Magnetoelectric effects in superconductors due to spin-orbit scattering: a non-linear $\sigma$-model description

TL;DR

This paper develops a generalized nonlinear sigma-model for diffusive superconductors that incorporates magnetoelectric effects caused by spin-orbit scattering, including the spin-Hall effect and spin-current swapping, validated through multiple derivations.

Contribution

It introduces a new sigma-model framework accounting for spin-orbit effects in superconductors, correcting previous derivations and including all leading-order invariants.

Findings

Derived new terms in the sigma-model from the Keldysh path integral.

Confirmed the model by deriving the Usadel equation from quantum kinetic equations.

Corrected a recent derivation of the Usadel equation with magneto-electric effects.

Abstract

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