Hofstadter butterflies in magnetically modulated graphene bilayer: an algebraic approach

TL;DR

This paper explores the Hofstadter spectrum in magnetically modulated bilayer graphene, revealing algebraic relations to monolayer graphene and demonstrating topological quantization of Hall conductivity.

Contribution

It introduces an algebraic approach to relate Hofstadter spectra of bilayer and monolayer graphene under magnetic modulation, and derives the Harper-Hofstadter equation for this system.

Findings

Hofstadter spectrum expressed in terms of monolayer graphene spectrum

Topological quantization of Hall conductivity demonstrated

Quantized Hall plateaus are equally spaced across quantum numbers

Abstract

It has been shown that Bernal stacked bilayer graphene (BLG) in a uniform magnetic field demonstrates integer quantum Hall effect with a zero Landau-level anomaly \cite{Geimbilayer}. In this article we consider such system in a two dimensional periodic magnetic modulation with square lattice symmetry. It is shown algebraically that the resulting Hofstadter spectrum can be expressed in terms of the corresponding spectrum of monolayer graphene in a similar magnetic modulation. In the weak-field limit, using the tight-binding model, we also derive the Harper-Hofstadter equation for such BLG system in a periodic magnetic modulation. We further demonstrate the topological quantisation of Hall conductivity in such system and point out that the quantised Hall plateaus are equally spaced for all quantum numbers for the quantised Hall conductivity.