Competition between valley symmetry breaking and intervalley coherence, and their respective routes to Kohn-Luttinger superconductivity in twisted bilayer graphene*

We study the tight competition between valley symmetry breaking and intervalley coherence at integer filling fractions of magic angle twisted bilayer graphene under hydrostatic pressure, finding that different electronic states are very close in energy reflecting the emergent U(4) symmetry. We apply a self-consistent real-space Hartree-Fock approximation that accounts for the screened long-range Coulomb interaction as well as the on-site Hubbard interaction U, being able to include all remote bands of the electronic spectrum. In this way, we show that Kramers intervalley coherence is the dominant symmetry breaking pattern at the charge neutrality point. On the other hand, at integer filling -2 (+2) the ground state displays preferentially valley symmetry breaking (intervalley coherence). We investigate this asymmetry between the electron and hole-doped regimes, concluding that valley symmetry breaking and intervalley coherence imply different routes to superconductivity relying on a Kohn-Luttinger instability of the electron system.