Pairing symmetry and spontaneous vortex-antivortex lattice in superconducting twisted-bilayer graphene: 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 moire supercells. With increasing the pairing potential, the system evolves from the mixed nontopological 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 lattices accompanying spontaneous supercurrent are induced by the twist. The superconducting order parameter is nonuniform in the moire unit cell. Nevertheless, the superconducting gap in the local density of states is identical in the unit cell. The twist-induced vortices and nontopological 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.