Physical Principles Underlying the Fractional Quantum Hall Effect

Abstract:  I review an approach to the theory of the Quantum Hall Effect (QHE) somewhat analogous to Landau's theory of phase transitions. After a short recapitulation of some of the discoveries leading to the QHE, I describe some of the main experimental facts and data and speculate about possible applications.  I then recall the basic equations of the electrodynamics of incompressible Hall fluids (IHF). Among other things, these equations imply the existence of chiral edge currents and elucidate their relation to the 2D chiral anomaly, and they enable us to determine the effective action of an IHF.  The effective action constrains the large-scale quantum theory of an IHF, which I then study in terms of conserved current densities.  This leads to a transparent understanding of the "quantization" of the Hall conductivity and enables one to give a list of possible values of the Hall conductivity that compares well with experimental data.  If time permits I also describe the quantum theory of the edge degrees of freedom of an IHF and then sketch some general ideas underlying my approach that have found interesting applications to, e.g., the theory of topological insulators.