Quantum Symmetries of Topological Phases of Matter

物质拓扑相的量子对称性

• 批准号: 2205962
• 负责人: Eric Rowell
• 金额: $ 19.49万
• 依托单位: Texas A&M University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-09-01 至 2025-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2205962&HistoricalAwards=false
• 关键词: Quantum; Symmetries; Topological; Phases; Matter

Ordinary phases of matter (gas, liquid, solid) are distinguished by their symmetries: transformations that leave them unchanged--for example a ninety degree rotation of a cubic solid. Exotic quantum states of matter present under extreme conditions (low temperatures, strong magnetic fields) exhibit symmetries that resist a simple geometric description. Rather, their symmetries are understood through topology: qualitative geometry in which angles and lengths are ignored. The study of topological states is as important for their application to quantum technologies as for understanding the physical world. Of particular interest are the applications in quantum information. The investigator will study the mathematical symmetries of these topological phases of matter for the purpose of classifying and distinguishing them, probing their properties, and understanding how they are related through phase transitions. Some emphasis will be on how the properties of these phases of matter might find utility in quantum technologies. The investigator will employ theoretical and computational methods to study mathematical models for topological phases of matter. While two-dimensional topological phases of matter have been well-studied, many important questions remain. Three-dimensional systems with topological features are playing an increasingly substantial role yet are not as well-studied from a rigorous mathematical perspective. Two key themes in quantum symmetries are braided fusion categories and motion group representations: the first models the topologically invariant features of topological phases of matter, and the second encodes the topological dynamics of anyons and loop-like excitations. Understanding how the models are related through symmetries and phase transitions will provide a clearer picture of the landscape of topological phases of matter. In a complementary direction, the investigator will develop methods to understand the physically relevant representations of the braid group and higher dimensional generalizations. In three dimensions there is tension between the sensitivity of the topological invariants and the physically motivated assumption of unitarity. This challenge will be met through the study of non-semisimple categories and representations.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.

物质的普通相态（气、液、固）以它们的对称性来区分，即不改变它们的变化——例如，一个立方固体旋转90度。在极端条件下（低温、强磁场）物质的奇异量子态表现出的对称性无法用简单的几何描述。相反，它们的对称性是通过拓扑来理解的：在拓扑中，角度和长度被忽略。拓扑态的研究对于量子技术的应用和对物理世界的理解同样重要。特别令人感兴趣的是在量子信息中的应用。研究者将研究这些物质拓扑相的数学对称性，以便对它们进行分类和区分，探测它们的性质，并了解它们如何通过相变相互关联。一些重点将放在物质的这些相的性质如何在量子技术中找到效用。研究者将采用理论和计算方法来研究物质拓扑相的数学模型。虽然物质的二维拓扑相已经得到了很好的研究，但许多重要的问题仍然存在。具有拓扑特征的三维系统正发挥着越来越重要的作用，但从严格的数学角度来看，还没有得到很好的研究。量子对称性的两个关键主题是编织融合类别和运动群表示：第一个模型描述了物质拓扑相的拓扑不变特征，第二个编码了任意子和环状激励的拓扑动力学。了解这些模型是如何通过对称和相变联系起来的，将为物质的拓扑相提供更清晰的图景。在一个互补的方向上，研究者将开发方法来理解辫群的物理相关表示和更高维度的概括。在三维空间中，拓扑不变量的敏感性与物理动机的统一性假设之间存在张力。这一挑战将通过对非半简单类别和表示的研究来解决。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。

项目成果

Reconstructing braided subcategories of SU(N)

重建 SU(N) 的编织子类别

• DOI: 10.1016/j.jalgebra.2023.08.005
• 发表时间: 2023
• 期刊: Journal of Algebra
• 影响因子: 0.9
• 作者: Feng, Zhaobidan;Ming, Shuang;Rowell, Eric C.
• 通讯作者: Rowell, Eric C.