The localization and the quantum Hall effect on the Hofstadter butterfly

Motivated by the recent experimental attempts to detect the Hofstadter butterfly, we numerically calculate the Hall conductivity in a modulated two-dimensional (213) electron system with disorder to study the localization. Remarkably we find that one subband can contain more than two critical energies where the states are extended, which suggests the existence of the Hall plateau in the middle of the band. We also show that we can change the number of the critical energies introducing the anisotropy of the periodic potential, which is expected to be feasible in a real experiment.