Diagrammatic perturbation approach to moiré bands in twisted bilayer graphene

We develop a diagrammatic perturbation theory to account for the emergence of moiré bands in the continuum model of twisted bilayer graphene. Our framework is build on treating the moiré potential as a perturbation that transfers electrons from one layer to another through the exchange of the three wave vectors that define the moiré Brillouin zone. By working in the two-band basis of each monolayer, we analyze the one-particle Green's function and introduce a diagrammatic representation for the scattering processes. We then identify the moiré-induced self-energy, relate it to the quasiparticle weight and velocity of the moiré bands, and show how it can be obtained by summing irreducible diagrams. We also connect the emergence of flat bands to the behavior of the static self-energy at the magic angle. In particular, we show that a vanishing Dirac velocity is a direct consequence of the relative orientation of the momentum transfer vectors, suggesting that the origin of magic angles in twisted bilayer graphene is intrinsically connected to its geometrical properties. Our approach provides a diagrammatic framework that highlights the physical properties of the moiré bands.