Geometric and conventional contribution to superfluid weight in twisted bilayer graphene*

We calculate the superfluid weight for twisted bilayer graphene (TBLG) taking into account both the conventional contribution and the contribution arising from the quantum geometry of the bands. We find that both contributions are larger than one would expect by treating the bands as well-isolated, that at the magic angle the geometric contribution is larger than the conventional one, and that for small deviations away from the magic angle the conventional contribution is larger than the geometric one. Our results show that, despite the flatness of the bands the superfluid weight in TBLG is finite and consistent with experimental observations. We also show how the superfluid weight can be tuned by varying the chemical potential and the twist angle opening the possibility to tune the nature of the superconducting transition between the standard BCS transition and the Berezinskii-Kosterlitz-Thouless transition.