Critical phenomena and thermodynamic geometry of charged Gauss-Bonnet AdS black holes

In this paper, we study the phase structure and equilibrium state space geometry of charged topological Gauss-Bonnet black holes in $d$-dimensional anti–de Sitter spacetime. Several critical points are obtained in the canonical ensemble, and the critical phenomena and critical exponents near them are examined. We find that the phase structures and critical phenomena drastically depend on the cosmological constant $\Lambda$ and dimensionality $d$. The result also shows that there exists an analogy between the black hole and the van der Waals liquid gas system. Moreover, we explore the phase transition and possible property of the microstructure using the state space geometry. It is found that the Ruppeiner curvature diverges exactly at the points where the heat capacity at constant charge of the black hole diverges. This black hole is also found to be a multiple system, i.e., it is similar to the ideal gas of fermions in some range of the parameters, while to the ideal gas of bosons in another range.

Figure 1

Heat capacity $C_Q$ with $r_h$ and $Q$ at $k=1$. (a) for $d=6$, $\Lambda =-1$, and (b) for $d=6$, $\Lambda =-0.49$.

Figure 2

The divergent behavior of the heat capacity as a function of $r_h$. (a) for $d=6$, $\Lambda =-1$. (b) for $d=6$, $\Lambda =-0.49$. The range in shadow has a temperature $T<0$.

Figure 3

$Q\mathrm{\text{−}}\Phi$ diagram (isotherm) near the four critical points located at $r_{c1}$, $r_{c2}$, $r_{c3}$, and $r_{c4}$, respectively.

Figure 4

Gibbs free energy as a function of the temperature $T$ for $\Lambda =-1$. The “swallow tail” behavior appears when $Q<Q_{c1}$.

Figure 5

Heat capacity $C_Q$ with $r_h$ and $Q$ at $k=-1$. (a) for small $\Lambda$, and (b) for large $\Lambda$.

Figure 6

Four different divergent behaviors of the heat capacity $C_Q$ in the different ranges of the cosmological constant $\Lambda$.

Figure 7

The divergent behavior of $C_Q$ for $k=-1$, $d=6$, and $\Lambda =-4$. (a) for the small charge case, and (b) for the large charge case.

Figure 8

Behavior of the Ruppeiner curvature at $d=6$. (a) for $k=1$, $\Lambda =-1$, (b) for $k=1$, $\Lambda =-0.49$, (c) for $k=-1$, $\Lambda =-4$.