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 ${\sigma }_{xy}$ and the diagonal conductance ${\sigma }_{yy}$ satisfy $|{\sigma }_{xy}+e^2\nu /h|\le const.L^{-1/12}$ and $|{\sigma }_{yy}|\le const.L^{-1/12}$. Here $e^2/h$ is the universal conductance with the charge $-e$ of electron and the Planck constant $h$; $\nu$ is the filling factor of the Landau level, and $L$ is the linear dimension of the system. In the thermodymanic limit, our results show ${\sigma }_{xy}=-e^2\nu /h$ and ${\sigma }_{yy}=0$. The former implies that integral and fractional filling factors $\nu$ with a gap lead to, respectively, integral and fractional quantizations of the Hall conductance.

LaTeX, 62 pages, no figures, typos corrected, some references added, discussion on the standard time-dependent vector potential added, accepted for publication in J. Stat. Phys