Theoretical Studies of Quantum Systems with Strong Interations

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，重点是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缘态;量子杂质中的非阿贝尔干涉现象，和几何分析的奇异性和奇异模式所产生的哈密顿驱动的非-平衡过程，如拉普拉斯增长和扩散限制聚集模型。研究主题统一的目标是发展一个理论，非平衡和干扰过程与潜在的空间共形对称性。这个项目是关于几何分析的奇异性所产生的非平衡过程中使用的进展下取得的先前NSF的支持。许多重要的非平衡系统都表现出共形对称性，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奇异模式发生在小尺度上。反过来，奇点又产生了在大尺度上可见的分形非平衡模式。起源，统计和正则化的奇异性和分形几何的随机模式的驱动过程包括一个主题的研究。 另一个主题是分数量子霍尔边缘态的非线性量子流体力学。重点给出了分数电荷激发作为边缘孤子的拓扑表现。研究的最后一个主题是在受控人工制造的量子纳米器件中实现的非阿贝尔干涉现象，这些器件表现出过屏蔽的多通道近藤机制。PI将通过培训和指导研究生和本科生研究生，并为本科生的研究经验做出了新的贡献。该奖项由材料研究部和物理部资助。它支持理论凝聚态物理研究和教育，与数学物理和数学的接口。该研究的重点是推进我们对复杂非平衡过程的理解。PI的一个重要方面？他的工作涉及生长过程，显示雪花状的手指，从一个阶段渗透到另一个阶段，就像合金和半导体结构的生长一样。PI试图从根本上理解这些指法模式如何在生长过程中出现。利用看似不同的研究领域之间的微妙联系，PI还将研究在高磁场中电子液滴边缘出现的新物质状态。PI还将寻求一种方法来观察不寻常的新物质状态，拓扑状态，在称为量子点的特殊原子微小结构中。这些国家的理论预测存在于电子液体局限于两个维度和高磁场; PI？的提议提供了一个新的竞技场，在其中研究这些可能的新的物质状态，这可能使我们能够利用量子力学状态来执行计算。量子计算被认为为计算机性能的巨大改进提供了机会，至少在一些重要问题上，密码学就是一个例子。PI将通过培训和指导研究生和本科研究生，并为本科生的研究经验做出新的贡献，整合教育和研究。拟议研究的结果将增进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.