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 (h) over bar (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 Gamma = (h) over bar/(2 tau). Analytic formulae 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 d sigma(xy)/dB and B sigma(2)(xx). 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.