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

TL;DR

This paper offers a new interpretation of the Diophantine equation for the quantum Hall effect in Hofstadter's butterfly, linking it to Landau quantization of massless Dirac and free fermions within the recursive energy spectrum.

Contribution

It introduces a novel physical interpretation of the Diophantine equation by connecting it to Landau quantization of different fermion types in the Hofstadter spectrum.

Findings

Identification of two quantization mechanisms in energy gaps

Connection between Diophantine equation and fermion types

Insight into the recursive structure of 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.