Geometry, topology, and dynamics in quantum Hall effects and related phenomena

非技术总结该奖项支持关于无序在材料性质中的作用的理论研究和教育。在各种应用中使用的真实材料不可避免地会有一些缺陷、杂质和其他种类的无序。了解无序对材料性能的影响是很重要的。虽然无序可能会产生不良影响，但它也可能导致全新的行为。PI的研究兴趣在于这第二类现象。例如，无序可以使电子在材料中散射。有时，无序对电子的多次散射会导致它们被捕获或局域在样品的某些地方。局域电子不能移动来导电和导热。这种陷阱现象被称为安德森本土化，以诺贝尔奖获得者P·W·安德森的名字命名。具有局域电子的固体是绝缘体，但如果改变系统的参数，电子可能会离域，并像金属一样导电。该奖项支持对无序驱动的金属和绝缘体之间的转变的研究。其中一个这样的转变是所谓的量子霍尔效应中的平台转变。量子霍尔效应是电子在很强磁场中由于无序而产生的安德森局域化的壮观表现。这些效应导致了半导体和石墨烯中电子的霍尔电导率的极其精确的量子化。霍尔电导率测量系统在垂直于施加电压的方向上导电的好坏。这种量子化是现代电阻标准的基础。用于量子霍尔测量的真实实验样品的一个重要特征是它们的有限大小和它们的边界所起的关键作用。PI研究的一个重要部分是详细了解量子霍尔系统边界附近的精细结构。技术总结该奖项支持无序材料的理论研究和教育。描述安德森相变临界性质的易于分析的理论仍然难以确定；这是无序电子系统领域的一个突出问题。量子霍尔效应的发现开启了一门新的研究学科，无论是实验还是理论，不断激发着新的思想和发展。量子霍尔效应是物质拓扑相的第一个例子，它的研究是当前非常活跃的研究领域。PI的研究项目将集中在整数和分数量子霍尔效应的几何和拓扑方面，以及它们与与实验相关的有限样本边界上的边态动力学的关系。具体项目包括：1)基于几何物体的经典统计力学的映射和共形约束的二维整数量子霍尔跃迁和其他无序临界点的理论；2)D类中的局域化和安德森相变，具有破裂的时间反转和自旋旋转对称性的超导体，以及相关的随机键伊辛模型；3)具有结构无序的网络模型，以及随机表面上的其他无序系统；4)整数和分数量子霍尔态边界附近的结构和动力学；分数量子霍尔态和黎曼表面上的霍尔粘性；5)分数量子霍尔态和非线性边缘动力学的精细结构和出现的共形对称性。这些项目将汇集不同物理和数学领域的想法，包括局域化、随机系统的统计力学、临界现象、共形场论、弦理论、微分和复几何、随机矩阵模型、复分析、概率论、分形学和可积系统。PI的研究通过将结果传达给不同的研究社区并促进不同领域的从业者之间的合作，有助于拉近这些领域的距离。这项研究涉及国际合作。这些项目将为研究生和博士后提供研究和培训机会。

NONTECHNICAL SUMMARYThis award supports theoretical research and education on the role of disorder in the properties of materials. Real materials that are used in all sorts of applications inevitably have some imperfections, impurities, and other kinds of disorder. It is important to understand how properties of materials are affected by the disorder. While disorder can have undesirable effects, it can also lead to qualitatively new behaviors. The PI's research interests lie in this second class of phenomena.For example disorder can scatter electrons in materials. Sometimes multiple scattering of electrons by disorder leads to them being trapped, or localized, in certain places in a sample. Localized electrons cannot move to conduct electricity and heat. This trapping phenomenon is called Anderson localization, after the Nobel Prize winner P. W. Anderson. A solid with localized electrons is an insulator, but if one changes parameters of the system, electrons can become delocalized and conduct electricity like a metal. This award supports research on transitions between metals and insulators driven by disorder.One such transition is the so-called plateau transition in quantum Hall effects. Quantum Hall effects are spectacular manifestations of Anderson localization of electrons due to disorder in the presence of a very strong magnetic field. The effects lead to extremely precise quantization of Hall conductivity of electrons in semiconductors and graphene. The Hall conductivity measures the how well the system conducts electricity in the direction perpendicular to that determined by the applied voltage. This quantization is the basis for the modern standard of resistance.An important feature of real experimental samples used in quantum Hall measurements is their finite size and key role played by their boundaries. A significant part of the PI's research is aimed at a detailed understanding of the fine structure of quantum Hall systems near their boundaries. TECHNICAL SUMMARYThis award supports theoretical research and education on disordered materials. Identification of an analytically tractable theory describing critical properties at Anderson transitions remains elusive; it is an outstanding problem in the area of disordered electronic systems. The discovery of quantum Hall effects has opened a new research discipline, experimental and theoretical, continues to stimulate new ideas and developments. Quantum Hall effects are the first examples of topological phases of matter, whose study is an enormously active current research area.The PI's research projects will focus on geometric and topological aspects of integer and fractional quantum Hall effects, and their relation to the dynamics of edge states at the boundaries of finite samples relevant to experiments. Particular projects include the investigation of:1) the theory of the integer quantum Hall transition and other disordered critical points in two dimensions based on mappings to classical statistical mechanics of geometric objects, and conformal restriction; 2) localization and Anderson transitions in class D, superconductors with broken time reversal and spin rotation symmetries, and related random bond Ising models; 3) network models with structural disorder, and other disordered systems on random surfaces;4) structure and dynamics near boundaries of integer and fractional quantum Hall states; fractional quantum Hall states and Hall viscosity on Riemann surfaces; 5) fine structure and emergent conformal symmetry of fractional quantum Hall states and non-linear edge dynamics.These projects will bring together ideas from various fields of physics and mathematics including localization, statistical mechanics of random systems, critical phenomena, conformal field theory, string theory, differential and complex geometry, random matrix models, complex analysis, probability theory, fractals, and integrable systems. The PI's research contributes to bringing these fields closer by communicating the results to various research communities and promoting collaborations between practitioners in diverse areas. The research involves international collaborations. The projects will provide research and training opportunities for graduate students and postdocs.