Supermoir? low-energy effective theory of twisted trilayer graphene

Stacking three monolayers of graphene with a twist generally produces two moire patterns. A moire of moire 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 moire length scale. In each valley of the underlying graphene, the theory comprises one Dirac cone at the FM point of the moire Brillouin zone and two weakly gapped points at KM and K'M. The velocities and small gaps exhibit a spatial dependence in the moire-of-moire 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.