From the chiral model of TBG to the Bistritzer–MacDonald model

We analyze the splitting of exact flat bands in the chiral model of the twisted bilayer graphene (TBG) when the AA′/BB′ coupling of the full Bistritzer–MacDonald model is taken into account. The first-order perturbation caused by the AA′/BB′ potential is the same for both bands and satisfies interesting symmetries, in particular it vanishes on the line defined by the K points. The splitting of the flat bands is governed by the quadratic term which vanishes at the K points.