A magnetic Hofstadter butterfly and its topologically quantized Hall conductance

TL;DR

This paper explores the energy spectrum of Dirac fermions in graphene with periodic magnetic modulation, revealing a magnetic Hofstadter butterfly pattern and topologically quantized Hall conductance, extending quantum Hall concepts.

Contribution

It demonstrates the existence of a magnetic Hofstadter butterfly and topologically quantized Hall conductance in magnetically modulated graphene, extending quantum Hall theory to this system.

Findings

Energy spectrum exhibits Hofstadter butterfly pattern.

Hall conductance is topologically quantized.

Extension of quantum Hall effect to magnetic superlattices.

Abstract

The energy spectrum of massless Dirac fermions in graphene under two dimensional periodic magnetic modulation having square lattice symmetry is calculated. We show that the translation symmetry of the problem is similar to that of the Hofstadter or TKNN problem and in the weak field limit the tight binding energy eigenvalue equation is indeed given by Harper Hofstadter hamiltonian. We show that due to its magnetic translational symmetry the Hall conductivity can be identified as a topological invariant and hence quantized. We thus extend the idea of Quantum Hall Effect to magnetically modulated two dimensional electron system. Finally we indicate possible experimental systems where this may be verified.