Fundamental relation between longitudinal and transverse conductivities in the quantum Hall system (original) (raw)

We investigate the relation between the diagonal (σxx) and offdiagonal (σxy) 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 /(2τ ) 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 Γ = /(2τ ). Analytic formulas are obtained for both σxx and σxy within the framework of this simple approximation at low temperatures. By examining the leading terms in σxx and σxy, we find a proportional relation between dσxy/dB and Bσ 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.