Supermoiré low-energy effective theory of twisted trilayer graphene

Abstract

Stacking three monolayers of graphene with a twist generally produces two moiré patterns. A moiré of moiré structure then emerges at larger distance where the three layers periodically realign. We devise here an effective low-energy theory to describe the spectrum at distances larger than the moiré lengthscale. In each valley of the underlying graphene, the theory comprises one Dirac cone at the ${\bf Γ}_M$ point of the moiré Brillouin zone and two weakly gapped points at ${\bf K}_M$ and ${\bf K}'_M$. The velocities and small gaps exhibit a spatial dependence in the moiré-of-moiré unit cell, entailing a non-abelian connection potential which ensures gauge invariance. The resulting model is numerically solved and a fully connected spectrum is obtained, which is protected by the combination of time-reversal and twofold-rotation symmetries.