The low-temperature phases of matter and transitions between these phases in disordered systems in general, and in particular the quantum Hall regime in a two-dimensional electron gas with a high perpendicular magnetic field, will be studied using theoretical methods. Specific phases of interest include the compressible and incompressible liquids in the quantum Hall effect. This work will treat their elementary excitations, both in the bulk and at the edge, disorder effects, and universal critical properties of the transitions between distinct phases. Other phases to be considered include quantum spin systems with quenched disorder. The theoretical methods used will be mainly analytical, frequently based on many-body theory and both perturbative and non-perturbative quantum field theoretic techniques. While the emphasis is on fundamental properties, applications to experiment will be made when possible. Characterizing the phases and transitions theoretically will help determine which ones occur in experimental systems. This is a renewal of a prior Presidential Young Investigator award. %%% This grant renewal of a prior Presidential Young Investigator award continues the support of a forefront theorist in many-body theoretical physics. This project will focus on studying low-temperature behavior of electron states in a two-dimensional electron gas in a high perpendicular magnetic field-the quantum Hall regime. Additional work will be carried out on understanding the behavior of quantum spin systems. This work will enhance our understanding of the fractional quantum Hall effect, the central topic of the 1998 Nobel Prize in physics, and in general improve our theoretical understanding of many-body effects that are critical in determining the properties of a range of electronic and magnetic materials of technological importance. ***
