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Theoretical Studies of Quantum Systems with Strong Interaction: Geometry and Topology of Quantum States and Flows

强相互作用量子系统的理论研究：量子态和流动的几何和拓扑

基本信息

批准号：
1949963
负责人：
Pavel Wiegmann
金额：
$ 33万
依托单位：
University of Chicago
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-09-01 至 2024-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1949963&HistoricalAwards=false
关键词：
Theoretical Studies Quantum Systems Strong

项目摘要

NONTECHNICAL SUMMARYThis award supports fundamental research and education in the properties of electrons inside what are called "topological" solids. Whereas in metals, electrons interact with each other only weakly, so that their motion resembles that of molecules in a gas, in these newly appreciated materials, the electrons interact strongly, forming a "quantum liquid." Just as the science of hydrodynamics can describe the turbulent motion of water and other familiar liquids, this project develops a hydrodyanmics of quantum liquids.In the last decade, materials physicists have studied the topological nature of quantum fluids, referring to unusual global properties, including resistanceless current flows around edges, that are robust against sample imperfections and impurities. Most theoretical studies in this subject are focused on properties of the ground (lowest-energy) state and on linear-transport theory, according to which, for example, the electrical current in a wire is proportional to the applied voltage. However, it has became apparent that the fundamental features of quantum fluids lie in hydrodynamics, beyond linear-response theory, and involve not just topology but also geometry. Geometric, hydrodynamic, properties reflect a response of a quantum system to local bending of the sample. The project aims to push forward two novel research directions in quantum materials physics: (i) a geometric theory of quantum topological fluids, specifically of the fractional quantum Hall effect, and (ii) a hydrodynamic description of motions of such fluids.Hydrodynamics is, perhaps, the most developed branch of physics, where fundamental laws interwind with vast applications to phenomena at all scales. At the same time, hydrodynamics hosts notoriously difficult unsolved problems. One is turbulence; another is turning classical hydrodynamics into a quantum theory. Both problems are commonly considered intractable. At the same time, nature confronts us with experimentally accessible and beautiful quantum fluids like superfluid helium, which flows without resistance, and the fractional quantum Hall effect. Earlier attempts to quantize hydrodynamics, going back to Landau and Feynman, led to the semiclassical theory. Today, the emergence of electronic and atomic quantum fluids calls for a full scope of quantization. This research brings advanced methods of theoretical physics combined with methods of modern geometry to material science with a focus on experimentally-measurable signatures of geometric phenomena in flows. Graduate students will receive broad training in theoretical techniques, advanced mathematical methods, and communication skills.TECHNICAL SUMMARYThe award supports research into two novel themes in condensed matter theory: the geometric and hydrodynamic approaches to the theory of quantum fluids. The motivating case study is the fractional quantum Hall effect (FQHE). The theory targets precise quantization of transport coefficients and the fundamental role of geometry in quantized fluids. The work also aims to develop experimental settings for observing effects of geometry in semiconductors, cold gases, classical chiral flows, and chiral metafluids. Another theme of the work is the search for the local conformal symmetries in chiral quantum liquids and at the same time in classical turbulent flows. A related theme investigates physical applications of quantum anomalies, primarily the gravitational anomaly, in chiral quantum flows. Finally, the work aims to develop a hydrodynamic approach to non-linear flows in quantum systems with topological characterizations and search for the relation with turbulence.The work exploits advanced methods of modern geometry, conformal field theory, anomalies of quantum field theory, and hydrodynamics adapted to study quantum materials. The foci of the project are quantum systems with a topological characterization. The transport coefficients in such systems are quantized with unmatched precision. A reason for that is that these coefficients are topological invariants of holomorphic bundles. Recent developments show that the quantization is related to the geometry of quantum states, focusing on local properties. In turn, topological properties are merely the global reflection of geometric properties. The geometric properties describe a transformation of quantum states under a variation of the underlying metric. As a result, they govern the motion of the quantum fluid and in the end determine its hydrodynamics. The work will advance understanding of non-linear aspects of non-equilibrium states in topological quantum systems. There are many indications that the nature of flows in topological quantum fluids is deeply connected with that of turbulent flows in classical fluids. The work provides insights into the geometry of turbulent flows and builds a platform for the engineering of topological metamaterials. Another target of the project is the quantization of hydrodynamics, a long-standing fundamental problem of quantum theory, often considered intractable. Recent understanding of the role of the gravitational anomaly in the FQHE suggests a clear path to overcoming difficulties of quantization. The project seeks to develop a comprehensive scheme of quantization of hydrodynamics of two- and three-dimensional chiral incompressible flows. The work adapts the methods of modern geometry to materials science and non-equilibrium statistical mechanics, forging links between different disciplines. The proposed work has an interdisciplinary character: it addresses fundamental problems of materials science and at the same time contributes to the fields of hydrodynamics and modern geometry. Education and mentoring are integral parts of the proposal. The work provides a quality platform to attract and to train theoretically minded students and to prepare them for careers in academia and science-related industry. Students will receive broad training in a full scope of research: analytical reasoning, mastering advanced mathematical methods by applying them to physical systems, projecting theoretical research to realistic materials, and communication with experimentalists and researchers in adjacent fields.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.

该奖项支持在所谓的“拓扑”固体内部的电子特性的基础研究和教育。在金属中，电子之间的相互作用很弱，因此它们的运动类似于气体中的分子，而在这些新发现的材料中，电子之间的相互作用很强，形成了一种“量子液体”。就像流体力学可以描述水和其他熟悉的液体的湍流运动一样，这个项目发展了量子液体的流体力学。在过去的十年里，材料物理学家研究了量子流体的拓扑性质，指的是不寻常的全局特性，包括边缘周围的无电阻电流，这些特性对样品缺陷和杂质具有很强的抵抗能力。这方面的大多数理论研究都集中在接地（最低能量）状态的特性和线性传输理论上，例如，根据线性传输理论，导线中的电流与施加的电压成正比。然而，很明显，量子流体的基本特征在于流体力学，超越了线性响应理论，不仅涉及拓扑，还涉及几何。几何、流体力学性质反映了量子系统对样品局部弯曲的响应。该项目旨在推动量子材料物理学的两个新的研究方向：(i)量子拓扑流体的几何理论，特别是分数量子霍尔效应；（ii）这种流体运动的流体力学描述。流体力学也许是物理学中最发达的分支，它的基本定律与各种尺度的现象的广泛应用交织在一起。与此同时，流体力学也存在着一些难以解决的问题。一个是湍流；另一个是将经典流体力学转变为量子理论。这两个问题通常被认为是难以解决的。与此同时，大自然向我们展示了实验上可以接近的、美丽的量子流体，比如无阻力流动的超流氦，以及分数量子霍尔效应。早期对流体力学量子化的尝试，可以追溯到朗道和费曼，导致了半经典理论。今天，电子和原子量子流体的出现要求全面的量子化。本研究将先进的理论物理方法与现代几何方法结合到材料科学中，重点研究流动中几何现象的实验可测量特征。研究生将在理论技术、高级数学方法和沟通技巧方面接受广泛的训练。该奖项支持凝聚态理论中两个新主题的研究：量子流体理论的几何和流体动力学方法。激励案例研究是分数量子霍尔效应（FQHE）。该理论的目标是输运系数的精确量子化和几何在量子化流体中的基本作用。该工作还旨在开发实验设置，以观察半导体，冷气体，经典手性流动和手性元流体中的几何效应。工作的另一个主题是寻找手性量子液体的局部共形对称性，同时在经典湍流中。一个相关的主题是研究量子异常的物理应用，主要是引力异常，在手性量子流中。最后，本工作旨在发展具有拓扑表征的量子系统中非线性流动的流体动力学方法，并寻找与湍流的关系。这项工作利用了现代几何、共形场论、量子场论的异常和流体力学的先进方法来研究量子材料。该项目的重点是具有拓扑特征的量子系统。这类系统的输运系数以无与伦比的精度被量化。一个原因是这些系数是全纯束的拓扑不变量。最近的发展表明，量子化与量子态的几何形状有关，重点是局部性质。反过来，拓扑性质仅仅是几何性质的全局反映。几何性质描述了量子态在底层度规变化下的变换。因此，它们控制着量子流体的运动，并最终决定其流体力学。这项工作将促进对拓扑量子系统中非平衡态的非线性方面的理解。有许多迹象表明，拓扑量子流体中的流动性质与经典流体中的湍流性质有着密切的联系。这项工作提供了对湍流几何的见解，并为拓扑超材料的工程建立了一个平台。该项目的另一个目标是流体力学的量子化，这是量子理论中一个长期存在的基本问题，通常被认为是难以解决的。最近对引力异常在FQHE中的作用的理解为克服量子化困难提供了一条明确的途径。该项目旨在开发一个全面的方案量化流体力学的二维和三维手性不可压缩流动。该作品将现代几何方法应用于材料科学和非平衡统计力学，在不同学科之间建立了联系。提议的工作具有跨学科的特点：它解决了材料科学的基本问题，同时对流体力学和现代几何领域做出了贡献。教育和指导是该提议的组成部分。这项工作提供了一个高质量的平台，以吸引和培养理论思维的学生，并为他们在学术界和科学相关行业的职业生涯做好准备。学生将接受全面的研究训练：分析推理，掌握先进的数学方法并将其应用于物理系统，将理论研究应用于现实材料，并与邻近领域的实验人员和研究人员进行交流。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。