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

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.

arXiv comment13 pages + supplementary material

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• Superconducting diode effect in diffusive superconductors and Josephson junctions with Rashba spin-orbit coupling
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• Nonreciprocal superconducting transport and the spin Hall effect in gyrotropic structures
• Magnetoelectric effects in diffusive two-dimensional superconductors studied by the nonlinear $\sigma$ model
• Dynamical Hall responses of disordered superconductors
• Spin-split superconductivity in spin-orbit coupled hybrid nanowires with ferromagnetic barriers
• Interfacial spin-orbit coupling in superconducting hybrid systems
• Interplay between Superconductivity and Altermagnetism in Disordered Materials and Heterostructures
• Superconducting Diode Effect in Weak Localization Regime
• Quantum Altermagnetic Instability in Disordered Metals
• Josephson anomalous vortices