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

Abstract

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 $-i \hbar /(2 τ)$ 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 $Γ= \hbar /(2 τ)$. Analytic formulas 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 $\mathrm{d}σ_{xy}/\mathrm{d}B$ and $B σ_{xx}^2$. 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.