CAREER: Geometry and Interference in Strongly Correlated Systems

该奖项由材料研究部的材料理论项目和数学科学部的拓扑项目共同资助，隶属于美国国家科学基金会数学科学优先领域。该职业奖支持理论研究和教育，涉及应用拓扑方法和新的理论方法来解决凝聚态理论中的突出问题。该研究将为进一步研究强相关和无序系统的物理学奠定基础。研究重点包括：(1)在现有实验中计算超导线相滑移的概率，(2)在存在奇异构型的情况下开发有效的低能量描述，(3)将拓扑分析和瞬子演算应用于库仑阻塞问题，(4)开发一种流体动力学方法作为非线性玻色子化来计算相关函数的渐近行为。相对于凝聚态理论中广泛使用的对称性分析，拓扑分析尚未被凝聚态学界广泛接受。这种类型的分析有可能解决或帮助解决许多尚未解决的问题。所提出的研究将促进拓扑方法在凝聚态理论中的应用，并将加强与量子场论和数学物理的联系。所考虑的一些问题深深植根于现代数学。在凝聚态中进行拓扑研究对凝聚态理论和数学物理都是有益的。教育部分包括将与研究相结合的几项活动：(i)关于凝聚态物理拓扑方法的回顾和讲座将填补研究生（更普遍地说，凝聚态物理）教育的重要空白。这些方法在现代物理学中起着越来越重要的作用。创建一门关于使用拓扑方法的新课程，并在互联网上提供辅助材料，将使物理界和学生的范围更广。（ii）将开设一门新的量子磁学研究生课程，以加强研究生教育，并汇集来自不同研究小组的学生。（iii）举办学生座谈会，并制定吸引访客逗留较长时间的策略，以加强对不同研究小组的研究生和本科生的教育，并促进他们在职业生涯的早期阶段积极参与研究。（iv）将为每门必修研究生课程设置“最小问题集”，以建立物理系的共同核心。它将在网上提供，并可能为系外的学生和教师提供资源。该奖项由材料研究部的材料理论项目和数学科学部的拓扑项目共同资助，隶属于美国国家科学基金会的数学科学优先领域。该职业奖支持理论研究和教育，涉及应用基于几何和拓扑的先进数学方法来解决凝聚态理论中的突出问题。PI将把这些方法与其他先进的理论技术结合起来，并将重点放在从这个角度来看似乎已经成熟的一系列问题上。对这些问题的研究将为解决由强电子-电子相互作用和无序产生的电子态的性质这一众所周知的困难和重要问题奠定基础。相对于凝聚态理论中广泛使用的对称性分析，拓扑分析尚未被凝聚态学界广泛接受。这种类型的分析有可能解决或帮助解决许多尚未解决的问题。本研究将促进拓扑方法在凝聚态理论中的应用，并将加强与量子场论和数学物理的联系。所考虑的一些问题深深植根于现代数学。在凝聚态中进行拓扑研究对凝聚态理论和数学物理都是有益的。教育部分包括将与研究相结合的几项活动：(i)关于凝聚态物理拓扑方法的回顾和讲座将填补研究生（更普遍地说，凝聚态物理）教育的重要空白。这些方法在现代物理学中起着越来越重要的作用。创建一门关于使用拓扑方法的新课程，并在互联网上提供辅助材料，将使物理界和学生的范围更广。（ii）将开设一门新的量子磁学研究生课程，以加强研究生教育，并汇集来自不同研究小组的学生。（iii）举办学生座谈会，并制定吸引访客逗留较长时间的策略，以加强对不同研究小组的研究生和本科生的教育，并促进他们在职业生涯的早期阶段积极参与研究。（iv）将为每门必修研究生课程设置“最小问题集”，以建立物理系的共同核心。它将在网上提供，并可能为系外的学生和教师提供资源

This CAREER award is co-funded by the Materials Theory program of the Division of Materials Research and the Topology program of the Division of Mathematical Sciences under the umbrella of the NSF-wide Mathematical Sciences Priority Area. This CAREER award supports theoretical research and education involving the application of topological methods and new theoretical methods to outstanding problems in condensed matter theory. The research will establish a basis for further studies of the physics of strongly correlated and disordered systems. Research thrusts include: (i) calculating the probability of phase slips in superconducting wires with applications to existing experiments, (ii) developing an effective low energy description in the presence of singular configurations, (iii) applying topological analysis and instanton calculus to the Coulomb blockade problem, (iv) developing a hydrodynamic approach as a non-linear bosonization to calculate the asymptotic behavior of correlation functions. In contrast to the symmetry analysis broadly used in condensed matter theory, topological analysis has not yet been widely accepted by the condensed matter community. This type of analysis has the potential to solve or to help solve many open problems. The proposed research will promote the use of topological methods in condensed matter theory and will strengthen connections with quantum field theory and mathematical physics. Some of the problems considered are deeply rooted in modern mathematics. Pursuing topological studies in condensed matter should turn out mutually beneficial for both condensed matter theory and mathematical physics.The educational component involves several activities that will be integrated with the research: (i) A review and lectures on topological methods in condensed matter physics will fill an important gap in graduate student (and more generally, condensed matter physics) education. These methods play an increasingly important role in modern physics. Creation of a new course on the use of topological methods with supporting materials on the Internet will reach a broader segment of the physics community and students. (ii) A new graduate course on Quantum magnetism will be developed to enhance graduate education and to bring together students from different research groups. (iii) Student symposia will be organized which, together with a strategy for attracting visitors to stay for longer times, will enhance education of graduate and undergraduate students from different research groups and will facilitate their active participation in research at earlier stages of their careers. (iv) A 'minimal set' of problems will be created for every mandatory graduate course to establish a common core for the physics department. It will be available online and may provide a resource for students and teaching faculty outside of the department.%%%This CAREER award is co-funded by the Materials Theory program of the Division of Materials Research and the Topology program of the Division of Mathematical Sciences under the umbrella of the NSF-wide Mathematical Sciences Priority Area. This CAREER award supports theoretical research and education involving the application of advanced mathematical methods based on geometry and topology to outstanding problems in condensed matter theory. The PI will combine these methods with other advanced theoretical techniques and focus on a set of problems that seem ripe for advance from this viewpoint. Work on these problems will lay a foundation to attack the notoriously difficult and important problem of the nature of electronic states that arise from strong electron-electron interactions and disorder.In contrast to the symmetry analysis broadly used in condensed matter theory, topological analysis has not yet been widely accepted by the condensed matter community. This type of analysis has the potential to solve or to help solve many open problems. The proposed research will promote the use of topological methods in condensed matter theory and will strengthen the connections with quantum field theory and mathematical physics. Some of the problems considered are deeply rooted in modern mathematics. Pursuing topological studies in condensed matter should turn out mutually beneficial for both condensed matter theory and mathematical physics.The educational component involves several activities that will be integrated with the research: (i) A review and lectures on topological methods in condensed matter physics will fill an important gap in graduate student (and more generally, condensed matter physics) education. These methods play an increasingly important role in modern physics. Creation of a new course on the use of topological methods with supporting materials on the Internet will reach a broader segment of the physics community and students. (ii) A new graduate course on Quantum magnetism will be developed to enhance graduate education and to bring together students from different research groups. (iii) Student symposia will be organized which, together with a strategy for attracting visitors to stay for longer times, will enhance education of graduate and undergraduate students from different research groups and will facilitate their active participation in research at earlier stages of their careers. (iv) A 'minimal set' of problems will be created for every mandatory graduate course to establish a common core for the physics department. It will be available online and may provide a resource for students and teaching faculty outside of the department.***