Time-reversal symmetry breaking vs spin-polarized phase from screening of Coulomb interaction in twisted bilayer graphene

Relying on a self-consistent Hartree-Fock approach in real space, we discuss the way in which the on-site Hubbard repulsion and the screened long-range Coulomb interaction compete to drive twisted bilayer graphene into opposite symmetry breaking patterns. While keeping constant the Hubbard coupling but allowing for variations in the strength of the screened Coulomb interaction, we obtain in general a phase with time-reversal symmetry breaking at strong coupling, which then gives way to a phase with spin-SU(2) symmetry breaking as the on-site Hubbard interaction prevails for sufficiently large screening. We illustrate this trend in the case of magic-angle twisted bilayer graphene at half-filling of the first valence band, where we find a transition from valley symmetry breaking to spin-SU(2) symmetry breaking as the screening of the Coulomb interaction is increased. A similar competition can also be observed in twisted graphene bilayers with flat bands at larger twist angles under pressure, where the strong-coupling regime of the screened Coulomb interaction favors a Chern insulator phase, which turns into a spin-polarized phase as the Hubbard interaction prevails for very large screening.