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·p perturbation theory and openly available, combines the accuracy of density functional theory 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 band structure 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.