CAREER: Geometry and topology of quantum materials

非技术总结这个职业奖项支持在快速发展的量子材料领域的研究和教育。量子材料是指那些电子的集体行为产生了非凡的性质，在从自旋电子学到量子计算的量子技术中具有潜在应用的材料。在一个范式转换的发现中，拓扑学，数学中描述物体变形时保持不变的性质的领域，被认为导致了非凡的材料，例如那些内部是绝缘体，但通过其表面和边缘上的状态具有无耗散导电的材料，这些状态是拓扑学所要求存在的。较少被探索但同样引人注目的是拓扑在确定在每种材料中发现哪些相的电子物质方面所起的关键作用，以及它们可能是长期受到追捧的奇异物质电子态的可能性。超导的例子是电子在整个材料或相中以恰好为零的电阻流动，其中电子似乎解离成具有非常规性质的较小实体，称为分馏相。PI将研究拓扑和量子几何--量子态抽象表示的几何性质--如何影响物质中电子的性质，特别是当材料中存在不可避免的缺陷或“污垢”时。对缺陷的仔细控制可以实现具有所需性质的电子物质的相。通过这项研究，PI旨在深入了解独特的量子现象的稳定性，并为拓扑量子材料的表征和预测提供新的数学工具，并与实验合作发现新的量子材料。在这个项目中，研究和教育通过多种努力相结合，专注于研究生，将加强和多样化他们对物质量子性质的培训。该活动将包括一个训练营，涵盖拓扑材料的基本知识和研究其电子性质的计算技术。国际和平研究所还将组织一个研讨会，展示来自不同背景的有才华的年轻研究人员在量子材料方面的工作。技术总结这个职业奖项支持旨在研究量子几何对固态电子集体性质的影响的理论研究和教育。PI将探索电子波函数的动量空间织构如何影响干净和肮脏材料中的电子行为，从而影响长程相干、传输和奇异激发的出现。该项目着眼于量子材料的新视角，目的是从格林函数算符的结构的角度统一量子几何现象，特别是当它投射到各种空间缺陷时，其拓扑健壮的零点。这种方法的目标是构建一种工具，可以有效地应用于识别具有和不具有平移对称性的系统中的非平凡几何，从而为1)表征无序拓扑晶体物质的行为；2)为寻找具有特殊物理性质的新材料提供指导；3)识别源于基态非平凡几何的稳健的物理响应。通过在电子结构理论、化学和材料科学中的应用，这种方法将对新兴的拓扑物质领域做出重大贡献。这项活动还包括建立材料拓扑性质计算的训练营，涵盖带论、群论和密度泛函理论的要点，并允许学生获得模拟各种数量的实验和技术相关性的实践经验。PI还旨在建立一个位于纽约的研讨会，以集中来自不同背景的优秀年轻研究人员。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 SUMMARY This CAREER award supports research and education in the rapidly evolving field of quantum materials. Quantum materials refer to those in which the collective behavior of electrons gives rise to extraordinary properties with potential applications in quantum technologies from spintronics to quantum computing. In a paradigm shifting discovery, topology, the field in mathematics that describes properties which remain unchanged when objects are deformed, was recognized to result in remarkable materials such as those that are insulators in the interior but possess dissipationless electric conduction through states on their surfaces and edges that are required to exist by topology. Less explored but equally remarkable is the crucial role that topology plays in determining which phases of electronic matter are found in each material, and the possibility they might be long sought-after exotic electronic states of matter. Examples are superconductivity where electrons flow with exactly zero resistance in the entire material or phases where electrons seem to dissociate into smaller entities with unconventional properties, called fractionalized phases.The PI will study how topology and quantum geometry, the geometric properties of an abstract representation of quantum states, can influence the nature of electrons in matter, particularly when unavoidable imperfections or “dirt” are present in materials. Careful control of defects may enable phases of electronic matter with desired properties to be realized. Through this research, the PI aims to gain insights into the stability of unique quantum phenomena, as well as offer new mathematical tools for the characterization of topological quantum materials and for the prediction and, working with experiment, the discovery of new ones.In this project, research and education are integrated through multiple efforts focusing on graduate students that will enhance and diversify their training in quantum properties of matter. The activity will include a bootcamp covering essentials of topological materials and computational techniques for studying their electronic properties. The PI will also organize a workshop to showcase talented young researchers from diverse backgrounds working on quantum materials. TECHNICAL SUMMARYThis CAREER award supports theoretical research and education aimed to study the influence of quantum geometry in the collective properties of electrons in the solid state. The PI will explore how the momentum space textures of electron wavefunctions affect electron behavior in both clean and dirty materials, influencing long-range coherence, transport, and the emergence of exotic excitations. The project focuses on a new perspective on quantum materials, with the aim to unify quantum geometric phenomena from the point of view of the structure of the Green’s function operator, in particular its topologically robust zeros when projected to various spatial defects. The goal of this approach is to construct tools that can be efficiently applied to identify nontrivial geometry both in systems with and without translational symmetry, therefore opening the possibility to 1) characterize the behavior of disordered topological crystalline matter; 2) offer guidelines for the search of new materials with exceptional physical properties; 3) identify robust physical responses that stem from the nontrivial geometry of the ground state. This approach is set to contribute significantly to the burgeoning field of topological matter, with applications in electronic structure theory, chemistry, and materials science. This activity also includes establishing a bootcamp for the computation of materials topological properties covering the essentials of band theory, group theory, and density functional theory, and allowing students to gain "hands-on" experience simulating various quantities of experimental and technological relevance. The PI also aims to establish a New York City based workshop to spotlight excellent young researchers from diverse backgrounds.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.