Geometry, topology, and dynamics in quantum Hall effects and related phenomena

Ohio State University, The, Columbus OH

Investigators

Abstract

NONTECHNICAL SUMMARY This award supports theoretical research and education on the role of disorder in the properties of materials. Real materials that are used in all sorts of applications inevitably have some imperfections, impurities, and other kinds of disorder. It is important to understand how properties of materials are affected by the disorder. While disorder can have undesirable effects, it can also lead to qualitatively new behaviors. The PI's research interests lie in this second class of phenomena. For example disorder can scatter electrons in materials. Sometimes multiple scattering of electrons by disorder leads to them being trapped, or localized, in certain places in a sample. Localized electrons cannot move to conduct electricity and heat. This trapping phenomenon is called Anderson localization, after the Nobel Prize winner P. W. Anderson. A solid with localized electrons is an insulator, but if one changes parameters of the system, electrons can become delocalized and conduct electricity like a metal. This award supports research on transitions between metals and insulators driven by disorder. One such transition is the so-called plateau transition in quantum Hall effects. Quantum Hall effects are spectacular manifestations of Anderson localization of electrons due to disorder in the presence of a very strong magnetic field. The effects lead to extremely precise quantization of Hall conductivity of electrons in semiconductors and graphene. The Hall conductivity measures the how well the system conducts electricity in the direction perpendicular to that determined by the applied voltage. This quantization is the basis for the modern standard of resistance. An important feature of real experimental samples used in quantum Hall measurements is their finite size and key role played by their boundaries. A significant part of the PI's research is aimed at a detailed understanding of the fine structure of quantum Hall systems near their boundaries. TECHNICAL SUMMARY This award supports theoretical research and education on disordered materials. Identification of an analytically tractable theory describing critical properties at Anderson transitions remains elusive; it is an outstanding problem in the area of disordered electronic systems. The discovery of quantum Hall effects has opened a new research discipline, experimental and theoretical, continues to stimulate new ideas and developments. Quantum Hall effects are the first examples of topological phases of matter, whose study is an enormously active current research area. The PI's research projects will focus on geometric and topological aspects of integer and fractional quantum Hall effects, and their relation to the dynamics of edge states at the boundaries of finite samples relevant to experiments. Particular projects include the investigation of: 1) the theory of the integer quantum Hall transition and other disordered critical points in two dimensions based on mappings to classical statistical mechanics of geometric objects, and conformal restriction; 2) localization and Anderson transitions in class D, superconductors with broken time reversal and spin rotation symmetries, and related random bond Ising models; 3) network models with structural disorder, and other disordered systems on random surfaces; 4) structure and dynamics near boundaries of integer and fractional quantum Hall states; fractional quantum Hall states and Hall viscosity on Riemann surfaces; 5) fine structure and emergent conformal symmetry of fractional quantum Hall states and non-linear edge dynamics. These projects will bring together ideas from various fields of physics and mathematics including localization, statistical mechanics of random systems, critical phenomena, conformal field theory, string theory, differential and complex geometry, random matrix models, complex analysis, probability theory, fractals, and integrable systems. The PI's research contributes to bringing these fields closer by communicating the results to various research communities and promoting collaborations between practitioners in diverse areas. The research involves international collaborations. The projects will provide research and training opportunities for graduate students and postdocs.