  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, rather 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.
