Hall plateau diagram for the Hofstadter butterfly energy spectrum

We extensively study the localization and the quantum Hall effect in the Hofstadter butterfly, which emerges in a two-dimensional electron system with a weak two-dimensional periodic potential. We numerically calculate the Hall conductivity and the localization length for finite systems with the disorder in general magnetic fields, and estimate the energies of the extended levels in an infinite system. We obtain the Hall plateau diagram on the whole region of the Hofstadter butterfly, and propose a theory for the evolution of the plateau structure with increasing disorder. There we show that a subband with the Hall conductivity $n{e}^{2}∕h$ has $\ensuremath{\mid}n\ensuremath{\mid}$ separated bunches of extended levels, at least for an integer $n\ensuremath{\leqslant}2$. We also find that the clusters of the subbands with identical Hall conductivity, which repeatedly appear in the Hofstadter butterfly, have a similar localization property.