Holographic CFT phase transitions and criticality for charged Gauss-Bonnet AdS black holes in the ensemble at fixed $(C, \mathcal{V}, \tilde{Q}, \tilde{\mathcal{A}})$

TL;DR

The paper investigates the holographic dual of extended black hole thermodynamics for charged Gauss–Bonnet AdS black holes in $d=4$ and $d=5$, highlighting how the GB coupling $\alpha$ and its boundary partner $\mathcal{A}$ enter the boundary CFT thermodynamics. It develops the holographic dictionary in the ensemble fixed at $$(C, \mathcal{V}, \tilde{Q}, \tilde{\mathcal{A}})$$, derives the relevant thermodynamic potentials $E$, $S$, $T$, $\Phi$, $p$, and the holographic Smarr relation, and discusses the necessity of constraining intermediate variables (e.g., setting $y=1$) to render the analysis tractable. Through numerical exploration, the authors reveal a rich, nontrivial phase structure in the CFT free energy $G$, including zeroth- and first-order transitions and a loop-like non-monotonic behavior that challenges conventional free-energy criticality. They analyze conjugate pairs $(\mu,C)$, $(p,\mathcal{V})$, $(\tilde{T},\tilde{S})$, and $(\tilde{\Phi},\tilde{Q})$, showing van der Waals-like features in several thermodynamic planes while highlighting unusual bounded behavior in $\mu-C$. The results emphasize the subtle dependence of holographic phase structure on the GB coupling and the chosen ensemble, and they motivate a deeper holographic interpretation of the dual CFT in this unconventional setup.

Abstract

We study the holographic dual of the extended thermodynamics of spherically symmetric, charged Gauss-Bonnet AdS black holes in the context of the AdS/CFT correspondence. Compared to Einstein's theory of gravity, Gauss-Bonnet gravity introduces higher-order curvature terms. The coupling constants of these higher-order curvature terms $α$ can serve as new thermodynamic quantities, which will also be dual to thermodynamic quantities on the boundary CFT, a feature not present in the CFT dual to Einstein's gravity previously. Based on the holographic dictionary, we studied the critical behavior and phase transition of the CFT description of the charged Gauss-Bonnet black holes in $d=4$ and $d=5$ in the ensemble at fixed $(C, \mathcal{V}, \tilde{Q}, \tilde{\mathcal{A}})$. The interesting behaviour of free energy stems from the fact that the constraints we introduced to handle the gravitational constant on CFT and the AdS radius differ from conventional approaches. Using the criticality equation, we numerically found the critical points of the zeroth-order and first-order phase transition for $\tilde{\mathcal{A}}$. The relationships between conjugate thermodynamic pairs (equation of state) were also examined. In the case of the $p-\mathcal{V}$, $\tilde{T}-\tilde{S}$ and $\tildeΦ-\tilde{Q}$ conjugate pairs, characteristics that are analogous to the first-order phase transition of van der Waals fluids were found.

Figures (21)

• Figure 1: Free energy $G$ vs. temperature $\tilde{T}$ plot in $d = 4$ for the fixed $(C, \mathcal{V}, \tilde{Q}, \tilde{\mathcal{A}})$ ensemble($y=1$). we plot different values of $\tilde{\mathcal{A}}$ for fixed $C$, $\mathcal{V}$ and $\tilde{Q}$, the parameters are $\mathcal{V} = 1$, $C=2$, $\tilde{Q} = 10$ and $\tilde{\mathcal{A}}=0.005, 0.01, 0.022, 0.0345, 0.04$(blue, orange, green, red, purple).
• Figure 2: $\mathcal{C}_{C, \mathcal{V}, \tilde{Q},\mathcal{\tilde{A}}}$ vs. $x$ plot in $d=4$ for the fixed $(C, \mathcal{V}, \tilde{Q}, \tilde{\mathcal{A}})$ ensemble($y=1$). we plot different values of $\tilde{\mathcal{A}}$ for fixed $C$, $\mathcal{V}$ and $\tilde{Q}$, the parameters are $\mathcal{V} = 1$, $C=2$, $\tilde{Q} = 10$ and $\tilde{\mathcal{A}}=0.005, 0.022, 0.0345, 0.04$(blue, orange, green, red).
• Figure 3: The solutions to \ref{['eq:3.9']} and \ref{['eq:3.10']} with $\mathcal{V}=1$ , $C=2$ and $\tilde{Q}=10$ for $y\tilde{\mathcal{A}}$ in the range $[0, 0.04]$. The blue dots represent the solution to \ref{['eq:3.9']}, and the orange dots represent the solution to \ref{['eq:3.10']}.
• Figure 4: Temperature $\tilde{T}$ vs. $x$ plot in $d=4$ for the fixed $(C, \mathcal{V}, \tilde{Q}, \tilde{\mathcal{A}})$ ensemble($y=1$). we plot different values of $\tilde{\mathcal{A}}$ for fixed $C$, $\mathcal{V}$ and $\tilde{Q}$, the parameters are $\mathcal{V} = 1$, $C=2$, $\tilde{Q} = 10$ and $\tilde{\mathcal{A}}=0.005, 0.022, 0.0345, 0.04$(blue, orange, green, red).
• Figure 5: Free energy $G$ vs. $x$ plot in $d=4$ for the fixed $(C, \mathcal{V}, \tilde{Q}, \tilde{\mathcal{A}})$ ensemble($y=1$). we plot different values of $\tilde{\mathcal{A}}$ for fixed $C$, $\mathcal{V}$ and $\tilde{Q}$, the parameters are $\mathcal{V} = 1$, $C=2$, $\tilde{Q} = 10$ and $\tilde{\mathcal{A}}=0.005, 0.022, 0.0345, 0.04$(blue, orange, green, red).