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

量子霍尔效应及相关现象中的几何、拓扑和动力学

• 批准号: 1508255
• 负责人: Ilya Gruzberg
• 金额: $ 31.5万
• 依托单位: Ohio State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-08-15 至 2018-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1508255&HistoricalAwards=false
• 关键词: Geometry; topology; dynamics; quantum; Hall

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.

非技术性总结该奖项支持理论研究和教育的作用，无序的性质的材料。用于各种应用的真实的材料不可避免地存在一些缺陷、杂质和其他类型的无序。重要的是要了解材料的性质如何受到无序的影响。虽然混乱可能会产生不良影响，但它也可能导致新的行为。PI的研究兴趣在于这第二类现象。例如，无序可以散射材料中的电子。有时，无序状态下电子的多次散射会导致它们被捕获或局限在样品的某些地方。定域电子不能移动以传导电和热。这种俘获现象被称为安德森局域化，以诺贝尔奖赢家P.W.安德森。具有局域电子的固体是绝缘体，但如果改变系统的参数，电子可以变得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跃迁的研究。其中一种跃迁是量子霍尔效应中所谓的平台跃迁。量子霍尔效应是电子在强磁场中无序化而产生的安德森局域化现象的壮观表现。这些效应导致半导体和石墨烯中电子的霍尔电导率极其精确的量化。霍尔电导率测量系统在垂直于由施加电压确定的方向上导电的程度。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的实验样品的一个重要特征是它们的有限尺寸和它们的边界所起的关键作用。PI研究的一个重要部分旨在详细了解量子霍尔系统在其边界附近的精细结构。该奖项支持无序材料的理论研究和教育。描述安德森跃迁临界性质的易于分析的理论的识别仍然难以捉摸;这是无序电子系统领域的一个悬而未决的问题。量子霍尔效应的发现开辟了一个新的研究学科，实验和理论，不断激发新的想法和发展。量子霍尔效应是物质拓扑相的第一个例子，其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，以及它们与实验相关的有限样品边界处边缘态动力学的关系。具体项目包括：1）基于几何对象的经典统计力学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研究; 2）D类、时间反转和自旋旋转对称性破缺的超导体的局域化和安德森跃迁，以及相关的随机键伊辛模型; 3）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的边界结构和动力学，分数量子霍尔态和黎曼面上的霍尔粘性; 5）分数量子霍尔态的精细结构和涌现共形对称性以及非线性边缘动力学。这些项目将汇集物理和数学各个领域的思想，包括随机系统的局部化，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、分形和可积系统。PI的研究有助于通过将结果传达给各个研究社区并促进不同领域从业者之间的合作，使这些领域更加紧密。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。