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，立方体固体旋转90度。物质在极端条件(低温、强磁场)下的奇异量子态表现出对称性，这种对称性抵制了简单的几何描述。相反，它们的对称性是通过拓扑学来理解的：定性几何，其中忽略了角度和长度。拓扑态的研究对于它们在量子技术中的应用和对理解物理世界一样重要。特别令人感兴趣的是它在量子信息中的应用。研究人员将研究这些物质拓扑相的数学对称性，目的是对它们进行分类和区分，探索它们的性质，并了解它们是如何通过相变联系在一起的。一些重点将放在这些相物质的性质如何在量子技术中找到用途上。研究人员将使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在发挥着越来越重要的作用，但从严格的数学角度进行研究的人还不多。量子对称性中的两个关键主题是辫子融合范畴和运动群表示：第一个主题模拟物质拓扑相的拓扑不变特征，第二个主题编码任意子和环状激发的拓扑动力学。了解这些模型是如何通过对称性和相变联系在一起的，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，研究人员将开发方法来理解辫子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和物理驱动的一元性假设之间存在着张力。这一挑战将通过对非半简单类别和代表性的研究来应对。这一奖项反映了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。

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.