Twisted Bilayer Graphene in Commensurate Angles

Abstract: The recent discovery of ``magic angles" in twisted bilayer graphene (TBG) has spurred extensive research into its electronic properties. The primary tool for studying this thus far has been the famous Bistritzer-MacDonald model (BM model), which relies on several approximations. This work aims to build the first steps in studying magic angles without using this model. Thus, we study a 2d model for TBG in both AA and AB stacking \emph{without} the approximations of the BM model in the continuum setting, using two copies of a potential with the symmetries of graphene, either sharing a common origin (in AA stacking) or with shifted origins (in AB stacking), and twisted with respect to each other. Our results hold for a wide class of potentials in both stacking types. We describe the angles for which the two twisted lattices are commensurate and prove the existence of Dirac cones in the vertices of the Brillouin zone for such angles. Furthermore, we show that for small potentials, the slope of the Dirac cones is small for commensurate angles that are close to incommensurate angles. This work is the first to establish the existence of Dirac cones for twisted bilayer graphene (for either stacking) in the continuum setting without relying on the BM model. This work is the first in a series of works to build a more fundamental understanding of the phenomenon of magic angles.