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

非技术总结该奖项支持理论研究和教育，以识别和分类称为拓扑相的物质的新阶段。物质相的概念使人们能够解释由相同的基本成分组成的系统中截然不同的物理性质，例如液体和固体冰形式的水。固态材料通常也可以出现在几个阶段中的一个阶段--例如，金属与绝缘体，或磁性与非磁性--它们的许多用途来自于利用这些阶段的不同物理特性。50多年前建立了一个理论框架，用于理解材料成分进入相的组织，即材料在转变为相时经历的“有序化”，并被证明非常成功，直到发现了奇异的新相--称为分数量子霍尔效应态。当电子被限制在二维空间并被置于巨大的垂直磁场中时，这些状态就会被发现，它们不符合旧的理论框架，并具有惊人的新物理性质，目前被冠以“拓扑顺序”的名义。这项研究的目的是促进对拓扑相的基本理解，以及当存在额外的对称性时如何丰富拓扑有序相的分类。例如：拓扑绝缘子与普通绝缘子的不同之处在于表面必须存在金属状态，因为颠倒时间方向会导致状态看起来与原始状态相同。这种时间反转对称性不符合简单的铁磁体，即时间反转与磁性方向相反的铁磁体。这项研究将解决重要的问题：存在其他对称性或这些对称性的组合会产生什么有趣的性质？它们有多健壮？有没有在传统固态材料中没有类似的相？这种相位的空间分离部分的量子力学连接性能否被用来为未来的信息技术存储或操纵量子信息？该奖项还支持教育和外展活动，包括：在高级研究生水平上开发一门新的一学期课程，重点是拓扑秩序；PI将通过在石溪大学组织的一个项目向高中生介绍与该研究领域有关的讲座；PI将每月组织一次针对更广泛社区的物理系列讲座，将邀请外部演讲者向听众介绍该领域和相关领域的尖端研究。技术总结该奖项支持理论研究和教育，以识别和分类物质的新阶段，称为拓扑阶段，特别关注对称如何丰富可能的拓扑阶段。尽管拓扑序是稳健的并且独立于任何全局对称性，但对称性的包含会导致额外的相，例如最近发现的拓扑绝缘体，它们只有在存在时间反转对称性时才是稳健的。PI将使用分析技术，包括代数拓扑学中的技术，来追求几个研究方向：1)对物理上真实的对称群，例如连续对称、反么正对称和空间对称群，分类对称丰富的相，推广有限离散对称群的结果；2)分类包括基本费米子自由度的对称丰富相和对称保护相。这项研究将集中在通过带框架可获得的这类相的稳定性，以及寻找内在的强相互作用相。3)构造多体不变量，以区分这种分类中的状态。4)根据拓扑量子场论，构建一个统一的数学框架，以解决上述分类问题。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，重点是高级研究生水平的拓扑秩序；PI将通过在石溪大学组织的一个项目，向高中生介绍与该研究领域有关的讲座；国际物理学会将每月组织一次针对更广泛社区的物理系列讲座，邀请外部演讲者向观众介绍这一领域及相关领域的尖端研究。

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.