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

Abstract

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.

6 pages, 4 figures, To be published in J. Phys. Soc. Jpn

References in corpus (1)

• Unconventional Integer Quantum Hall effect in graphene

Cited by in corpus (2)

• Fractal defect states in the Hofstadter butterfly
• Chern insulator with large Chern numbers. Chiral Majorana fermion liquid