Non-Abelian gauge potentials in graphene bilayers

We study the effect of spatial modulations in the interlayer hopping of graphene bilayers, such as those that arise upon shearing or twisting. We show that their single-particle physics, characterized by charge accumulation and recurrent formation of zero-energy bands as the pattern period L increases, is governed by a non-Abelian gauge potential arising in the low-energy electronic theory due to the coupling between layers. We show that such gauge-type couplings give rise to a potential that, for certain discrete values of L, spatially confines states at zero energy in particular regions of the Moir\'e patterns. We also draw the connection between the recurrence of the flat zero-energy bands and the non-Abelian character of the potential.