Supermoiré low-energy effective theory of twisted trilayer graphene

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