Constrained Random Phase Approximation of the effective Coulomb interaction in lattice models of twisted bilayer graphene

Recent experiments on twisted bilayer graphene show the urgent need for establishing a low-energy lattice model for the system. We use the constrained random phase approximation to study the interaction parameters of such models taking into account screening from the moire bands left outside the model space. Based on an atomic-scale tight-binding model, we develop a numerically tractable approximation to the polarization function and study its behavior for different twist angles. We find that the polarization has three different momentum regimes. For small momenta, the polarization is quadratic, leading to a linear dielectric function expected for a two-dimensional dielectric material. For large momenta, the polarization becomes independent of the twist angle and approaches that of uncoupled graphene layers. In the intermediate momentum regime, the dependence on the twist-angle is strong. Close to the largest magic angle the dielectric function peaks at a momentum of $~1/(4 \: nm)$ attaining values of 25, meaning very strong screening at intermediate distances. We also calculate the effective screened Coulomb interaction in real space and give estimates for the on-site and extended interaction terms for the recently developed hexagonal-lattice model. For free-standing TBG the effective interaction decays slower than $1/r$ at intermediate distances $r$, while it remains essentially unscreened at large enough $r$.