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

We derive, from first principles, the complete Luttinger liquid theory of abelian 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 dual symmetry and obtain a complete classification of quasiparticle operators and tunneling exponents. The chiral anomaly on the edge is used to obtain unambiguously the Hall conductance. In resolving the problem of counter-flowing edge modes, we find that the long range Coulomb interactions play a fundamental role. In order to set up a theory for arbitrary ν we use the idea of a two-dimensional network of percolating edge modes. We derive an effective, single mode Luttinger liquid theory for tunneling processes into the edge which yields a continuous tunneling exponent 1/ν. The network approach is also used to re-derive the instanton vacuum theory for plateau transitions.