Quantum Phases in Condensed Matter

这是一笔赠款，用于资助理论凝聚态物理的研究，以及教育活动。这项研究将集中在当前非常感兴趣的两个领域：分数量子霍尔效应的场论和相变的重整化群。一本关于场论在凝聚态物质中的应用的教科书也正在编写中。自从二维电子气的霍尔电导被量子化为分数这一显著发现以来，一直有源源不断的理论工作来解释这一现象的许多方面。尽管有令人印象深刻的知识体系，但仍然缺乏一个微观理论来描述所有这些分数，从磁场中的电子开始，受制于库仑相互作用和随机基板势。Chern-Simons场论是最有希望的选择之一，在场论中，人们将电子映射为玻色子或费米子，平均而言，要么看不到场，要么看不到填充整数个朗道能级的较弱的场。一个基于这些理论的框架已经被开发出来。它从上述起点以连贯的方式再现了一些特殊分数(其中净场为零)的许多已知结果，并有望为其他分数产生更多的结果。这种分析在许多方面与使用的其他技术非常不同，它要么与现有的一些结果不一致，要么提供一个完全独立的证实，特别是对于半满的情况。前段时间，一个费米子的重整化群方案被设计出来，在这个方案中，人们系统地消除费米面附近的模，以提取低能物理。在这个方案中，朗道的费米液体看起来是一个固定点，而不稳定性，也就是说，超导电性出现了相关的流动。遗憾的是，该方法仅限于零耦合时发生的跃迁。有限耦合的扩展通常涉及开始时的费米子以及由费米子形成的玻色场。将研究一种方案，在该方案中，两个高能模式被并排消除，而到目前为止，该方案被证明是难以捉摸的。这笔助学金将用于支持凝聚态理论的研究以及研究生教科书的编写。这项研究集中在场论技术在当前凝聚态物理中感兴趣的问题上的应用。这些问题包括分数量子霍尔效应的研究--一种新的物质状态。拟议工作的成功完成应该会对凝聚态和材料物理学产生广泛的影响。***

Dmr-9800626 Shankar This is a grant to fund research in theoretical condensed matter physics, as well as educational activities. The research will focus on two areas of great current interest: field theory of the fractional quantum Hall effect and renormalization group for phase transitions. A textbook is also being written on the application of field theory to condensed matter. Since the remarkable discovery that the Hall conductance of a two-dimensional electron gas is quantized to some fractional values, there has been a steady stream of theoretical work aimed at explaining many aspects of this phenomenon. While there is an impressive body of knowledge, one still lacks a single microscopic theory that will describe all these fractions, starting from just electrons in a magnetic field, subject to Coulomb interactions and the random substrate potential. Chern-Simons field theories, in which one maps electrons into either bosons or fermions which, on the average, see either no field or a weaker field in which they fill an integer number of Landau levels, are one of the most promising options. A framework based on these theories has been developed. It reproduces in a coherent way many known results for some special fractions (in which the net field vanishes) from the above mentioned starting point and promises to yield more results for the other fractions. Being very different in many ways from other techniques used, this analysis will either disagree with some existing results or provide a totally independent confirmation, especially for the half-filled case. Some time ago a renormalization group scheme for fermions was worked out in which one systematically eliminates modes near the Fermi surface to extract the low energy physics. In this scheme Landau's Fermi liquid appears as a fixed point, while instabilities, to say, superconductivity emerge as relevant flows. Unfortunately, the method has been limited to transitions that occur at zero couplin g. Extension to finite coupling typically involves the fermions one begins with as well as bosonic fields formed out of them. A scheme in which high energy modes of both are eliminated side by side, and which has so far proven elusive, will be investigated. %%% This grant will support research on condensed matter theory as well as the writing of a graduate textbook. The research focuses on the application of field theoretical techniques to problems of current interest in condensed matter physics. These problems include the study of the fractional quantum Hall effect - a new state of matter. Successful completion of the proposed work should have wide ranging impact on condensed matter and materials physics. ***