Attractive electron-electron interactions from internal screening in magic angle twisted bilayer graphene

Twisted bilayer graphene (tBLG) has emerged as a new platform for studying electron correlations. tBLG exhibits correlated insulating and superconducting states at twist angles close to the largest magic angle, with the phase diagram resembling that of cuprates, which suggests an unconventional mechanism for superconductivity. We have studied the effect of internal screening on electron-electron interactions and Hubbard parameters in undoped tBLG, within the random phase approximation (RPA) and constrained RPA (cRPA). Near the magic angle, the flattening of bands drastically increases the RPA dielectric constant of tBLG and the abrupt change in the band velocity as function of the band energy gives rise to attractive regions in the RPA screened interaction in real space, which could be intimately connected to the reported correlated-insulator and superconducting phases. The cRPA polarizability was used to parametrize a twist-angle-dependent Keldysh model which exhibits a dramatic enhancement with decreasing twist angle, and to calculate the extended Hubbard parameters of a downfolded Hamiltonian. We find that the extended Hubbard parameters depend sensitively on the twist angle and the on-site Hubbard U is a not simply a linear function of twist angle.