Topological quantum matter and crystalline symmetry

非技术总结该奖项支持关于理解由于量子系统中晶体对称性的存在而可能出现的新类型现象的理论研究。由于原子的周期性排列，量子材料通常存在于晶体环境中。因此，所讨论的材料通常也具有晶体对称性，包括空间的旋转、反射和平移，这意味着在这些操作下晶体结构保持不变。在这些对称性的存在下，量子材料可以容纳新类型的物理现象。例如，如果量子材料具有旋转对称性，那么某些类型的晶体缺陷就可能积累量子电荷，这些电荷是电子电荷的一小部分。该项目将结合数学物理和量子场论的方法，以进一步提高我们描述和分类具有晶体对称性的量子系统所表现出的健壮、量子化属性的能力。该项目还将开发在各种环境下实验探索这些现象的方法，包括二维量子材料、光子系统和嘈杂的中型量子计算机。该项目具有很强的教育成分，因为它完全支持一名研究生。PI将进一步开发并公开发布他关于拓扑量子物质和机器学习的专题研究生课程的讲稿。此外，PI将通过其机构的外联方案和积极的社交媒体存在，在高中一级继续积极开展各种形式的外联活动。正在进行的行业合作将通过该项目在量子计算中的应用，进一步扩大该项目的社会影响。技术总结该奖项支持发展具有晶体对称性的物质的拓扑相理论的理论研究。这一理论将应用于强相互作用区域，从而超越了单粒子能带理论，使用了拓扑量子场理论的工具。该项目有几个不同的组成部分，直接建立在上一个项目期取得的重大进展的基础上。在第一个组成部分中，Pi的群将他们的二维强相互作用晶体拓扑相的理论推广到所有17个壁纸群，该理论适用于保壳壁纸群。第二部分是对分数级陈氏绝缘体和量子自旋液体模型进行数值计算，特别是使用投影量子波函数，以了解新的晶体对称性保护的拓扑不变量的出现及其物理性质。这些不变量包括通过部分对称操作和通过在晶格缺陷处积累的分数电荷获得的不变量。第三个部分是开发实验方法来探索PI最近在二维量子材料、光子系统和嘈杂的中型量子计算机中发现的晶体拓扑不变量。最后，理论上的进展将被扩展到发展一种用晶体对称性增强的临界现象理论。该项目有很强的教育成分，因为它完全支持研究生。PI将进一步开发并公开发布他关于拓扑量子物质和机器学习的专题研究生课程的讲稿。此外，PI将通过其机构的外联方案和积极的社交媒体存在，在高中一级继续积极开展各种形式的外联活动。正在进行的行业合作将通过其在量子计算中的应用来进一步扩大该项目的社会影响。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 SUMMARYThis award supports theoretical research on understanding new kinds of phenomena that can arise due to the presence of crystalline symmetry in quantum systems. Quantum materials typically exist in a crystalline environment due to the periodic arrangement of atoms. Hence, the material under question often also has a crystallographic symmetry that includes rotations, reflections, and translations of the space, meaning that the crystal structure remains the same under such operations. In the presence of these symmetries, quantum materials can host new types of physical phenomena. For example, if a quantum material has a rotational symmetry, then it is possible for certain kinds of crystal defects to accumulate quantized electric charges that are fractions of the charge of the electron. The project will incorporate methods from mathematical physics and quantum field theory to further our ability to characterize and classify the robust, quantized properties exhibited by quantum systems with crystalline symmetry. The project will also develop methods to experimentally probe these phenomena in a variety of settings, including two-dimensional quantum materials, photonic systems, and noisy intermediate scale quantum computers.The project has a strong educational component, as it fully supports a graduate student. The PI will further develop and publicly release lecture notes for his special topics graduate courses on topological quantum matter and on machine learning. In addition, the PI will remain active in various forms of outreach, at the high school level through outreach programs at his institution and through an active social media presence. Ongoing industry collaborations will further broaden the societal impacts of the project through its applications to quantum computing.TECHNICAL SUMMARYThis award supports theoretical research in developing the theory of topological phases of matter with crystalline symmetry. This theory will apply in the regime of strong interactions, and thus go beyond single-particle band theory, using the tools of topological quantum field theory. The project has several different components and builds directly on significant progress from the last project period. In the first component, the PI’s group will extend their theories of two-dimensional strongly interacting crystalline topological phases, which applied to the case orientation-preserving wallpaper groups, to include all 17 wallpaper groups. The second component is to perform numerical calculations for models of fractional Chern insulators and quantum spin liquids, specifically using projected quantum wave functions, to understand the emergence of new crystalline symmetry-protected topological invariants and their physical properties. These include invariants obtained through partial symmetry operations and through fractional charges accumulated at lattice defects. The third component is to develop methods to experimentally probe the PI’s recently discovered crystalline topological invariants in two-dimensional quantum materials, photonic systems, and noisy intermediate scale quantum computers. Finally, the theoretical advances will be extended to develop a theory of critical phenomena enhanced with crystalline symmetry.The project has a strong educational component, as it fully supports a graduate student. The PI will further develop and publicly release lecture notes for his special topics graduate courses on topological quantum matter and on machine learning. In addition, the PI will remain active in various forms of outreach, at the high school level through outreach programs at his institution and through an active social media presence. Ongoing industry collaborations will further broaden the societal impacts of the project through its applications to quantum computing.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.