Dynamical critical behavior in the integer quantum Hall effect

TL;DR

This study numerically analyzes the dynamical scaling of the integer quantum Hall effect, revealing a dynamical exponent different from the spatial dimension, which suggests new insights into quantum critical behavior without electron interactions.

Contribution

It provides a numerical determination of the dynamical exponent in the integer quantum Hall effect, showing it may differ from the spatial dimension in a non-interacting system.

Findings

Dynamical exponent z ≈ 1.19 was estimated.

The dynamical exponent may differ from the spatial dimension d=2.

Results suggest non-interacting electrons exhibit unique critical dynamics.

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 $\sigma_{xx}(\omega)$. This quantity is calculated numerically in the lowest Landau level for various values of the Fermi energy $E$, of the frequency $\omega$, and of the system size $L$. Data for the width $W(\omega,L)$ of the peak are analyzed by means of the dynamical finite-size scaling law $W(\omega,L)\approx L^{-1/\nu}f\bigl(\omega L^z\bigr)$, where $\nu$ is the static critical exponent of the localization length, and $z$ is the dynamical exponent. A fit of the data, assuming $\nu=2.33$ 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.