Nonlinear <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>σ</mml:mi></mml:math> model for disordered systems with intrinsic spin-orbit coupling
Pauli Virtanen, F. S. Bergeret, I. V. Tokatly
Abstract
We derive the nonlinear $\ensuremath{\sigma}$ 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.