Dissipative transport in transition regions between quantum Hall plateaus

The paper deals with the quantum Hall effect for noninteracting electrons and in particular with dissipative transport in the transition regions between the Hall plateaus. First, the quasiclassical approach (the guiding-center approximation) is extended to include transport in the transition regions. Next, it is shown that the results remain valid when the highly restrictive quasiclassical condition is relaxed: the magnetic length need not be short compared with the characteristic variation length of the potential. The main results for the Ohmic conductivity ${\sigma }_{\mathrm{xx}}$ in the low-temperature limit are the following: (i) For a finite electric field the transition regions, i.e., the extended-state intervals, have a finite width and ${\sigma }_{\mathrm{xx}}$ is of order ($e^2$/h)f(1-f), where f is the occupied fraction of extended states. (ii) For an infinitesimal electric field the transition regions shrink to discrete points at which ${\sigma }_{\mathrm{xx}}$ is of order $e^2$/h, consistent with the scaling theory of the quantum Hall effect. The paper also discusses some general properties of the electron spectrum, as well as the notion of a Landau band, its occupation, and stability.