The Hall conductivity $\sigma_{ \rm xy}$ of a two-dimensional electron system is quantized in units of $e^2/h$ when the Fermi level is located in the mobility gap between two Landau levels. We consider the deviation of $\sigma_{ \rm xy}$ from a quantized value caused by the thermal activation of electrons to the extended states for the case of a long range random potential. This deviation is of the form $\sigma_{ \rm xy}^*\exp(-\Delta/T)$. The prefactor $\sigma_{ \rm xy}^*$ is equal to $e^2/h$ at temperatures above a characteristic temperature $T_2$. With the temperature decreasing below $T_2$, $\sigma_{ \rm xy}^*$ decays according to a power law: $\sigma_{ \rm xy}^* = \frac{ e^2}{ h}(T/T_2)^\gamma$. Similar results are valid for a fractional Hall plateau near filling factor $p/q$ if $e$ is replaced by the fractional charge $e/q$.
Categories: cond-mat