(Mis-)handling gauge invariance in the theory of the quantum Hall effect. III. The instanton vacuum and chiral-edge physics

Abstract

The concepts of an instanton vacuum and F invariance are used to derive a complete effective theory of massless edge excitations in the quantum Hall effect. Our theory includes the effects of disorder and Coulomb interactions, as well as the coupling to electromagnetic fields and statistical gauge fields. The results are obtained by studying the strong-coupling limit of a Finkelstein action, previously introduced for the purpose of unifying both integral and fractional quantum Hall regimes. We establish the fundamental relation between the instanton vacuum approach and the completely equivalent theory of chiral edge bosons. In this paper we limit the analysis to the integral regime. We show that our complete theory of edge dynamics can be used as an important tool to investigate long-standing problems such as long-range, smooth disorder, and Coulomb interaction effects. We introduce a two-dimensional network of chiral-edge states and tunneling centers (saddle points) as a model for smooth disorder. This network is then used to derive a mean-field theory of the conductances, and we work out the characteristic temperature $(T)$ scale at which the transport crosses over from mean-field behavior at high $T$ to the critical behavior plateau transitions at much lower $T.$ The results explain the apparent lack of scaling which is usually seen in the transport data taken from arbitrary samples at finite $T.$ Second, we address the problem of electron tunneling into the quantum Hall edge. We show that the tunneling density of states near the edge is affected by the combined effects of the Coulomb interactions and the smooth disorder in the bulk. We express the problem in terms of an effective Luttinger liquid with conductance parameter $(g)$ equal to the filling fraction (\ensuremath{\nu}) of the Landau band. Hence, even in the integral regime, our results for tunneling are completely non-Fermi-liquid-like, in sharp contrast to the predictions of single-edge theories.