# Covariant conservation laws and the spin Hall effect in Dirac-Rashba systems

## TL;DR

This paper provides a theoretical framework using symmetry relations to understand spin Hall effects in Dirac-Rashba systems, revealing conditions for their suppression and restoration under disorder and interactions.

## Contribution

It introduces exact symmetry relations (Ward identities) that constrain spin dynamics and predict the emergence of spin Hall effects in Dirac-Rashba systems with various interactions.

## Key findings

- Scalar impurities suppress spin Hall currents in Rashba-only systems.
- Adding spin-valley interactions restores finite spin Hall conductivity.
- Ward identities serve as a benchmark for numerical methods and experimental predictions.

## Abstract

We present a theoretical analysis of two-dimensional Dirac-Rashba systems in the presence of disorder and external perturbations. We unveil a set of exact symmetry relations (Ward identities) that impose strong constraints on the spin dynamics of Dirac fermions subject to proximity-induced interactions. This allows us to demonstrate that an arbitrary dilute concentration of scalar impurities results in the total suppression of nonequilibrium spin Hall currents when only Rashba spin-orbit coupling is present. Remarkably, a finite spin Hall conductivity is restored when the minimal Dirac-Rashba model is supplemented with a spin-valley interaction. The Ward identities provide a systematic way to predict the emergence of the spin Hall effect in a wider class of Dirac-Rashba systems of experimental relevance and represent an important benchmark for testing the validity of numerical methodologies.