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

凝聚态系统有隙相中对称性和拓扑的相互作用

• 批准号: 1519579
• 负责人: Lukasz Fidkowski
• 金额: $ 24.67万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-01-01 至 2018-03-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1519579&HistoricalAwards=false
• 关键词: Interplay; symmetry; topology; gapped; phases

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.

非技术总结该奖项支持理论研究和教育，以确定和分类称为拓扑相的物质的新阶段。 物质相的概念允许人们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，例如水的液态和固态冰形式。 固态材料通常也可以是几种相之一-例如金属与绝缘体，或磁性与非磁性-并且它们的大部分有用性源于利用这些相的不同物理特性。 50多年前建立了一个理论框架，用于理解材料成分组织成相，即材料转变成相时所经历的“排序”，并证明非常成功，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放置在大的垂直磁场中时发现的这些状态不符合旧的理论框架，并且具有惊人的新物理性质，目前以“拓扑秩序”的名义。 本研究的目的是推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。 举例来说，请注意：拓扑绝缘体与普通绝缘体的不同之处在于表面必须存在金属状态，而拓扑绝缘体之所以能够存在，是因为反转时间方向会导致看起来与原始状态相同的状态。这种时间反转对称性并不适用于简单的铁磁体，因为时间反转会翻转磁性的方向。 这项研究将解决重要的问题：什么有趣的属性可以出现与其他对称的存在，或与这些对称的组合？ 它们有多强大？ 在传统的固态材料中，是否有任何这样的相没有类似物？ 这些相的空间分离部分的量子力学连接性是否可以用来存储或操纵未来信息技术的量子信息？ PI将使用分析技术和新颖的数学方法来研究这些问题。该奖项还支持教育和推广活动，包括：在高级研究生水平上开发一个新的一学期课程，重点是拓扑秩序; PI将通过斯托尼布鲁克大学组织的一个项目向高中生提供与研究领域相关的讲座; PI将针对更广泛的社区组织一个月一次的物理学系列讲座，该讲座将邀请外部演讲者向观众介绍这一领域和相关领域的前沿研究。技术概述该奖项支持理论研究和教育，以识别和分类物质的新相，称为拓扑相，特别关注对称性如何丰富可能的拓扑相。 虽然拓扑序是稳健的，并且独立于任何全局对称性，但包含对称性可能会导致额外的相位，例如最近发现的拓扑绝缘体，只有当存在时间反演对称时才是稳健的。 PI将使用分析技术，包括代数拓扑学，追求几个研究方向：1）对物理上现实的对称群（例如连续，反酉和空间对称）进行分类，推广有限离散对称群的结果。 这项研究将集中在这些阶段的稳定性，通过带框架，以及在寻找固有的强相互作用的阶段。3）构建多体不变量，区分在这样的分类状态。 4）以拓扑量子场论为基础，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：在高级研究生水平上开发一个新的一学期课程，重点是拓扑秩序; PI将通过斯托尼布鲁克大学组织的一个项目，向高中生提供与研究领域有关的讲座; 2 PI将组织一个月一次的针对更广泛社区的物理学讲座系列，这将引入外部演讲者，向听众介绍该领域和相关领域的前沿研究。