Theoretical Studies of Quantum Systems with Strong Interactions

该奖项由材料研究部，数学科学部和物理部资助。该项目属于NSF范围内的数学科学优先领域的保护伞下福尔斯。该奖项支持专注于非平衡过程中产生的奇点的理论研究。在PI之前的奖项下取得的进展将用于将该领域扩展到简并量子系统中的共形动力学和非线性效应领域。PI旨在定义和发展共形不变动力学过程的理论。许多远离平衡态的重要系统表现出与临界现象的共形不变性相似的共形不变性。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，奇异性又产生了在大尺度上可见的分形非平衡模式。拉普拉斯增长中的指状不稳定性，临界系统中的分形团簇，简并和相干量子系统中的流体动力学不稳定性是研究的主题。该研究的主题包括起源，统计和奇异性的正则化，以及动力学过程随机模式的分形几何。PI将解决在既定领域和新兴趋势的长期存在的问题，这些问题包括：-非平衡经典和量子过程中的奇点统计;-临界系统的随机几何;-随机增长和聚集;和-相关量子系统中的非线性输运和奇异性。特别强调随机Loewner演化，一个新兴的领域，提供了新的工具，并提出了新的问题，在二维临界和分形结构出现的结果，随机增长现象。拟议的研究结果将提高知识和理解复杂的非技术摘要：该奖项由材料研究部、数学科学部和物理部资助。该项目属于NSF范围内的数学科学优先领域的保护伞下福尔斯。该奖项支持与数学接口的理论凝聚态物理研究，致力于推进我们对复杂凝聚态系统的理解。研究重点是非平衡过程。PI工作的一个重要方面涉及显示雪花状指状物的生长过程，这些指状物从一个相渗透到另一个相，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是如何在生长过程中出现的。PI还计划利用数学和理论物理学的最新进展来研究随机几何结构发挥重要作用的其他过程，并研究限制在二维空间的物质如何通过相变进行重组。PI将通过培训和指导研究生和本科研究生来整合教育和研究，并为本科生和数学教育者计划的研究经验做出新的贡献。拟议研究的结果将提高对复杂凝聚态系统的认识和理解。

TECHNICAL SUMMARY:This award is funded by the Division of Materials Research, the Division of Mathematical Sciences and the Physics Division. This project falls under the umbrella of the NSF-wide Mathematical Sciences Priority Area. This award supports theoretical research focused on singularities arising in non-equilibrium processes. Advances made under the PIs previous award will be used to extend the field into the domain of conformal kinetic and non-linear effects in degenerate quantum systems.The PI aims to define and develop a theory of conformally invariant kinetic processes. Many important systems far from equilibrium show conformal invariance 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 turn singularities give rise to fractal non-equilibrium patterns visible at a large scale. Fingering instability in Laplacian Growth, fractal clusters in critical systems, hydrodynamic instability in degenerate and coherent quantum systems are subjects of the study. Themes of the study include the origin, statistics, and regularization of singularities, and fractal geometry of stochastic patterns of kinetic processes. The PI will address long-standing problems in established fields and emerging trends, these include:- Statistics of singularities in non-equilibrium classical and quantum processes;- Stochastic geometry of critical systems;- Stochastic growth and aggregation; and- Nonlinear-transport and singularities in correlated quantum systems.Special emphasis will be given to Stochastic Loewner Evolution, an emerging field that provides new tools and poses new questions for criticality in two dimensions and to fractal structures emerging as results of stochastic growth phenomena.The results of the proposed research will enhance knowledge and understanding of complex condensed matter systems.NON-TECHNICAL SUMMARY:This award is funded by the Division of Materials Research, the Division of Mathematical Sciences and the Physics Division. This project falls under the umbrella of the NSF-wide Mathematical Sciences Priority Area. This award supports theoretical condensed matter physics research at an interface with mathematics that is focused on advancing our understanding of complex condensed matter systems. The research focuses on non-equilibrium processes. An important aspect of the PIs 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. The PI also plans to capitalize on recent advances in mathematics and theoretical physics to study other processes in which random geometric structures play an important role and to study how matter restricted to two-dimensions reorganizes itself through a phase transition. 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, and Mathematics Educators programs. The results of the proposed research will enhance knowledge and understanding of complex condensed matter systems.