Linear response study of collisionless spin drag

In this work we are concerned with the understanding of the collisionless drag or entrainment between two superfluids, also called Andreev-Bashkin effect, in terms of current response functions. The drag density is shown to be proportional to the cross transverse current-current response function, playing the role of a normal component for the single-species superfluid density. We can in this way link the existence of finite entrainment with the exhaustion of the energy-weighted sum rule in the spin channel. The formalism is then used to reproduce some known results for a weakly interacting Bose-Bose mixture. We include the drag effect to determine the beyond mean-field correction on the speed of sound and on the spin dipole excitations for a homogeneous and a trapped weakly interacting gas, respectively. Finally, we show that the response to a quick dipole perturbation on one of the species induces a dipole moment on the other species which is proportional to the drag at short times.