Pairing symmetry and spontaneous vortex-antivortex lattice in superconducting twisted bilayer graphene: A Bogoliubov-de Gennes approach*

We study the superconducting pairing symmetry in twisted bilayer graphene by solving the Bogoliubov-de Gennes equation for all electrons in Moir\'{e} supercells. With increasing the pairing potential, the system evolves from the mixed non-topological $d+id$ and $p+ip$ phase to the $s+p+d$ phase via the first order phase transition. In the time-reversal symmetry breaking $d+id$ and $p+ip$ phase, vortex and antivortex lattice accompanying spontaneous supercurrent are induced by the twist. The superconducting order parameter is nonuniform in the Moir\'{e} unit cell. Nevertheless, the superconducting gap in the local density of states is identical in the unit cell. The twist induced vortices and non-topological nature of the mixed $d+id$ and $p+ip$ phase are not captured by the existing effective models. Our results suggest the importance of long-range pairing interaction for effective models.