Theoretical Studies of Quantum Systems with Strong Interactions

技术概述材料研究部、物理部和数学科学部为本奖项提供资金。该奖项支持理论凝聚态物理和统计物理一般领域的理论研究和教育，重点是量子液体和统计力学中的几何非平衡现象。该研究强调了奇点和非平衡过程中产生的不稳定性的作用以及奇点的量子和统计方面。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，重点关注边缘态和分数量子霍尔态与保形不变性的关系。PI将重点关注分数量子霍尔体系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态孤子的激励的分数电荷的拓扑表现。PI还将研究可以观察到边缘孤子的现实实验设置。第二个研究领域集中在驱动过程中的奇点和紧急共形对称性。PI将研究Lapacian生长中的指法不稳定性，其基础是在小尺度上出现的非平衡模式的奇点会产生在大尺度上可见的分形非平衡模式。这项工作将建立在粘性冲击理论的基础上，该理论是在先前的国家科学基金支持下发展起来的。研究的第三个主题是关键现象中几何对象的统计。本研究解决了长期存在的临界现象中不同相的临界簇或域的面积长度分布问题。该研究解决了材料研究的重要问题，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了贡献。该奖项还支持培训和指导研究生和本科生以及一名博士后。研究成果将整合到研究生水平的现代动力学课程和凝聚态物理的传统课程中。材料研究部、物理部和数学科学部为本奖项提供资金。该奖项支持与数学和数学物理相结合的研究。它侧重于非线性量子动力学的新兴领域，强调几何在量子电子和原子液体的非平衡过程中的作用。类似的几何现象也出现在统计力学的关键现象和材料的生长和聚集过程中。这项研究的一个重点是发展一种特殊量子液体的理论描述。量子液体与我们更熟悉的液体的不同之处在于，它们的性质受量子力学的影响。PI将专注于一种特殊的量子液体，一种量子霍尔液体，当电子被限制在半导体人造材料结构的二维空间并暴露在高磁场中时产生。电子自我组织的方式导致了包裹大部分液体的“边缘状态”。“边缘状态”也与金属状态有关，这种金属状态出现在被称为拓扑绝缘体的一类特殊绝缘材料的表面。PI将以一种强调几何作用的方式发展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接近于一种相转变为另一种相的几何模式，例如水到冰。PI将使用复杂的数学方法来确定临界回路的数量，其中包含具有特定面积和边界长度的一个相位。这些循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关键行为方法，例如二维相变。该提议的更广泛影响是将材料科学的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，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。它需要数学的复杂性和对相关量子态和统计力学的基本方面的深刻理解。

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.