Spin-fluctuation-induced pairing in twisted bilayer graphene

We investigate the interplay of magnetic fluctuations and Cooper pairing in twisted bilayer graphene from a purely microscopic model within large-scale tight-binding approach resolving \AA ngstr\"om scale. For local onsite repulsive interactions using random-phase approximation for spin fluctuations, we derive effective interaction that use self-consistent solutions Bogoliubov-de-Gennes equations superconductivity. study predominant types as function strength, temperature band filling. large regions this parameter space, find chiral $d$-wave regimes, spontaneously breaking time-reversal symmetry, separated by instabilities at integer fillings. Interestingly, is strongly concentrated AA moir\'e unit cell exhibits phase windings multiples $2\pi$ around these superconducting islands, i.e. pinned vortices. The spontaneous circulating current creates distinctive field pattern. This signature should be measurable state-of-the-art experimental techniques.