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. ***
