CAREER: Geometry and topology of quantum materials

该职业奖支持快速发展的量子材料领域的研究和教育。量子材料是指电子的集体行为产生非凡特性的材料，具有从自旋电子学到量子计算等量子技术的潜在应用。在一个范式转换的发现中，拓扑学，即描述物体变形时保持不变的特性的数学领域，被认为可以产生非凡的材料，例如那些在内部是绝缘体的材料，但通过其表面和边缘的状态具有无耗散的导电性，这是拓扑学所要求的。较少被探索但同样值得注意的是，拓扑学在决定每种材料中发现的电子物质的相位方面起着至关重要的作用，并且它们可能是人们长期追求的奇异电子物质状态。例如超导性，电子在整个材料中以零电阻流动，或者电子似乎分解成具有非常规性质的更小的实体，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特性）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。仔细地控制缺陷可以使具有期望性质的电子物质的相得以实现。通过这项研究，PI旨在深入了解独特量子现象的稳定性，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，研究和教育通过多方面的努力结合在一起，重点放在研究生身上，这将增强和多样化他们在物质量子特性方面的训练。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特性的计算技术。PI还将组织研讨会，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。本职业奖支持理论研究和教育，旨在研究量子几何对固态电子集体性质的影响。PI将探索电子波函数的动量空间结构如何影响干净和脏材料中的电子行为，影响远程相干性，输运和外来激发的出现。该项目侧重于量子材料的新视角，旨在从格林函数算子的结构角度统一量子几何现象，特别是当投影到各种空间缺陷时，它的拓扑鲁棒零。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或不具有平移对称性的系统中的非平凡几何，从而开启以下可能性：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能的新材料提供指导；3)识别来自基态非平凡几何结构的稳健物理响应。这种方法将在电子结构理论、化学和材料科学的应用中为新兴的拓扑物质领域做出重大贡献。该活动还包括建立一个材料拓扑性质计算训练营，涵盖带理论、群论和密度泛函理论的基本内容，并让学生获得模拟各种数量的实验和技术相关的“动手”经验。该计划还旨在建立一个以纽约为基地的研讨会，以吸引来自不同背景的优秀年轻研究人员。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.