An alternative perspective to observe the critical phenomena of dilaton AdS black holes

Abstract: The critical phenomena of dilaton AdS black holes are probed from a totally different perspective other than the $P-v$ criticality and the $q-U$ criticality discussed in the former literature. We investigate not only the two point correlation function but also the entanglement entropy of dilaton AdS black holes. We achieve this goal by solving the equation of motion constrained by the boundary condition numerically and we concentrate on $\delta L$ and $\delta S$ which have been regularized by subtracting the terms in pure AdS with the same boundary region. For both the two point correlation function and the entanglement entropy, we consider $4\times2\times2=16$ cases due to different choices of parameters. The van der Waals like behavior can be clearly witnessed from all the $T-\delta L$ ($T-\delta S$) graphs for $q<q_c$. Moreover, the effects of dilaton gravity and the spacetime dimensionality on the phase structure of dilaton AdS black holes are disclosed. Furthermore, we discuss the stability of dilaton AdS black holes by applying the analogous specific heat definition and remove the unstable branch by introducing a bar $T=T_$. It is shown that the first order phase transition temperature $T_$ is affected by both $\alpha$ and $n$. The analogous equal area laws for both the $T-\delta L$ graph and the $T-\delta S$ graph are examined numerically. The relative errors for all the cases are small enough that we can safely conclude that the analogous equal area laws hold for $T-\delta L$ ($T-\delta S$) graph of dilaton AdS black holes.