CAREER: Bulk and boundary properties of topological matter

本职业奖支持旨在提高我们对物质拓扑相理解的理论研究。拓扑相，如分数量子霍尔态和拓扑绝缘体，提出了一个基本的概念挑战，因为它们不能用有序参数或对称破缺来表征。因此，需要开发新的概念和工具来理解这些系统。该奖项支持的研究将集中在这些新阶段的两个方面。首先，PI将研究物质拓扑相的体积和边界性质之间的一般关系。这种“体积-边界对应”对于理论和实验都很重要，因为探测这些系统最简单的方法之一是通过边界上的输运测量。其次，PI将研究这些阶段的具体微观模型，其双重目标是(a)确定实验实现和(b)理解这些状态的更深层次结构和数值方法的局限性。将探讨的一些主要主题包括：1)二维拓扑相保护边缘模式的条件，2)二维拓扑相的完全可溶晶格模型及其局限性，3)玻色子整数量子霍尔态的准现实模型，4)三维玻色子对称保护拓扑相的体积和表面性质。除了研究之外，该奖项还以两种方式支持教育。首先，PI将创建一个新的“拓扑与物理”研究生班。重要的是，本课程将涵盖凝聚态和高能物理的拓扑现象，目的是吸引两组学生的混合，从而促进两个子领域之间的相互作用。第二，PI将指导本科生的暑期研究项目。该奖项的其他好处将来自外展活动，包括将向参加芝加哥大学夏季REU项目的本科生提供关于量子计算和超导的几次讲座。该职业奖支持旨在提高我们对物质的奇异相（即“拓扑相”）的理解的理论研究。就像水分子的集合可以根据温度和压力表现为固体、液体或气体一样，材料中的电子可以根据不同的参数组织成金属、绝缘体、超导体和其他相。凝聚态物理的基本目标之一是表征和分析在这些多电子或多粒子系统中可能形成的不同相。70多年前，Lev Landau开发了一个强有力的框架来回答这类问题。他的理论以对称概念为基础，能够解释从超导体到磁体的各种相。尽管取得了这些成功，但现在很清楚，在一些材料中，量子效应可以导致物质的新阶段，从根本上超越朗道范式。最引人注目的例子被称为“拓扑相”。这些相具有丰富的内部结构，但与磁体或超导体等传统相不同，这种结构与对称性无关。相反，这些相的定义特征具有拓扑特征。因此，需要开发全新的概念和工具来理解这些系统。该奖项支持的研究旨在开发这种机器，最终目标是建立一个类似于朗道理论的一般理论框架，用于分析这些新的物质相。除了研究之外，该奖项还以两种方式支持教育。首先，PI将创建一个新的“拓扑与物理”研究生班。重要的是，本课程将涵盖凝聚态和高能物理的拓扑现象，目的是吸引两组学生的混合，从而促进两个子领域之间的相互作用。第二，PI将指导本科生的暑期研究项目。该提案的其他好处将来自外展活动，包括将为参加芝加哥大学夏季REU项目的本科生提供关于量子计算和超导的几次讲座。

TECHNICAL SUMMARYThis CAREER award supports theoretical research aimed at improving our understanding of topological phases of matter. Topological phases, such as fractional quantum Hall states and topological insulators, pose a fundamental conceptual challenge because they cannot be characterized using order parameters or symmetry breaking. Consequently, new concepts and tools need to be developed to understand these systems. The research supported by this award will focus on two aspects of these novel phases. First, the PI will investigate the general relationship between bulk and boundary properties of topological phases of matter. This 'bulk-boundary correspondence' is important both for theory and experiment since one of the easiest ways to probe these systems is via transport measurements at the boundary. Second, the PI will investigate concrete microscopic models for these phases, with the dual aims of (a) identifying experimental realizations and (b) understanding the deeper structure of these states and the limitations of numerical approaches. Some of the main topics that will be explored include:1) Conditions for protected edge modes for 2D topological phases,2) Exactly soluble lattice models for 2D topological phases and their limitations,3) Quasi-realistic models for bosonic integer quantum Hall states,4) Bulk and surface properties of 3D bosonic symmetry-protected topological phases. In addition to research, this award supports education in two ways. First, the PI will create a new graduate class on 'Topology and Physics.' Importantly, this class will cover topological phenomena in both condensed matter and high energy physics with the goal of attracting a mixture of students from both groups and thereby facilitating interactions between the two subfields. Second, the PI will supervise summer research projects for undergraduates. Other benefits of this award will come from outreach activities, including several talks on quantum computing and superconductivity that will be given to the undergraduate students participating in the summer REU program at the University of Chicago.NONTECHNICAL SUMMARYThis CAREER award supports theoretical research aimed at improving our understanding of exotic phases of matter known as 'topological phases.' Just as a collection of water molecules can behave like a solid, a liquid, or a gas depending on the temperature and pressure, the electrons in a material can organize into metals, insulators, superconductors and other phases depending on various parameters. One of the basic goals of condensed matter physics is to characterize and analyze the different phases that can form in these many-electron or many-particle systems. More than 70 years ago, Lev Landau developed a powerful framework for answering these types of questions. His theory, based on the concept of symmetry, was able to explain a vast array of phases -- everything from superconductors to magnets. Despite these successes it is now clear that, in some materials, quantum effects can lead to new phases of matter that are fundamentally beyond the Landau paradigm. The most dramatic examples are known as 'topological phases.' These phases have a rich internal structure, but unlike conventional phases like magnets or superconductors, this structure has nothing to do with symmetry. Instead, the defining features of these phases have a topological character. As a result, entirely new concepts and tools need to be developed to understand these systems. The research supported by this award aims to develop this machinery with the ultimate goal of building a general theoretical framework, analogous to Landau theory, for analyzing these novel phases of matter.In addition to research, this award supports education in two ways. First, the PI will create a new graduate class on 'Topology and Physics.' Importantly, this class will cover topological phenomena in both condensed matter and high energy physics with the goal of attracting a mixture of students from both groups and thereby facilitating interactions between the two subfields. Second, the PI will supervise summer research projects for undergraduates. Other benefits of this proposal will come from outreach activities, including several talks on quantum computing and superconductivity that will be given to the undergraduate students participating in the summer REU program at the University of Chicago.