Flat bands, Dirac cones, and higher-order band crossings in twisted multilayer graphene

For the chiral limit of two sheets of $n$-layer Bernal-stacked graphene established in the Physical Review Letters arXiv:2109.10325 and arXiv:2109.11514, we prove a trichotomy: depending on the twisting angle, we have either (1) generically, the band crossing of the first two bands is of order $n$; or (2) at a discrete set of magic angles, the first two bands are completely flat; or (3) for another discrete set of twisting parameters, the bands exhibit Dirac cones. This new mathematical discovery disproves the common belief in physics that such a twisted multilayer graphene model can only have higher order band crossings or flat bands, and it leads to a new type of topological phase transition.