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。量子材料是指那些电子的集体行为产生了非凡的性质，在量子技术中具有潜在的应用，从自旋电子学到量子计算。在一个范式转变的发现中，拓扑学，数学中描述物体变形时保持不变的属性的领域，被认为会导致显着的材料，例如那些内部是绝缘体，但通过拓扑学所需的表面和边缘上的状态具有无耗散导电的材料。较少探索但同样值得注意的是拓扑学在确定每种材料中发现的电子物质相以及它们可能是长期追求的奇异电子态物质的可能性方面发挥的关键作用。例如，电子在整个材料中以零电阻流动的超导性，或者电子似乎分解成具有非常规性质的更小实体的相，称为分数相。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使得能够实现具有期望性质的电子物质的相。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，并与实验一起发现新的量子材料。在这个项目中，研究和教育是通过多方面的努力，重点是研究生，将加强和多样化他们的培训，在量子特性的所谓了该活动将包括一个训练营，涵盖拓扑材料和计算技术的要点，以研究其电子特性。PI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。该职业奖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纹理如何影响清洁和肮脏材料中的电子行为，影响长程相干性，传输和奇异激发的出现。该项目侧重于量子材料的新视角，目的是从绿色函数算子结构的角度统一量子几何现象，特别是投影到各种空间缺陷时的拓扑鲁棒零点。这种方法的目标是构建工具，可以有效地应用于识别非平凡几何系统中的平移对称性和非平移对称性，从而打开了可能性1）表征无序拓扑晶体物质的行为; 2）提供指导方针，为寻找新的材料具有特殊的物理性能; 3）识别源于基态的非平凡几何形状的鲁棒物理响应。这种方法将为新兴的拓扑物质领域做出重大贡献，并在电子结构理论，化学和材料科学中应用。这项活动还包括建立一个训练营，用于计算材料拓扑性质，涵盖能带理论，群论和密度泛函理论的要点，并允许学生获得模拟各种实验和技术相关性的“动手”经验。PI还旨在建立一个以纽约市为基地的研讨会，以聚焦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.