Nonlinear $\sigma$-model for disordered systems with intrinsic spin-orbit coupling

TL;DR

This paper develops a nonlinear sigma-model incorporating spin-orbit coupling in disordered metals and superconductors, revealing the mechanisms behind spin-charge coupling and the spin-Hall effect, with applications to superconducting systems.

Contribution

It introduces a nonlinear sigma-model with SU(2) gauge fields for systems with intrinsic spin-orbit coupling, including higher-order gradient terms and a Usadel equation for superconductors.

Findings

Derived the nonlinear sigma-model with SU(2) gauge fields.

Identified the spin-charge coupling responsible for the spin-Hall effect.

Applied the model to describe anomalous supercurrents in Rashba superconductors.

Abstract

We derive the nonlinear $\sigma$-model to describe diffusive transport in normal metals and superconductors with intrinsic spin-orbit coupling (SOC). The SOC is described via an SU(2) gauge field, and we expand the model to the fourth order in gradients to find the leading non-Abelian field-strength contribution. This contribution generates the spin-charge coupling that is responsible for the spin-Hall effect. We discuss how its symmetry differs from the leading quasiclassical higher-order gradient terms. We also derive the corresponding Usadel equation describing the diffusive spin-charge dynamics in superconducting systems. As an example, we apply the obtained equations to describe the anomalous supercurrent in dirty Rashba superconductors at arbitrary temperatures.