Causality and stability analysis of relativistic spin hydrodynamics: Insights from a nonvanishing spin density background

We investigate the stability and causality of relativistic spin hydrodynamics in the presence of a nonvanishing spin density background, assuming that the spin chemical potential $\omega^{\mu\nu}$ is of leading order ($\omega^{\mu\nu} \sim \mathcal{O}(1)$) in the gradient expansion and is treated as a finite background in the linear perturbation analysis. It is found that within the first-order spin hydrodynamic framework, a finite spin density background modifies the dispersion relations, and modes propagating along different directions are controlled by distinct transport coefficients. Certain specific modes only appear in the $x$ direction. However, the modes in the large wave-vector limit exhibit acausal behavior. To address this issue, we subsequently adopt the framework of minimal causal spin hydrodynamics and derive the corresponding stability and causality conditions. The spin density background directly determines whether stability and causality can be satisfied simultaneously. In the small wave-vector limit, the results are similar to those of the first-order theory. In the large wave-vector region, however, significant differences emerge: the distinctions between different directions are no longer merely simple substitutions of transport coefficients, but involve more complex combinations. This indicates that the difference between modes in different directions increases with increasing wave vector.