Magic Angle Butterfly in Twisted Trilayer Graphene

Abstract

We consider a configuration of three stacked graphene monolayers with commensurate twist angles ${\theta }_{12}/{\theta }_{23}=p/q$, 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 ${\text{AA}}_{12}$ and ${\text{AA}}_{23}$ 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.

arXiv comment9 pages, 2 tables, 6 figures