Quantum Hall Effect on the Hofstadter Butterfly

TL;DR

This paper numerically investigates the quantum Hall effect in a disordered, modulated 2D electron system, revealing how anisotropy influences critical energies and Hall conductivity in the Hofstadter butterfly spectrum.

Contribution

It provides new insights into the behavior of critical energies and Hall conductivity trajectories under anisotropic modulation and disorder in the Hofstadter butterfly.

Findings

Identification of critical energies for extended states in butterfly subbands

Trajectories of these energies as disorder varies

Change in Hall conductivity with anisotropic modulation

Abstract

Motivated by recent experimental attempts to detect the Hofstadter butterfly, we numerically calculate the Hall conductivity in a modulated two-dimensional electron system with disorder in the quantum Hall regime. We identify the critical energies where the states are extended for each of butterfly subbands, and obtain the trajectory as a function of the disorder. Remarkably, we find that when the modulation becomes anisotropic, the critical energy branches accompanying a change of the Hall conductivity.