Interplay of symmetry and topology in gapped phases of condensed matter systems

该奖项支持理论研究和教育，以识别和分类被称为拓扑相的物质的新相。物质相的概念允许人们在由相同的基本成分组成的系统中解释截然不同的物理性质，例如水的液态冰和固态冰。固态材料通常也可以是几种相中的一种-例如金属与绝缘体，或磁性与非磁性-它们的用途很大程度上源于利用这些相的不同物理特性。50多年前，人们建立了一个理论框架，用来理解材料组成物的相组织，即材料在转变为相时所经历的“有序”，直到发现了奇异的新相——分数量子霍尔效应态——才被证明是非常成功的。当电子被限制在二维空间并被放置在巨大的垂直磁场中时，这些状态并不符合旧的理论框架，而是具有惊人的新物理性质，目前被称为“拓扑秩序”。本研究旨在促进对拓扑相的基本理解，以及当存在额外对称性时如何丰富拓扑有序相的分类。例如：拓扑绝缘体与普通绝缘体的不同之处在于其表面必须存在一种金属状态，而拓扑绝缘体之所以能够存在，是因为时间方向的逆转会导致一种看起来与原始状态相同的状态。简单的铁磁体不遵守这种时间反转对称，因为它的时间反转反转了磁的方向。这项研究将解决一些重要的问题：当存在其他对称或这些对称的组合时，会产生什么有趣的特性？它们有多健壮？在传统的固态材料中有没有类似的相？这些相的空间分离部分的量子力学连通性能否被利用来存储或操纵量子信息以用于未来的信息技术？PI将使用分析技术和新颖的数学方法来调查这些问题。该奖项还支持教育和推广活动，包括：开发一门新的一学期课程，专注于高级研究生水平的拓扑秩序；PI将通过石溪大学组织的一个项目向高中生讲授与研究领域相关的课程；PI将每月组织一次针对更广泛社区的物理系列讲座，这些讲座将邀请外部演讲者向听众介绍这一领域和相关领域的前沿研究。该奖项支持理论研究和教育，以识别和分类新的物质相，称为拓扑相，特别关注对称性如何丰富可能的拓扑相。尽管拓扑秩序是鲁棒的，并且独立于任何全局对称性，但对称性的包含可能导致额外的相位，例如最近发现的拓扑绝缘体，只有在时间反转对称性存在时才具有鲁棒性。PI将使用包括代数拓扑在内的分析技术，进行以下几个研究方向：1)对物理现实对称群（如连续对称、反酉对称和空间对称）的对称富相进行分类，推广有限离散对称群的结果。2)对涉及基本费米子自由度的富对称相和保护对称相进行分类。这项研究将集中于通过带框架可获得的这些相的稳定性，以及寻找固有的强相互作用相。3)构造多体不变量，以区分这种分类中的状态。特别是，PI将探索非平凡对称保护相与低维异常之间的联系。4)基于拓扑量子场论，构建统一的数学框架，解决上述分类问题。该奖项还支持教育和推广活动，包括：开发一门新的一学期课程，专注于高级研究生水平的拓扑秩序；PI将通过石溪大学组织的一个项目向高中生讲授与研究领域相关的课程；PI将每月组织一次针对更广泛社区的物理系列讲座，这些讲座将邀请外部演讲者向听众介绍这一领域和相关领域的前沿研究。

NON-TECHNICAL SUMMARYThis award supports theoretical research and education to identify and classify new phases of matter termed topological phases. The concept of a phase of matter allows one to account for drastically different physical properties in systems made out of the same underlying constituents, e.g. the liquid and solid ice forms of water. Solid state materials can also typically come in one of several phases - e.g. metal versus insulator, or magnetic versus non-magnetic - and much of their usefulness derives from exploiting the different physical properties characteristic of such phases. A theoretical framework for understanding the organization of constituents of materials into phases, "ordering" that materials undergo when they transform into a phase, was established over 50 years ago and had proven extremely successful until exotic new phases - called fractional quantum Hall effect states - were discovered. These states, found when electrons are confined to two dimensions and are placed in large perpendicular magnetic fields, did not fit into the old theoretical framework and possess striking new physical properties that currently go under the name of "topological order". This research is aimed to advance fundamental understanding of topological phases and how the classification of topologically ordered phases is enriched when additional symmetries are present. For example: topological insulators, which differ from ordinary insulators by the necessary existence of a metallic state at the surface, can exist because reversing the direction of time leads to a state that looks the same as the original one. This time reversal symmetry is not obeyed by a simple ferromagnet for which time reversal flips the direction of magnetism. The research will address important questions: What interesting properties can arise with other symmetries present, or with combinations of such symmetries? How robust are they? Are there any such phases that have no analogue in conventional solid-state materials? Can the quantum mechanical connectedness of spatially separated parts of such phases be exploited to store or manipulate quantum information for future information technology? The PI will investigate these questions using analytical techniques and novel mathematical methods.This award also supports education and outreach activities including: the development of a new one-semester course focused on topological order at the advanced graduate students level; the PI will present lectures related to the research area to high school students through a program organized at Stony Brook university; the PI will organize a once-a-month physics lecture series aimed at the broader community, which will bring in outside speakers to introduce the audience to cutting-edge research in this and related fields.TECHNICAL SUMMARYThis award supports theoretical research and education to identify and classify new phases of matter, termed topological phases, with a specific focus on how symmetries enrich possible topological phases. Although topological order is robust and independent of any global symmetries, the inclusion of symmetries can lead to additional phases, for example the recently discovered topological insulators, which are robust only when time reversal symmetry is present. The PI will use analytical techniques, including those from algebraic topology, to pursue several research directions:1) Classifying symmetry-enriched phases for physically realistic symmetry groups, e.g. continuous, anti-unitary, and spatial symmetries, generalizing results for finite discrete symmetry groups.2) Classifying both symmetry-enriched and symmetry-protected phases involving fundamental fermion degrees of freedom. This research will focus on the stability of such phases accessible through the band framework, as well as on the search for inherently strongly interacting phases.3) Constructing many-body invariants, which distinguish among the states in such a classification. In particular, the PI will explore the connection between non-trivial symmetry-protected phases and anomalies in one lower dimension.4) Constructing a unified mathematical framework, based on topological quantum field theory, to tackle the above classification problems.This award also supports education and outreach activities including: the development of a new one-semester course focused on topological order at the advanced graduate students level; the PI will present lectures related to the research area to high school students through a program organized at Stony Brook university; the PI will organize a once-a-month physics lecture series aimed at the broader community, which will bring in outside speakers to introduce the audience to cutting-edge research in this and related fields.