Quantum Symmetries of Topological Phases of Matter

物质的普通相（气体、液体、固体）通过它们的对称性来区分：使它们保持不变的变换-例如立方固体的90度旋转。 在极端条件下（低温、强磁场），物质的奇异量子态呈现出难以用简单几何描述的对称性。相反，它们的对称性是通过拓扑学来理解的：在拓扑学中，角度和长度被忽略了。 拓扑态的研究对于它们在量子技术中的应用和理解物理世界一样重要。 特别令人感兴趣的是在量子信息中的应用。研究人员将研究这些物质的拓扑相的数学对称性，以便对它们进行分类和区分，探索它们的属性，并了解它们如何通过相变相互关联。 一些重点将放在如何在量子技术中发现这些物质相的属性。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经得到了很好的研究，但仍然存在许多重要的问题。 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范畴和运动群表示：第一个模型的拓扑不变的特征的拓扑相位的物质，第二个编码的拓扑动力学的任意子和环状激发。 了解这些模型如何通过对称性和相变相互关联，将为物质的拓扑相提供更清晰的图景。 在一个互补的方向，研究人员将开发的方法来理解的辫子群和高维泛化的物理相关的表示。 在三维空间中，拓扑不变量的敏感性和物理上的么正性假设之间存在着张力。 这个奖项反映了NSF的法定使命，并被认为是值得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的支持。

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.