Quantum Hall Effect of Massless Dirac Fermions and Free Fermions in Hofstadter's Butterfly

We propose a new physical interpretation of the Diophantine equation of $\sigma_{xy}$ for the Hofstadter problem. First, we divide the energy spectrum, or Hofstadter's butterfly, into smaller self-similar areas called "subcells", which were first introduced by Hofstadter to describe the recursive structure. We find that in the energy gaps between subcells, there are two ways to account for the quantization rule of $\sigma_{xy}$, that are consistent with the Diophantine equation: Landau quantization of (i) massless Dirac fermions or (ii) free fermions in Hofstadter's butterfly.