Magnetoelectric effects in diffusive two-dimensional superconductors studied by the nonlinear $\sigma$ model

We describe a numerical approach to modeling magnetoelectric effects generated by spin-orbit coupling in inhomogeneous diffusive 2D superconductors. It is based on direct minimization of the free energy of diffusion modes, including their coupling to the spin-orbit field strength, described by a recently discovered $\sigma$-model action. We explain how to retain exact conservation laws in the discretized model, and detail the numerical procedure. We apply the approach to the spin-galvanic and the inverse spin-galvanic effects in finite-size 2D superconductors, and describe short-range oscillations of circulating currents and spin densities originating from the spin-orbit coupling.