Insensitivity of Quantized Hall Conductance to Disorder and Interactions

Abstract

A two-dimensional quantum Hall system is studied for a wide class of potentials including single-body random potentials and repulsive electron-electron interactions. We assume that there exists a non-zero excitation gap above the ground state(s), and then the conductance is derived from the linear perturbation theory with a sufficiently weak electric field. Under these two assumptions, we proved that the Hall conductance $σ_{xy}$ and the diagonal conductance $σ_{yy}$ satisfy $|σ_{xy}+e^2ν/h|\le{\rm const.}L^{-1/12}$ and $|σ_{yy}|\le{\rm const.}L^{-1/12}$. Here $e^2/h$ is the universal conductance with the charge $-e$ of electron and the Planck constant $h$; $ν$ is the filling factor of the Landau level, and $L$ is the linear dimension of the system. In the thermodymanic limit, our results show $σ_{xy}=-e^2ν/h$ and $σ_{yy}=0$. The former implies that integral and fractional filling factors $ν$ with a gap lead to, respectively, integral and fractional quantizations of the Hall conductance.