Fundamental relation between longitudinal and transverse conductivities in the quantum Hall system

We investigate the relation between the diagonal (${\sigma }_{xx}$) and off-diagonal (${\sigma }_{xy}$) components of the conductivity tensor in the quantum Hall system. We calculate the conductivity components for a short-range impurity potential using the linear response theory, employing an approximation that simply replaces the self-energy by a constant value $-i\mathrm{\hslash }\mathrm{/}(2\tau )$ with $\tau$ the scattering time. The approximation is equivalent to assuming that the broadening of a Landau level due to disorder is represented by a Lorentzian with the width $\mathrm{\Gamma }=\mathrm{\hslash }\mathrm{/}(2\tau )$. Analytic formulas are obtained for both ${\sigma }_{xx}$ and ${\sigma }_{xy}$ within the framework of this simple approximation at low temperatures. By examining the leading terms in ${\sigma }_{xx}$ and ${\sigma }_{xy}$, we find a proportional relation between $\mathrm{d}{\sigma }_{xy}\mathrm{/}\mathrm{d}B$ and $B{\sigma }_{xx}^2$. The relation, after slight modification to account for the long-range nature of the impurity potential, is shown to be in quantitative agreement with experimental results obtained in the GaAs/AlGaAs two-dimensional electron system at the low magnetic-field regime where spin splitting is negligibly small.