Incommensurability-induced sub-ballistic narrow-band-states in twisted bilayer graphene

We study the localization properties of electrons in incommensurate twisted bilayer graphene for small angles, encompassing the narrow-band regime, by numerically exact means. Sub-ballistic states are found within the narrow-band region around the magic angle. Such states are delocalized in momentum-space and follow non-Poissonian level statistics, in contrast with their ballistic counterparts found for close commensurate angles. Transport results corroborate this picture: for large enough systems, the conductance of samples with fixed width decreases with the system size in the longitudinal direction for incommensurate angles within the sub-ballistic regime. Our results show that incommensurability/quasiperiodicity effects are of crucial importance in the narrow-band regime. The incommensurate nature of a general twist angle must therefore be taken into account for an accurate description of magic-angle twisted bilayer graphene.