Theory and Application of Berry Phase Methods in Solids

非技术总结在过去的二十年里，人们越来越认识到，来自微分几何和拓扑的某些数学概念有时对于理解电子在晶体固体中的行为是至关重要的。电子由量子力学波函数来描述，这些波函数随动量变化的方式编码了关于固体的许多信息，特别是它的电和磁响应以及它们之间的耦合。当波函数在动量空间中发生扭曲时，就会产生所谓的“拓扑绝缘体”态，这在过去的十年里一直是人们研究的热点。根据定义，电流不能在绝缘子内部流动，但拓扑绝缘子有一个不同寻常的特性，即表面肯定有载流通道。本研究计划旨在进一步发展这种效应的形式理论，发明稳健和高效的计算算法来计算固体的相关性质，并对显示新的或增强的性质的材料进行计算搜索。该项目将导致算法的开发，这些算法最终将在开放源码包中实施，并向更广泛的电子结构社区提供。它还将有助于开发具有商业应用前景的新型材料，特别是涉及电和磁响应耦合的材料。将对初级研究人员(研究生和博士后)进行培训和指导，为科学工作队伍的发展做出贡献。技术总结本研究计划侧重于拓扑绝缘体或其他轨道电流起重要作用的材料的电子性质。其目的是(I)利用微分几何中的Berry相、Berry曲率和陈数的数学概念，进一步发展这类系统的形式理论；(Ii)发明与这些数学概念相关的准确和高效的计算材料性质的方法；以及(Iii)使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，从而潜在地导致技术应用。虽然最近的工作主要集中在时间反转不变的拓扑绝缘体，如Bi2Se3，但这里的重点将是量子反常霍尔或陈氏绝缘体，轴子绝缘体和Weyl半金属，其中时间反转对称性自发破缺。虽然陈氏绝缘体状态的可能性在25年前就已经被指出，但它直到最近才被实验证明，而且只有在低温下才被证明。将制定策略，从理论上确定实验合成可能达到的二维陈氏绝缘体状态，其间隙和居里温度足够大，足以接近室温操作。第二个和重叠的推力将是关于涉及宏观轨道电流的材料特性的理论和计算，包括体和表面反常霍尔效应、轨道磁化和轨道磁电耦合。作为一个交叉主题，计算材料搜索策略将被用来寻找可能具有期望的拓扑或磁电性质的候选材料和结构，特别是包括正常磁绝缘体表面及其与时间反转不变的拓扑绝缘体的界面的二维Chern绝缘态。该项目将导致算法的开发，最终将在开放源码包中实现，并提供给更广泛的电子结构社区。它还将有助于开发具有商业应用前景的新型材料，特别是涉及电和磁响应耦合的材料。将对初级研究人员(研究生和博士后)进行培训和指导，促进科学劳动力的发展。

NON-TECHNICAL SUMMARYIn the last two decades, there has been a growing appreciation that certain mathematical concepts from differential geometry and topology are sometimes central to the understanding of the behavior of electrons in crystalline solids. The electrons are described by quantum-mechanical wavefunctions, and the manner in which these vary with momentum encodes many kinds of information about the solid, especially its electrical and magnetic responses and their coupling with each other. When the wavefunctions become twisted in momentum space, this results in so-called "topological insulator" states, which have been the focus of intense research interest in the last decade. By definition, electric currents cannot flow in the interior of an insulator, but a topological insulator has the unusual property that there are guaranteed to be current-carrying channels at the surfaces. The present research program is designed to further develop the formal theory of such effects, to invent robust and efficient computational algorithms for computing the related properties of solids, and to carry out a computational search for materials displaying new or enhanced properties. The project will lead to the development of algorithms that will ultimately be implemented in open-source code packages and made available to the wider electronic-structure community. It will also contribute to the development of novel materials that are promising for commercial applications, especially ones involving the coupling of electrical and magnetic responses. Training and mentorship of junior researchers (graduate students and postdocs) will take place, contributing to scientific workforce development.TECHNICAL SUMMARYThis research program is focused on the electronic properties of topological insulators or other materials in which orbital currents play an important role. The objectives are (i) to further develop the formal theory of such systems, making use of the mathematical concepts of Berry phase, Berry curvature, and Chern number from differential geometry; (ii) to invent 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. While much recent work has concentrated on time-reversal invariant topological insulators such as Bi2Se3, the emphasis here will be on quantum anomalous Hall or Chern insulators, axion insulators, and Weyl semimetals, in which time-reversal symmetry is spontaneously broken. While the possibility of the Chern-insulator state was pointed out already 25 years ago, it has only recently been demonstrated experimentally, and that only at low temperature. Strategies will be developed for theoretically identifying possible two-dimensional Chern-insulator states accessible to experimental synthesis, with gaps and Curie temperatures large enough to approach room-temperature operation. A second and overlapping thrust will be on the theory and calculation of materials properties that involve macroscopic orbital currents, including bulk and surface anomalous Hall effects, orbital magnetization, and orbital magnetoelectric couplings. As a cross-cutting theme, computational materials search strategies will be used to identify promising candidate materials and structures that may exhibit the desired topological or magnetoelectric properties, especially including two-dimensional Chern-insulator states at surfaces of normal magnetic insulators and their interfaces with time-reversal-invariant topological insulators.The project will lead to the development of algorithms that will ultimately be implemented in open-source code packages and made available to the wider electronic-structure community. It will also contribute to the development of novel materials that are promising for commercial applications, especially ones involving the coupling of electrical and magnetic responses. Training and mentorship of junior researchers (graduate students and postdocs) will take place, contributing to scientific workforce development.