Dynamical critical behavior in the integer quantum Hall effect

Abstract

We investigate dynamical scaling properties in the integer quantum Hall effect for non-interacting electrons at zero temperature, by means of the frequency-induced peak broadening of the dissipative longitudinal conductivity $σ_{xx}(ω)$. This quantity is calculated numerically in the lowest Landau level for various values of the Fermi energy $E$, of the frequency $ω$, and of the system size $L$. Data for the width $W(ω,L)$ of the peak are analyzed by means of the dynamical finite-size scaling law $W(ω,L)\approx L^{-1/ν}f\bigl(ωL^z\bigr)$, where $ν$ is the static critical exponent of the localization length, and $z$ is the dynamical exponent. A fit of the data, assuming $ν=2.33$ is known, yields $z=1.19\pm 0.13$. This result indicates that the dynamical exponent in the integer quantum Hall effect may be different from the pertinent space dimension ($d=2$), even in the absence of interactions between electrons.