CAREER: Strongly correlated systems through the lens of topological phases

该职业奖支持理论研究和教育，以调查固态材料中可能出现的物质状态。凝聚态物理的终极目标之一就是从理论上理解并在实验室中实现物质的所有可能相。区分物质的不同相的传统方法是基于对称的思想。例如，不同的固体可以通过形成它们的原子的不同晶体对称性来分类，例如立方体。类似的方法也可以应用于相互作用的电子在材料中产生的许多相。然而，这种方法本质上是基于经典物理学的直觉，在实践中不足以描述由量子力学实现的物质的众多阶段。事实上，这种方法甚至不允许人们从理论上区分金属和绝缘体。更令人惊讶的是，存在着具有相同对称性的不同绝缘体。该项目的中心目标是研究这些量子绝缘体和它们所承载的大量现象，其中包括保护导电免受物质无序存在的通常有害影响的表面。这样的理解可以得出关于材料中电子复杂行为的一般性结论，并最终为量子计算机的构建块提供材料平台。除了它自己的领域，这个项目还会对其他几个科学领域产生影响，比如数学和高能物理。除了研究之外，该项目还包括针对K-12、本科生和研究生的多管齐下的教育和推广计划。K-12社区将通过参与nsf资助的Theorynet项目（该项目将理论家引入课堂）和波士顿物理圈（Boston Physics Circle）的共同组织（该组织为高中生准备物理竞赛）来实现。本科生社区将通过直接参与研究和流行的研究报告来服务。这个项目的一个主要组成部分是研究生的培训，他们将参与研究的各个方面。该职业奖支持理论研究和教育，以理解强相互作用电子系统的基本原理。在过去的十年里，在理解物质的间隙相方面取得了令人印象深刻的进展。对称性和拓扑学的思想在这种相的物理学中起着重要的作用。部分进展是由对称保护拓扑（SPT）相的研究推动的，这是拓扑绝缘体的相互作用推广。SPT的许多有趣的物理现象发生在它的边界上，它保证携带非平凡对称保护的边缘模式。令人惊讶的是，玻色子的SPT相的完整分类，以及最近提出的具有任意相互作用的费米子的SPT相。然而，对这些相的表面态的理解仍然有限，特别是对费米子的强相互作用的SPT相。研究二维和三维spt的边界并将结果应用于其他问题是本项目的主要目标。特别是，PI将研究：i)支持任意子激发的三维SPT的间隙对称边界，ii)二维费米子SPT及其一维CFT边界，iii)应用SPT推导三维（相互作用）绝缘体存在的严格电子填充约束，iv)存在猝灭无序的SPT表面状态。这项研究将极大地促进我们对物质强相互作用相的理解（不仅仅是spt）。特别是，它将加深我们对对称异常的理解，这可能导致量子场论之间新的对偶性的猜想。除了它自己的领域，这个项目还会对其他几个科学领域产生影响，比如数学和高能物理。除了研究之外，该项目还包括针对K-12、本科生和研究生的多管齐下的教育和推广计划。K-12社区将通过参与nsf资助的Theorynet项目（该项目将理论家引入课堂）和波士顿物理圈（Boston Physics Circle）的共同组织（该组织为高中生准备物理竞赛）来实现。本科生社区将通过直接参与研究和流行的研究报告来服务。这个项目的一个主要组成部分是研究生的培训，他们将参与研究的各个方面。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

NONTECHNICAL SUMMARYThis CAREER award supports theoretical research and education towards investigating possible states of matter that can appear in solid-state materials. One of the ultimate goals of condensed matter physics is to theoretically understand and realize in the laboratory all possible phases of matter. The traditional approach to distinguishing different phases of matter from one another is based on ideas of symmetry. For instance, different solids can be classified by the distinct crystal symmetries the atoms that form them attain, e.g. cubic. A similar approach can also be applied to the many phases interacting electrons can create in materials.However, that approach is essentially based on intuition from classical physics and is in practice insufficient for describing a multitude of phases of matter enabled by quantum mechanics. In fact, that approach does not even allow one to theoretically distinguish a metal from an insulator. Even more surprising, there exist distinct insulators with the same symmetry. The central goal of this project is the study of these quantum insulators and the vast array of phenomena they host, which include surfaces on which electric conduction protected from the usually detrimental effects of material disorder exists. Such understanding could lead to general conclusions about the complex behavior of electrons in materials and could eventually provide material platforms for the building blocks of quantum computers.Beyond its own field, this project can have impact on several other areas of science, such as mathematics and high-energy physics. In addition to the research, the project includes a multipronged educational and outreach program targeting K-12, undergraduate, and graduate students. The K-12 community will be reached through participation in the NSF-funded Theorynet program that brings theorists into classrooms, and through co-organization of the Boston Physics Circle, which prepares high-school students for physics competitions. The undergraduate community will be served by direct involvement in research and by popular research presentations. A major component of this project is the training of graduate students who will be involved in all aspects of the research.TECHNICAL SUMMARYThis CAREER award supports theoretical research and education towards understanding fundamentals of strongly interacting electron systems. In the last decade there has been an impressive progress in the understanding of gapped phases of matter. Ideas of symmetry and topology play an important role in the physics of such phases. Part of the progress has been driven by the study of symmetry-protected topological (SPT) phases, which are interacting generalizations of topological insulators. Much of the interesting physics of an SPT occurs at its boundary, which is guaranteed to carry nontrivial symmetry-protected edge modes. Surprisingly, a full classification of SPT phases of bosons and, more recently, fermions with arbitrary interactions has been proposed. However, the understanding of surface states of such phases is still limited, particularly for strongly interacting SPT phases of fermions. The study of boundaries of 2D and 3D SPTs and the application of results to other problems is a major goal of this project. In particular, the PI will investigate: i) Gapped symmetric boundaries of 3D SPTs supporting anyon excitations, ii) 2D fermion SPTs and their 1D CFT boundaries, iii) Application of SPTs to derive tight electron filling constraints on the existence of (interacting) insulators in 3D, iv) SPT surface states in the presence of quenched disorder.This research will significantly advance our understanding of strongly interacting phases of matter (beyond just SPTs). In particular, it will deepen our understanding of symmetry anomalies, which may lead to the conjecture of new dualities between quantum field theories.Beyond its own field, this project can have impact on several other areas of science, such as mathematics and high-energy physics. In addition to the research, the project includes a multipronged educational and outreach program targeting K-12, undergraduate, and graduate students. The K-12 community will be reached through participation in the NSF-funded Theorynet program that brings theorists into classrooms, and through co-organization of the Boston Physics Circle, which prepares high-school students for physics competitions. The undergraduate community will be served by direct involvement in research and by popular research presentations. A major component of this project is the training of graduate students who will be involved in all aspects of the research.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.