Quasinormal Modes and Van der Waals like phase transition of charged AdS black holes in Lorentz symmetry breaking massive gravity

Using the quasinormal modes of a massless scalar perturbation, we investigate the small/large black hole phase transition in the Lorentz symmetry breaking massive gravity. We mainly focus on two issues: i) the sign change of slope of the quasinormal mode frequencies in the complex-$\omega$ diagram; ii) the behaviors of the imaginary part of the quasinormal mode frequencies along the isobaric or isothermal processes. For the first issue, our result shows that, at low fixed temperature or pressure, the phase transition can be probed by the sign change of slope. While increasing the temperature or pressure to some certain values near the critical point, there will appear the deflection point, which indicates that such method may not be appropriate to test the phase transition. In particular, the behavior of the quasinormal mode frequencies for the small and large black holes tend to the same at the critical point. For the second issue, it is shown that the non-monotonic behavior is observed only when the small/large black hole phase transition occurs. Therefore, this property can provide us with an additional method to probe the phase transition through the quasinormal modes.