The fractional quantum Hall effect: Chern-Simons mapping, duality, Luttinger liquids and the instanton vacuum

We derive, from first principles, the complete Luttinger liquid theory of a belian quantum Hall edge states. This theory includes disorder and Coulomb interactions as well as the coupling to external electromagnetic fields. We introduce a theory of spatially separated edge modes, find an enlarged dua l symmetry smd obtain a complete classification of quasiparticle operators and tunneling exponents. The chiral anomaly on the edge is used to obtain u nambiguously the Hall conductance. In resolving the problem of counter-flow ing edge modes, we find that the long range Coulomb interactions play a fun damental role. In order to set up a theory for arbitrary nu we use the idea of a two-dimensional network of percolating edge modes. We derive an effec tive, single mode Luttinger liquid theory for tunneling processes into the edge which yields a continuous tunneling exponent 1/nu. The network approac h is also used to re-derive the instanton vacuum theory for plateau transit ions. (C) 1999 Elsevier Science B.V. All rights reserved.