Continuum model of the twisted graphene bilayer

The continuum model of the twisted graphene bilayer [Lopes dos Santos, Peres, and Castro Neto, Phys. Rev. Lett. 99, 256802 (2007)] is extended to include all types of commensurate structures. The essential ingredient of the model, the Fourier components of the spatially modulated hopping amplitudes, can be calculated analytically for any type of commensurate structures in the low-twist-angle limit. We show that the Fourier components that could give rise to a gap in the sublattice exchange symmetric (SE-even) structures discussed by Mele [Phys. Rev. B 81, 161405 (2010)] vanish linearly with angle, whereas the amplitudes that saturate to finite values, as $\ensuremath{\theta}\ensuremath{\rightarrow}0$, ensure that all low-angle structures share essentially the same physics. We extend our previous calculations beyond the validity of perturbation theory to discuss the disappearance of Dirac cone structure at angles below $\ensuremath{\theta}\ensuremath{\lesssim}1$${}^{\ensuremath{\circ}}$.