Symmetry preservation in commensurate twisted bilayers

Symmetry plays a key role in materials hosting Dirac electrons and underpins our ability to completely flatten the Dirac cone through the tuning of physical parameters such as twisting in van der Waals heterostructures. The emergent moiré patterns in twisted bilayers at small twist angles appear, at first glance, to be independent of the initial stacking order and hence are only shifted when one layer is translated with respect to the other. However, when the twist angle is large, differences can be seen at the level of both the lattice and electronic structure in the case of twisted bilayer graphene and twisted bilayer kagome. In this work, we show that the rotational and dihedral symmetry of high-symmetry triangular bilayers is preserved for all commensurate twist angles with a sixfold symmetric twist center. Hence, we demonstrate that the exact symmetry of small twist angle systems depends upon the initial stacking of the bilayer. We further apply the principles of our method to twisted bilayer graphene with a threefold symmetric twist center to recover the results of [E. J. Mele, ]. Finally, we outline how our method can be generalized to commensurate multilayer systems.