Theory and Application of Berry Phase Methods in Solids

非技术总结对材料电学性质的更好理解是许多现代技术发展的基础，特别是在电子、磁性和光学材料方面。近年来，一类基于几何相或Berry相的数学方法已经开始对我们理解材料的电子结构和计算其性质的能力产生深远的影响。这项研究计划集中在这些方法的进一步发展，无论是在形式理论水平上还是在计算实现方面。许多工作是针对拓扑材料，即量子力学电子波函数在一定意义上扭曲的那些材料。这种材料的特征通常是某些块状性质(即，那些只依赖于晶体样品内部的性质)，但这些性质在其表面上有特定的表现。这个项目的一个重要主题是更好地理解这种“块体-边界对应”，特别是当它适用于磁性和磁电性质时。该奖项还支持对研究生的培训和指导，为他们的职业发展和科学的劳动力发展做出贡献。此外，在这个项目中开发的算法和计算机代码将以开放源代码的形式贡献给更广泛的电子结构研究社区。技术总结该奖项支持对材料电子性质的理论和计算研究，特别强调其基本的数学描述涉及基于Berry相和曲率的几何量。这些都是宏观轨道电流发挥重要作用的典型性质，包括电极化、轨道磁化、反常霍尔电导率和轨道磁电耦合。对这些性质的恰当的数学描述是晶体材料的拓扑绝缘体和半金属相理论的最新进展。这一研究活动的目标包括：(1)进一步发展电子结构的形式理论，特别是关于几何量的描述；(2)深入研究块体和表面性质之间的关系；(3)发明和传播与这些数学概念有关的准确和有效的计算材料性质的方法；以及(4)利用计算方法来确定这些性质可以表现出来的有希望的新材料或结构。第二个重点将是在突然水平上的电子动力学，如跨越拓扑相界的量子猝灭，以及在绝热水平上，与自旋-轨道耦合磁性材料中声子的动力学有关。方法将包括正式的理论发展；简单模型哈密顿的实施和测试；以及第一原理电子结构计算。研究生将接受这些方法的指导和培训，为他们的教育和职业发展做出贡献。在该项目中开发的算法和计算机代码将以开放源代码的形式提供，以造福于更广泛的电子结构研究社区。该计划还承诺识别和评估最终可能用于商业应用的新电子材料。该奖项反映了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。

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.