Theoretical Studies of Quantum Systems with Strong Interactions

强相互作用量子系统的理论研究

• 批准号: 1206648
• 负责人: Pavel Wiegmann
• 金额: $ 40万
• 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: 2012
• 资助国家: 美国
• 起止时间: 2012-09-01 至 2017-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1206648&HistoricalAwards=false
• 关键词: Theoretical; Studies; Quantum; Systems; Strong

TECHNICAL SUMMARYThe Division of Materials Research, the Physics Division, and the Division of Mathematical Sciences contribute funds to this award. This award supports theoretical research and education in the general areas of theoretical condensed matter physics, and statistical physics focusing on geometrical non-equilibrium phenomena in quantum liquids and statistical mechanics. The study emphasizes a role of singularities and instabilities arising in non-equilibrium processes and quantum and statistical aspects of singularities. The first area of research concerns non-linear quantum hydrodynamics of fractional quantum Hall states with a focus on edge states and the relation of fractional quantum Hall states to conformal invariance. The PI will focus on the quantum non-linear hydrodynamics of electronic liquid in the fractional quantum Hall regime and especially on a topological manifestation of the fractional charge of excitations as solitons on fractional quantum Hall edge states emerging as a result of non-linear dynamics. The PI will also investigate realistic experimental settings where edge solitons can be observed. The second area of research focuses on singularities and emergent conformal symmetry in driven processes. PI will study fingering instability in Lapacian Growth building on the idea that singularities of non-equilibrium patterns occurring at small scales give rise to fractal non-equilibrium patterns visible at a large scale. The work will be built on the theory of viscous shocks developed under prior NSF support.The third theme of research focuses on the statistics of geometrical objects in critical phenomena. This study addresses the long-standing problem of the area-length distribution of critical clusters or domains of different phases in critical phenomena.The research addresses important problems of material research, and simultaneously contributes to emergent fields in mathematical physics and mathematics by synthesizing problems and methods of different disciplines.This award also supports training and mentoring graduate and undergraduate students and a postdoctoral fellow. Research results will be integrated into graduate level courses in modern dynamics, and traditional courses in condensed matter physics. NON-TECHNICAL SUMMARYThe Division of Materials Research, the Physics Division, and the Division of Mathematical Sciences contribute funds to this award. This award supports research at the interface with mathematics, and mathematical physics. It focuses on the emergent field of non-linear quantum dynamics with an emphasis on the role of geometry in non-equilibrium processes in quantum electronic and atomic liquids. Similar geometrical phenomena also emerge in critical phenomena of statistical mechanics and processes of growth and aggregation of materials. One thrust of the research focuses on developing a theoretical description of a special kind of quantum liquid. Quantum liquids differ from more familiar liquids in that their properties are dominated by the effects of quantum mechanics. The PI will focus on a particular kind of quantum liquid, a quantum Hall liquid that results when electrons confined to two dimensions in an artificial materials structure made of semiconductors and exposed to a high magnetic field. The way the electrons organize themselves leads to an 'edge state' that wraps around the bulk of the liquid. The 'edge state' is also related to a metallic state that arises at the surfaces of a particular class of insulating materials, known as topological insulators. The PI will develop a hydrodynamic theory of this liquid in a way that emphasizes the role of geometry. Another thrust of the research concerns the investigation of geometrical patterns that emerge close to the transformation of one phase into another, for example water to ice. The PI will use sophisticated mathematical methods to determine the number of critical loops containing one phase in the presence of the other with a particular area and length of boundary. The loops are examples of fluctuating random geometrical shapes and patterns that appear not only in phase transitions but also in growth processes of materials and disordered systems. This effort reflects a new approach to critical behavior, for example phase transformations, in two-dimensions. The broader impact of the proposal is a dissemination of methods of material science to the fields of non-equilibrium statistical mechanics, non-linear physics, probability theory and complex analysis, forging links between the disciplines. Interdisciplinary research area is particularly well suited for the training of graduate students and postdoctoral fellows. It requires mathematical sophistication and deep understanding of fundamental aspects of correlated quantum states and statistical mechanics.

材料研究部、物理部和数学科学部为该奖项提供资金。该奖项支持理论凝聚态物理学和统计物理学一般领域的理论研究和教育，重点关注量子液体和统计力学中的几何非平衡现象。这项研究强调了在非平衡过程中产生的奇异性和不稳定性以及奇异性的量子和统计方面的作用。第一个研究领域涉及分数量子霍尔态的非线性量子流体力学，重点是边缘态和分数量子霍尔态与共形不变性的关系。PI将专注于分数量子霍尔制度中电子液体的量子非线性流体动力学，特别是作为非线性动力学结果出现的分数量子霍尔边缘状态上的孤立子的激发分数电荷的拓扑表现。PI还将研究可以观察到边缘孤子的实际实验设置。研究的第二个领域集中在驱动过程中的奇点和涌现共形对称性。 PI将研究Lapacian增长中的指进不稳定性，其基础是在小尺度上发生的非平衡模式的奇异性会产生在大尺度上可见的分形非平衡模式。这项工作将建立在美国国家科学基金会先前支持下发展的粘性激波理论基础上。第三个研究主题集中在临界现象中几何物体的统计。本研究旨在解决长期存在的临界现象中不同相态的临界团簇或畴的面积长度分布问题，解决材料研究中的重要问题，同时通过综合不同学科的问题和方法，为数学物理和数学的新兴领域做出贡献。该奖项还支持培养和指导研究生和本科生以及博士后研究员。研究成果将被纳入现代动力学研究生课程和凝聚态物理学传统课程。非技术摘要材料研究部、物理部和数学科学部为该奖项提供资金。该奖项支持与数学和数学物理接口的研究。它侧重于非线性量子动力学的新兴领域，强调量子电子和原子液体中非平衡过程中几何形状的作用。类似的几何现象也出现在统计力学的临界现象以及材料的生长和聚集过程中。 研究的一个重点是发展一种特殊量子液体的理论描述。量子液体与更熟悉的液体的不同之处在于，它们的性质由量子力学的影响所主导。PI将专注于一种特殊的量子液体，一种量子霍尔液体，当电子被限制在由半导体制成的人造材料结构中的二维空间并暴露在高磁场中时产生。电子组织自身的方式导致了一种“边缘状态”，它包裹着液体的主体。 “边缘状态”也与金属状态有关，这种金属状态出现在特定类别的绝缘材料表面，称为拓扑绝缘体。PI将以强调几何作用的方式开发这种液体的流体动力学理论。这项研究的另一个重点是研究在一个相态转变为另一个相态时出现的几何图案，例如水变成冰。PI将使用复杂的数学方法来确定包含一个相位的关键回路的数量，其中另一个相位具有特定的边界面积和长度。 环是波动的随机几何形状和图案的例子，不仅出现在相变中，而且出现在材料和无序系统的生长过程中。 这一努力反映了一种新的方法来临界行为，例如相变，在二维。该建议的更广泛影响是将材料科学方法传播到非平衡统计力学、非线性物理学、概率论和复杂分析等领域，从而在学科之间建立联系。跨学科研究领域特别适合培养研究生和博士后研究员。它需要数学的复杂性和对相关量子态和统计力学基本方面的深刻理解。