Topological chiral superconductivity with spontaneous vortices and supercurrent in twisted bilayer graphene

We study d-wave superconductivity in twisted bilayer graphene and reveal phenomena that arise due to the moire superlattice. In the d-wave pairing, the relative motion (RM) of two electrons in a Cooper pair can have either d + id or d - id symmetry with opposite angular momenta. Due to the enlarged moire superlattice, the center-of-mass motion (COMM) can also carry a finite angular momentum while preserving the moire periodicity. By matching the total angular momentum, which has contributions from both the RM and the COMM, Cooper pairs with d + id and d - id RMs are intrinsically coupled in a way such that the COMM associated with one of the RMs has a spontaneous vortex-antivortex lattice configuration. Another phenomenon is that the chiral d-wave state carries spontaneous bulk circulating supercurrent. The chiral d-wave superconductors are gapped and also topological as characterized by an integer Chern number. Nematic d-wave superconductors, which could be stabilized, for example, by uniaxial strain, are gapless with point nodes.