CAREER: Disordered Systems and Stochastic Growth Phenomena

这个职业奖福尔斯属于NSF范围内的数学科学优先领域的保护伞，并支持理论凝聚态物理学的综合研究和教育。PI计划解决超导导线中的局部化与破坏自旋旋转对称性，量子霍尔效应的网络模型，无序超导体中的罕见波动，以及随机Loewner演化及其推广。自旋-旋转对称性破缺的超导线材呈现出非定域准粒子态，导致不寻常的热输运性质。这发生在一个临界点。PI计划使用散射理论和超对称方法来研究这个临界点的性质。量子霍尔跃迁是临界点的具有挑战性的例子。量子霍尔跃迁的网络模型及其推广将被研究，目的是理解这些跃迁的本质。适用于网络模型的超对称方法为研究提供了形式化的方法。无序超导体中准粒子态的密度和空间结构是我们理解这些迷人系统的重要组成部分。将结合最佳涨落方法和杂质对超导性抑制的自洽处理来研究无序结构中的罕见涨落和无序超导体中随之产生的低能态。PI计划研究随机Loewner演化及其推广。随机Loewner演化是研究二维临界性和随机增长现象所导致的分形结构的一个新发展。可能的应用包括长期存在的开放问题，如扩散限制聚集和其他增长现象。教育部分包括开发现代凝聚态物理学的研究生课程，指导研究生和本科生研究生，并为本科生，青年学者，和研讨会的小学专家和数学教育家计划，以及与专业人士在科学和工业博物馆（MSI）的工作。这些活动将与教育专业人员的反馈密切结合。课程将侧重于概念的理解，而不是理论技术，并将获得实验倾向的学生。REU和青年学者计划覆盖了代表性不足的群体，而MSI每年为大约50万在校学生提供服务。%这个职业奖福尔斯属于NSF范围内的数学科学优先领域。它支持理论凝聚态物理学的综合研究和教育。PI将使用先进的理论方法来研究凝聚态物理学核心的两个问题，这两个问题由随机性的共同主题联系在一起。第一种是随机环境中的电子，例如，在原子从规则的周期性晶格中随机移位的材料中。PI专注于有趣的物理案例，当电子的运动进一步限制在平面或导线上时，以及当涉及新的物质状态和它们之间的转换时，如超导或量子霍尔状态。第二个领域涉及材料的生长过程，比如雪花的形成过程。PI通过追求这些过程的数学描述的最新突破来寻求智力进步。该研究汇集了物理学和数学的思想，包括局部化，超导性，随机系统的统计力学，临界现象，复分析，概率论，分形和随机矩阵。该提案的教育和推广部分将把研究纳入从高中到研究生阶段的物理教学，为高中生提供新的课程材料，并帮助一个主要科学博物馆的专业人员保持科学更新。***

This CAREER award falls under the umbrella of the NSF-wide Mathematical Sciences Priority Area and supports integrated research and education in theoretical condensed matter physics. The PI plans to address localization in superconducting wires with broken spin-rotation symmetry, network models for quantum Hall effects, rare fluctuations in disordered superconductors, and stochastic Loewner evolution and its generalizations. Superconducting wires with broken spin-rotation symmetry exhibit delocalized quasiparticle states leading to unusual thermal transport properties. This happens at a critical point. The PI plans to study the nature of this critical point using scattering theory and supersymmetry methods. Quantum Hall transitions are challenging examples of critical points. Network models for quantum Hall transitions and their generalizations will be studied with an aim to understanding the nature of these transitions.The supersymmetry method adapted for network models provides the formal approach for the research. Density and spatial structure of quasiparticle states in disordered superconductors are essential parts of our understanding of these fascinating systems. The optimal fluctuation method and self-consistent treatment of the suppression of superconductivity by impurities will be combined to study rare fluctuations in disorder configurations and ensuing low-energy states in disordered superconductors. The PI plans to study stochastic Loewner evolution and its generalizations. The stochastic Loewner evolution is a recent development in the study of criticality in two dimensions and fractal structures emerging as results of stochastic growth phenomena. Possible applications include such long-standing open problems as diffusion-limited aggregation and other growth phenomena.The education component involves developing graduate level courses in modern condensed matter physics, supervision of graduate and undergraduate research students, and making novel contributions to the Research Experiences for Undergraduates, Young Scholars, and Seminars for Elementary Specialists And Mathematics Educators programs as well as working with professionals at the Museum of Science and Industry (MSI). These activities will be closely coupled with feedback from education professionals. Courses will focus on conceptual understanding rather than theoretical techniques, and will be accessible to experimentally inclined students. The REU and Young Scholars programs reach out to underrepresented groups, while the MSI each year serves about one half million school students. %%%This CAREER award falls under the umbrella of the NSF-wide Mathematical Sciences Priority Area. It supports integrated research and education in theoretical condensed matter physics. The PI will use advanced theoretical methods to study two problems at the heart of condensed matter physics connected by a common theme of randomness. The first involves electrons in a random environment, for example, in materials in which the atoms are randomly displaced from a regular periodic lattice. The PI focuses on interesting physical cases when the motion of the electrons is further restricted to a plane or a wire, and when novel states of matter and transformations among them are involved, like superconducting or quantum Hall states. The second area involves materials growth processes, like those at work in the formation of snowflakes. The PI seeks intellectual advances by pursuing a recent breakthrough in the mathematical description of these processes. The research brings together ideas from physics and mathematics including localization, superconductivity, statistical mechanics of random systems, critical phenomena, complex analysis, probability theory, fractals, and random matrices. The education and outreach component of the proposal will integrate research into teaching of physics from high school to graduate level, provide new course material for high school students, and help keep professionals at a major science museum scientifically up to date. ***