Magnetoelectric effects in superconductors due to spin-orbit scattering: Nonlinear σ -model description

We suggest a generalization of nonlinear $\sigma$-model for diffusive superconducting systems to account magnetoelectric effects due spin-orbit scattering. In the leading orders strength and gradient expansion it includes two additional terms responsible spin-Hall effect spin-current swapping. First, assuming delta-correlated disorder we derive new from Keldysh path integral representation generating functional. Then argue phenomenologically that they exhaust all invariants allowed in effective action order coupling (SOC). Finally, results are confirmed by direct derivation saddle-point (Usadel) equation quantum kinetic equations presence randomly distributed impurities with SOC. At this point correct recent Usadel magneto-electric does not resort Born approximation.