CAREER: Geometry and Interference in Strongly Correlated Systems

这个职业奖是由材料研究部的材料理论项目和数学科学部的拓扑学项目共同资助的，隶属于NSF全数学科学优先领域。该职业奖支持将拓扑学方法和新的理论方法应用于凝聚态理论中的突出问题的理论研究和教育。这项研究将为进一步研究强关联和无序系统的物理学奠定基础。研究的主要内容包括：(I)计算超导导线中相滑移的概率及其在现有实验中的应用；(Ii)在存在奇异组态的情况下发展有效的低能描述；(Iii)将拓扑分析和瞬子微积分应用于库仑阻塞问题；(Iv)发展流体动力学方法作为非线性玻色化来计算关联函数的渐近行为。与凝聚态理论中广泛使用的对称性分析相比，拓扑分析尚未被凝聚态广泛接受。这种分析有可能解决或帮助解决许多悬而未决的问题。这项拟议的研究将促进拓扑方法在凝聚态理论中的使用，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。从事凝聚态的拓扑学研究对凝聚态理论和数学物理都是有益的。教育部分包括几个将与研究相结合的活动：(I)凝聚态物理拓扑学方法的复习和讲座将填补研究生(更广泛地说，凝聚态物理)教育的一个重要空白。这些方法在现代物理学中扮演着越来越重要的角色。在互联网上开设一门关于使用拓扑学方法和辅助材料的新课程，将惠及更广泛的物理界和学生群体。(Ii)发展一项新的量子磁学研究生课程，以加强研究生教育，并把不同研究小组的学生汇聚在一起。(Iii)举办学生座谈会，配合吸引游客逗留更长时间的策略，加强对来自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)为每一门必修的研究生课程设立一套“最低限度的问题”，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。%这个职业奖是由材料研究部的材料理论项目和数学科学部的拓扑学项目共同资助的，隶属于NSF-Wide数学科学优先领域。该职业奖支持理论研究和教育，涉及将基于几何和拓扑学的先进数学方法应用于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，从这个角度着眼于一系列似乎成熟的问题。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引起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。与凝聚态理论中广泛使用的对称性分析相比，拓扑分析尚未被凝聚态社区广泛接受。这种分析有可能解决或帮助解决许多悬而未决的问题。这项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。从事凝聚态的拓扑学研究对凝聚态理论和数学物理都是有益的。教育部分包括几个将与研究相结合的活动：(I)凝聚态物理拓扑学方法的复习和讲座将填补研究生(更广泛地说，凝聚态物理)教育的一个重要空白。这些方法在现代物理学中扮演着越来越重要的角色。在互联网上开设一门关于使用拓扑学方法和辅助材料的新课程，将惠及更广泛的物理界和学生群体。(Ii)发展一项新的量子磁学研究生课程，以加强研究生教育，并把不同研究小组的学生汇聚在一起。(Iii)举办学生座谈会，配合吸引游客逗留更长时间的策略，加强对来自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)为每一门必修的研究生课程设立一套“最低限度的问题”，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.***