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The Eilenberger equation is a standard tool in the description ofsuperconductors with an arbitrary degree of disorder. It can be generalized tosystems with linear-in-momentum spin-orbit coupling (SOC), by exploiting theanalogy of SOC with a non-abelian background field. Such field mixes singletand triplet components and yields the rich physics of magnetoelectricphenomena. In this work we show that the application of this equation extendsfurther, beyond superconductivity. In the normal state, the linearizedEilenberger equation describes the coupled spin-charge dynamics. Moreover, itsresolvent corresponds to the so called Cooperons, and can be used to calculatethe weak localization corrections. Specifically, we show how to solve thisequation for any source term and provide a closed-form solution for the case ofRashba SOC. We use this solution to address several problems of interest forspintronics and superconductivity. Firstly, we study spin injection fromferromagnetic electrodes in the normal state, and describe the spatialevolution of spin density in the sample, and the complete crossover from thediffusive to the ballistic limit. Secondly, we address the so-calledsuperconducting Edelstein effect, and generalize the previously known resultsto arbitrary disorder. Thirdly, we study weak localization correction beyondthe diffusive limit, which can be a valuable tool in experimentalcharacterization of materials with very strong SOC. We also address theso-called pure gauge case where the persistent spin helices form. Our workestablishes the linearized Eilenberger equation as a powerful and a veryversatile method for the study of materials with spin-orbit coupling, whichoften provides a simpler and more intuitive picture compared to alternativemethods.

A Unified Description of Spin Transport, Weak Antilocalization and Triplet Superconductivity in Systems with Spin-Orbit Coupling