Ordered phases and superconductivity in two-dimensional electron systems subject to pair spin-orbit interaction

Pair spin-orbit interaction can emerge in strongly interacting systems characterized by a large spin-orbit coupling. Here we study the role of this interaction in stabilizing ordered and unconventional superconducting phases. We find that, if the system avoids superconductivity, the order realized is a combination of charge density and spin-vorticity waves. The latter is reminiscent of a loop-current state, albeit in the spin channel, rather than in the charge channel. If the system becomes superconducting, the order parameter assumes the form of a paired density wave, i.e., pairing occurs at finite momentum. Intriguingly, one of the possible pairings acquires a form analogous to Amperean superconductivity. However, the order parameter here is always a blend of paired density wave and Amperean pairing, rather than being purely one or the other.