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.