Hofstadter butterflies of bilayer graphene

TL;DR

This paper computes the electronic spectrum of bilayer graphene in magnetic fields, revealing how layer displacement affects Hofstadter butterfly patterns and Landau level splitting, with implications for understanding stacking-dependent electronic properties.

Contribution

It introduces a nonperturbative calculation method for bilayer graphene's spectrum considering arbitrary layer displacements, using a periodic gauge with singular flux vortices.

Findings

Hofstadter butterfly plots exhibit reduced symmetry based on layer displacement.

Zero-energy Landau level splitting varies significantly between Bernal and non-Bernal stacking.

The method accommodates arbitrary layer shifts in magnetic field calculations.

Abstract

We calculate the electronic spectrum of bilayer graphene in perpendicular magnetic fields nonperturbatively. To accommodate arbitrary displacements between the two layers, we apply a periodic gauge based on singular flux vortices of phase $2\pi$. The resulting Hofstadter-like butterfly plots show a reduced symmetry, depending on the relative position of the two layers against each other. The split of the zero-energy relativistic Landau level differs by one order of magnitude between Bernal and non-Bernal stacking.