Dynamical stability from quasi normal modes in 2nd, 1st and 0th order holographic superfluid phase transitions

We study a simple extension of the original Hartnoll, Herzog and Horowitz (HHH) holographic superfluid model with two nonlinear scalar self-interaction terms $\lambda |\psi|^4$ and $\tau |\psi|^6$ in the probe limit. Depending on the value of $\lambda$ and $\tau$, this setup allows us to realize a large spectrum of holographic phase transitions which are 2nd, 1st and 0th order as well as the ``cave of wind'' phase transition. We speculate the landscape pictures and explore the near equilibrium dynamics of the lowest quasinormal modes (QNMs) across the whole phase diagram at both zero and finite wave-vector. We find that the zero wave-vector results of QNMs correctly present the stability of the system under homogeneous perturbations and perfectly agree with the landscape analysis of homogeneous configurations in canonical ensemble. The zero wave-vector results also show that a 0th order phase transition cannot occur since it always corresponds to a global instability of the whole system. The finite wave-vector results show that under inhomogeneous perturbations, the unstable region is larger than that under only homogeneous perturbations, and the new boundary of instability match with the turning point of condensate curve in grand canonical ensemble, indicating a new explanation from the subsystem point of view. The additional unstable section also perfectly match the section with negative value of charge susceptibility.