Nematic superconductivity in twisted bilayer graphene

Twisted bilayer graphene displays insulating and superconducting phases caused by exceptional flattening of its lowest energy bands. Superconductivity with highest Tc appears near half filling of the valence band (n≈-2). The data show that in the large part of the superconducting dome near n=-2, threefold lattice rotation symmetry is broken in the superconducting phase, i.e., a superconductor is also a nematic. We perform a comprehensive analysis of superconductivity in twisted-bilayer graphene within an itinerant approach and present a mechanism for nematic superconductivity. We take as an input the fact that at dopings, where superconductivity has been observed, the Fermi energy lies in the vicinity of twist-induced Van Hove singularities in the density of states. We argue that the low-energy physics can be properly described by patch models with either six or twelve Van Hove points. We obtain pairing interactions for the patch models in terms of parameters of the microscopic model for the flat bands, which contains both local and twist-induced nonlocal interactions. We show that the latter give rise to attraction in different superconducting channels. For six Van Hove points, there is just one attractive d-wave channel, and we find chiral d±id superconducting order, which breaks time-reversal symmetry but leaves the lattice rotation symmetry intact. For twelve Van Hove points, we find two attractive channels, g and i waves, with almost equal coupling constants. We show that both order parameters are nonzero in the ground state and explicitly demonstrate that in this co-existence state the threefold lattice rotation symmetry is broken, i.e., the superconducting state is also a nematic. We find two possible nematic states, one is time-reversal symmetric, the other additionally breaks time-reversal symmetry. We apply our scenario to twisted bilayer graphene near n=-2 and argue that it is applicable also to other systems with two (or more) attractive channels with similar couplings as our reasoning for a nematic superconductivity is based on generic symmetry considerations.