentorship of graduate students by contributing to their career advancement and to the scientific workforce development. In addition, the algorithms and computer codes developed in this project will be contributed in open-source form for the benefit of the wider electronic-structure research community.TECHNICAL SUMMARYThis award supports theoretical and computational 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, and orbital magnetoelectric couplings. The proper mathematical description of these properties underlies much recent progress in the theory of topological insulator and semimetal phases of crystalline materials. The goals of this research activity include: (i) further development of the formal theory of electronic structure, especially concerning descriptions in terms of geometric quantities; (ii) in-depth studies of the relation between bulk and surface properties; (iii) invention and dissemination of accurate and efficient computational methods for computing materials properties related to these mathematical concepts; and (iv) utilization of computational methods to identify promising new materials or structures in which these properties can manifest themselves. A secondary focus will be on electronic dynamics at both the sudden level, as for quantum quenches across a topological phase boundary, and at the adiabatic level, in connection with the dynamics of phonons in spin-orbit coupled magnetic materials. Methods will include formal theoretical developments; implementation and testing in terms of simple model Hamiltonians; and first-principles electronic structure calculations. Graduate students will be mentored and trained in these methodologies, contributing to their education and career development. The algorithms and computer codes developed in this project will be contributed in open-source form for the benefit of the wider electronic-structure research community. The program also holds out promise for the identification and evaluation of new electronic materials that may ultimately be useful for 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.