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（气体、液体、固体）以其对称性来区分：使它们保持不变的变换——例如立方固体的九十度旋转。 在极端条件（低温、强磁场）下存在的奇异物质量子态表现出难以简单几何描述的对称性。相反，它们的对称性是通过拓扑来理解的：忽略角度和长度的定性几何。 拓扑态的研究对于其在量子技术中的应用与理解物理世界同样重要。 特别令人感兴趣的是量子信息中的应用。研究人员将研究物质这些拓扑相的数学对称性，以便对它们进行分类和区分，探究它们的性质，并了解它们如何通过相变相互关联。 重点将放在这些物质相的特性如何在量子技术中发挥作用。研究人员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得到充分研究，但仍然存在许多重要问题。 具有拓扑特征的三维系统正在发挥着越来越重要的作用，但从严格的数学角度对其进行的研究还不够充分。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：第一个模型模拟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发的拓扑动力学。 了解模型如何通过对称性和相变相互关联，将为物质拓扑相的景观提供更清晰的图景。 在一个互补的方向上，研究人员将开发方法来理解辫子组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驱动的幺正性假设之间存在着紧张关系。 这一挑战将通过非半简单类别和表示的研究来应对。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力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

Reconstruction of Modular Data from $${\text {SL}}_2({\mathbb {Z}})$$ Representations

从 $${ ext {SL}}_2({mathbb {Z}})$$ 表示重建模块化数据

• DOI: 10.1007/s00220-023-04775-w
• 发表时间: 2023-07
• 期刊: Communications in Mathematical Physics
• 影响因子: 2.4
• 作者: Ng, Siu;Rowell, Eric C.;Wang, Zhenghan;Wen, Xiao
• 通讯作者: Wen, Xiao

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: 2023-12
• 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.

G ‐crossed braided zesting

G — 交叉编织的热情

• DOI: 10.1112/jlms.12816
• 发表时间: 2023-10
• 期刊: Journal of the London Mathematical Society
• 作者: Delaney, Colleen;Galindo, César;Plavnik, Julia;Rowell, Eric C.;Zhang, Qing
• 通讯作者: Zhang, Qing