Twisted Bilayer Graphene Near Commensuration

It is well known that Twisted Bilayer Graphene (TBG) admits a continuum model for twist angles near $0$. In this talk, we derive a similar continuum model for TBG twisted near any commensurate angle. The model contains two additional parameters. One acts like an imaginary part for the AA-stacking parameter $w_0$ and the other provides an overall energy shift. For the first six distinct commensurate configurations (in order of the number of atoms per unit cell) we compute the model parameters. Additionally, we numerically observe flat bands at "magic angles" near each commensurate configuration. For at least the first nontrivial configuration (at which $\theta \approx 38.21^{\circ}$ or $21.79^{\circ}$) the energies and twist angles are on experimentally reasonable scales.