Hofstadter butterflies of bilayer graphene

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.

Figure 1

(Color online) Hofstadter butterfly of a bilayer graphene in the Bernal stacking configuration. The band structure at zero magnetic field is rotationally symmetric in good approximation for an area around the $K$ point and shows a split into four massive bands, with the two middle ones touching at $E_{\mathrm{F}}$. The density of states (DOS) of a finite-width ribbon [a pair of (200,0) zigzag ribbons] in the same configuration shows the SUSYLL emerging at finite magnetic field. The split of the SUSYLL (discussed below) is not visible due to the limited resolution of the plot.

Figure 2

(Color online) Hofstadter butterfly of a bilayer graphene in two differently shifted configurations. Top panel: $AA$ stacking (two layers exactly aligned). The band structure for this highly symmetric stacking (same rotational symmetry as for Bernal stacking in Fig. 1) shows the single-layer cone simply split up in energy. Bottom panel: Intermediate position between Bernal and $AA$ stacking. The rotational symmetry is broken and the bands split into two cones at different offsets from the $K$ point and different energies. The straight lines overlaid at the energy minimum and maximum are the regular Landau levels of the massive bands. Near $E_{\mathrm{F}}$, one can make out the parabolic traces of the relativistic Landau levels and the horizontal lines of the SUSYLLs (see text). Insets at the lower right of each panel: DOS of a finite-width ribbon shows the corresponding behavior in each case.

Figure 3

(Color online) Evolution of the split of the supersymmetric Landau level as a function of the displacement between the two graphene layers. Top panel: Magnitude of the split for displacements in two directions. The light spots correspond to Bernal stacking where the level is nearly-degenerate. Bottom panel: Same data along a cut at $\delta y=0$. The small remaining split at the Bernal stacking configuration originates in the long-range interlayer hoppings contained in the parametrization. The small discontinuities are caused by the cutoff $r_{\mathrm{cut}}$. The calculation here was done at $\Phi ={\Phi }_0∕256$, but proved to be independent of the magnetic field for values up to $\sim 0.05{\Phi }_0$.