There’s a lot of excitement about twisted bilayer graphene, which I wrote a bit about earlier. We have been going back to basics and studying the band structure of these systems, using some of the models which have been largely accepted, at least in broad terms, to describe them. These models take the form of two continuum coupled Dirac equations with a spatially periodic hopping term connecting them. The first thorough early study was by Bistritzer and MacDonald back in 2011. They discovered that (1) for nearly all angles the original Dirac points of persist, (2) at certain “magic” angles the Dirac velocity vanishes, and (3) near these magic angles the low energy bands become exceptionally narrow over the entire Brillouin zone.
What we showed in our recent preprint is that actually the vanishing Dirac velocity is just one of several topological phase transitions associated with merging/splitting/annihilation of Dirac points as the angle is varied. The above video shows how these points move around the moiré Brillouin zone as the angle is varied, in a range of angles close to the first magic angle (about 1 degree). The parameter shown at the upper right of the video is inversely proportional to the angle. Please see our preprint for more details.