Theoretical Studies of Quantum Systems with Strong Interations

技术总结该奖项由材料研究部和物理部资助。它支持理论研究和教育，专注于相干量子液体中的几何非平衡现象，特别是分数量子霍尔边态中的几何非平衡现象，量子杂质中的非阿贝尔干涉现象，以及哈密顿驱动的非平衡过程中出现的奇点和奇异图案的几何分析，如拉普拉斯生长和扩散限制聚集模型。研究主题统一的目标是发展一种具有潜在空间共形对称性的非平衡和干扰过程理论。这个项目涉及利用在以前的国家科学基金支持下取得的进展对非平衡过程中产生的奇点进行几何分析。许多重要的非平衡系统表现出共形对称性，因此具有类似于临界现象的共形不变性的可积结构。然而，非平衡过程是不同的。共形不变性不可避免地导致了小尺度奇异图案的出现。反过来，奇点又产生了大尺度上可见的非平衡分形图。被驱动过程的随机模式的奇点的起源、统计和正则化以及分形几何是本研究的一个主题。另一个主题是分数量子霍尔边态的非线性量子流体力学。重点讨论了分数电荷激发作为边缘孤子的一种拓扑表现。这项研究的最后一个主题是非阿贝尔干涉现象，如在展示超屏蔽多通道Kondo方案的可控人工制造的量子纳米器件中实现的非阿贝尔干扰现象。PI将通过培训和指导研究生和本科生的研究经验，将教育和研究结合起来，并为本科生的研究经验做出新的贡献。非技术性总结该奖项由材料研究部和物理部资助。它支持与数学物理和数学相结合的理论凝聚态物理研究和教育。这项研究的重点是促进我们对复杂的非平衡过程的理解。S的工作的一个重要方面涉及到生长过程，表现出雪花般的手指从一个相渗透到另一个相，就像在合金和半导体结构的生长中发生的那样。PI试图从根本上理解这些指法模式是如何在成长过程中出现的。利用看似不同的研究领域之间的微妙联系，PI还将研究强磁场中一滴电子边缘出现的新物质状态。PI还将寻求一种方法来观察物质的不寻常的新状态，即在称为量子点的特殊微小原子结构中的拓扑态。这些态在理论上被预言存在于二维强磁场中的电子液体中；皮埃尔？S的提议提供了一个新的平台来研究这些可能的新的物质状态，使我们能够利用量子力学状态来进行计算。量子计算被认为为计算机性能的大幅提高提供了机会，至少在一些重要问题上是这样，密码学就是一个例子。PI将通过培训和指导研究生和本科生，将教育和研究结合在一起，并为本科生的研究体验做出新的贡献。拟议的研究结果将提高人们对远未处于平衡状态的复杂凝聚态系统的认识和理解。

TECHNICAL SUMMARYThis award is funded by the Division of Materials Research and the Physics Division. It supports theoretical research and education focused on geometrical non-equilibrium phenomena in coherent quantum liquids, especially in Fractional QuantumHall edge states; non-Abelian interference phenomena in quantum impurities, and geometrical analysis of singularities and singular patterns arising in Hamiltonian driven non-equilibrium processes such as Laplacian Growth and diffusion limited aggregation models.The research topics are unified by the goal to develop a theory of non-equilibrium and interference processes with underlying spatial conformal symmetry. This project is concerned with geometric analyses of singularities arising in non-equilibrium processes using advances achieved under prior NSF support. Many important systems out of equilibrium show conformal symmetries and therefore integrable structures similar to conformal invariance of critical phenomena. However non-equilibrium processes are different. Conformal invariance inevitably leads to singular patterns occurring at small scales. In its turn singularities give rise to fractal non-equilibrium patterns visible at large scales. The origin, statistics, and regularization of singularities and fractal geometry of stochastic patterns of driven processes comprise one theme of the research. Another theme is non-linear quantum hydrodynamics of Fractional Quantum Hall edge states. The emphasis is given to a topological manifestation of a fractional charge excitation as an edge soliton. The last theme of the research is non-Abelian interference phenomena as realized in controlled artificially fabricated quantum nanodevices exhibiting over-screened multichannel Kondo regime.The PI will integrate education and research through training and mentoring graduate and undergraduate research students, and making novel contributions to the Research Experiences for Undergraduates.NON-TECHNICAL SUMMARYThis award is funded by the Division of Materials Research and the Physics Division. It supports theoretical condensed matter physics research and education at an interface with mathematical physics and mathematics. The research is focused on advancing our understanding of complex non-equilibrium processes. An important aspect of the PI?s work involves growth processes that display snowflake-like fingers that penetrate from one phase into another, as happens in the growth of alloys and semiconductor structures. The PI seeks a fundamental understanding of how these fingering patterns emerge in the growth process. Capitalizing on subtle connections between seemingly disparate areas of research, the PI will also study new states of matter that emerge at the edges of a droplet of electrons in a high magnetic field. The PI will also pursue an approach to observing unusual new states of matter, topological states, in special tiny structures of atoms called quantum dots. These states were theoretically predicted to exist in electronic liquids confined to two dimensions and in high magnetic field; the PI?s proposal provides a new arena in which to study these possible new states of matter that may enable us to exploit quantum mechanical states to perform computation. Quantum computing is believed to provide an opportunity for a vast improvement in computer performance, at least on some important problems, cryptography being one example. The PI will integrate education and research through training and mentoring graduate and undergraduate research students, and making novel contributions to the Research Experiences for Undergraduates. The results of the proposed research will enhance knowledge and understanding of complex condensed matter systems that are far from being in an equilibrium state.