Magic angle butterfly in twisted trilayer graphene

We consider a configuration of three stacked graphene monolayers with commensurate twist angles θ12/θ23=p/q, where p and q are coprime integers with 0<p<|q| and q can be positive or negative. We study this system using the continuum model in the chiral limit when interlayer coupling terms between AA12 and AA23 sites of the moiré patterns 12 and 23 are neglected. There are only three inequivalent displacements between the moiré patterns 12 and 23, at which the three monolayers' Dirac zero modes are protected. Remarkably, for these displacements and an arbitrary p/q we discover exactly flat bands at an infinite set of twist angles (magic angles). We provide theoretical explanation and classification of all possible configurations and topologies of the flat bands.