Stability studies of first order spin-hydrodynamic frameworks

We study the stability of first-order dissipative spin-hydrodynamic frameworks. We considered two different first-order dissipative spin-hydrodynamic frameworks. The first one considers the spin chemical potential ($\omega^{\alpha\beta}$) to be first order ($\mathcal{O}(\partial)$) in the hydrodynamic gradient expansion. The hydrodynamic gradient ordering of the spin chemical potential is a debatable issue within the frameworks of spin hydrodynamics. Therefore as a second choice, we also consider the spin hydrodynamic equations with $\omega^{\alpha\beta}\sim\mathcal{O}(1)$. We find that for both frameworks, at the level of linear perturbations some spin modes can be unstable. To remove these generic instabilities we consider the Frenkel condition. We argue that Frenkel condition helps get rid of the unstable solutions in both cases, but with a physical drawback for the case where $\omega^{\mu\nu}\sim\mathcal{O}(\partial)$.