We analyse 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 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.
Categories: math-ph cond-mat.mes-hall cond-mat.str-el math.AP math.MP