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

We investigate the relation between the diagonal (σ(xx)) and off-diagonal (σ(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 [Formula: see text] with τ 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 [Formula: see text]. Analytic formulae are obtained for both σ(xx) and σ(xy) within the framework of this simple approximation at low temperatures. By examining the leading terms in σ(xx) and σ(xy), we find a proportional relation between dσ(xy)/dB and Bσ(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.