Higher Order Magic in Twisted Four Layer Graphene

The process of stacking graphene multilayers with a small relative twist angle between each of the layers has been found to produce flat bands at a series of “magic angles.” Recent studies have investigated the effect that varying the number of layers has on the values of the magic angles calculated with identical twist angles. Fixing these angles is necessary for the system to have lattice periodicity, even within the Bistritzer-MacDonald (BM) model. Here, we consider a 4-layer structure with the angles between layers 1-2 and layers 3-4 identical, but with the angle between layers 2-3 unconstrained. Our analysis is made possible by treating the upper/lower bilayers in a low energy theory, with the bilayers coupled via a “higher-order” BM model. The results show that a duality exists between the two types of twist angles: if one pair of twist angles produces a flat band, then swapping the two angles also produces a flat band. This allows significantly larger magic angles than those produced when the twist angles are all the same.