Localization and hopping conductivity in the quantum Hall regime

The long-range asymptotic behavior of the two-particle Green function of a two-dimensional electronic system in the presence of a strong magnetic field and a Gaussian white-noise potential is studied. Away from the center of the Landau level we can show that weak disorder leads to an exponential tail of the Green function, i.e., G∼$e^{\mathrm{-}\alpha \Vert \mathrm{r}\mathrm{-}\mathrm{r}\mathcal{’}\Vert }$. The rate of the exponential decay is found when α is large to be α≊(‖lnW‖${/2)}^{1/2}$, where W parametrizes the strength of the disorder. At shorter distances the Gaussian behavior of the unperturbed system predominates, and so there is a crossover between the two. The hopping conductivity in the quantum Hall devices is also discussed, and it is shown that the temperature dependence of the exponent in the conductance is not a simple power law, although it approaches the usual Mott $T^{\mathrm{-}1/3}$ law as T→0.