Hall plateau diagram for the Hofstadter butterfly energy spectrum

TL;DR

This paper investigates the quantum Hall effect and localization in the Hofstadter butterfly spectrum, providing a comprehensive Hall plateau diagram and a theory for how disorder influences the structure.

Contribution

It presents a detailed numerical analysis of Hall conductivity and localization, and introduces a theory describing the evolution of plateau structures with increasing disorder.

Findings

Hall plateau diagram covering the entire Hofstadter butterfly

Subbands with Hall conductivity n e^2/h have |n| separated extended level groups

Clusters of subbands with same Hall conductivity share similar localization properties

Abstract

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 $|n|$ separated bunches of extended levels, at least for an integer $n \leq 2$. 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.