Network Models of the Quantum Hall Effect

Abstract

In the decade since the discovery of the quantum Hall effect (QHE) in two-dimensional electron systems in high magnetic field this phenomenon has received enormous attention both from the experimental and theoretical point of view (Prange and Girvin, 1987). Although many aspects of the effect are now understood based on this work, the actual transport theory of the effect, particularly in the transition regions between quantized plateaus, has remained incomplete. In the past few years, building on earlier work by Halperin (1982), a new approach to this transport theory, based on the scattering properties of states at the fermi surface has been developed by a number of authors (Streda et al., 1987; Jain and Kivelson, 1988a, 1988b; Biittiker, 1988; Baranger and Stone, 1989). In this S-matrix approach, which primarily evolved from applications to mesoscopic conductors, there are N states at the fermi level for every Landau level below e_F due to the presence of edges (which break the Landau level degeneracy), and typically N-1 of these states actually carry current along the edge. The S-matrix approach assumes the validity of the independent electron approximation and provides a physically intuitive picture which can account for the integer quantization of the Hall conductance.