An exact continuum model for low-energy electronic states of twisted bilayer graphene

We introduce a complete physical model for the single-particle electronic structure of twisted bilayer graphene (tBLG), which incorporates the crucial role of lattice relaxation. Our model, based on $k \cdot p$ perturbation theory, combines the accuracy of DFT calculations through effective tight-binding Hamiltonians with the computational efficiency and complete control of the twist angle offered by continuum models. The inclusion of relaxation significantly changes the bandstructure at the first magic-angle twist corresponding to flat bands near the Fermi level (the "low-energy" states), and eliminates the appearance of a second magic-angle twist. We show that minimal models for the low-energy states of tBLG can be easily modified to capture the changes in electronic states as a function of twist angle.