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

We analyse the splitting of exact flat bands in the chiral model of the twisted bilayer graphene (TBG) when the $A{A}^{\mathrm{\prime }}\mathrm{/}B{B}^{\mathrm{\prime }}$ coupling of the full Bistritzer–MacDonald model is taken into account. The first-order perturbation caused by the $A{A}^{\mathrm{\prime }}\mathrm{/}B{B}^{\mathrm{\prime }}$ 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.