Holographic Extended Thermodynamics of deformed AdS-Schwarzschild black hole

TL;DR

This work analyzes the thermodynamics of a gravitationally deformed AdS–Schwarzschild black hole within extended black hole thermodynamics and its holographic CFT dual. By applying the gravitational decoupling method, it uncovers a van der Waals–type first-order transition in the bulk for a finite α-range and confirms mean-field critical exponents, while also exposing Hawking–Page-like confinement–deconfinement phenomena on the boundary across three distinct ensembles. The boundary theory exhibits a Hawking–Page-type transition in (V, C) and unconventional behavior in (p, C) and (p, μ), with the GD deformation parameters α and β driving new scales and critical structure. Overall, the GD deformation modifies both bulk universality and boundary dynamics, revealing a rich holographic landscape of phase transitions controlled by deformation and central charge.

Abstract

We investigate the thermodynamics and phase structure of the deformed AdS-Schwarzschild black hole, generated via the gravitational decoupling (GD) method. In the bulk canonical ensemble, our results exhibit a van der Waals-type first-order phase transition in addition to the Hawking-Page transition, in the suitable parameter regime. Further, we compute the critical exponents characterising the bulk transition, confirming their consistency with mean-field theory predictions. Exploiting the exact holographic dictionary between extended black hole thermodynamics and the dual conformal field theory (CFT), we extend this analysis to the boundary and uncover a rich array of phase transitions and critical phenomena across three distinct thermodynamic ensembles. In particular, in the fixed $(\mathcal{V},C)$ ensemble, the dual CFT exhibits a Hawking-Page-type transition. However, in the fixed $(p,C)$ ensemble, the deformation parameter leads to a distinct thermodynamic behaviour in which multiple branches become unstable, leaving a single thermodynamically stable phase, thus marking a clear departure from the standard van der Waals scenario. Throughout, we emphasise the pivotal influence of the GD deformation parameter on the thermodynamic behaviour, and we elucidate its role in the confinement-deconfinement transitions characteristic of the deformed AdS-Schwarzschild geometry.

Figures (24)

• Figure 1: Plot of the $T$–$r_{h}$ relation for two distinct scenarios. (Left): For $\beta = 1$, two critical values of the deformation parameter are observed: $\alpha_{c_{1}} = 10.3$ and $\alpha_{c_{2}} = 16.2$. The red curve exhibits two inflection points, the blue curve exhibits one, and the magenta curve corresponds to the case with no inflection points. Here, $\alpha_{c}^{min} = 11.4998$ and $\alpha_{c}^{max} = 14.8394$. (Right): For $\beta = 0$, the critical value is $\alpha_{c} = 6.25$. In both cases, the AdS curvature scale is fixed at $l = 15$.
• Figure 2: Without loss of generality, for fixed values of $\alpha = 13.5$ and $\beta = 1$, there exist two critical temperatures, $T_{c_{1}} = 0.0068$ and $T_{c_{2}} = 0.0180$, such that the system exhibits a van der Waals-like phase transition only in the intermediate range $T \in (T_{c_{1}}, T_{c_{2}})$. Outside this interval, i.e., for $T < T_{c_{1}}$ or $T > T_{c_{2}}$, no first-order phase transition occurs. The characteristic swallowtail structure in the $P-v$ diagram appears within this temperature window. Furthermore, there exists a lower threshold temperature, $T_{0} = 0.014$, below which the pressure $P$ starts to become negative, i.e., for $T \leq T_{0}$ within some range of $v$ as shown.
• Figure 3: (Left): $P$--$v$ diagram for the charged AdS black hole with $\alpha = 6.25$, $\beta = 0$, showing a van der Waals-like first-order phase transition at the critical temperature $T_{c} = 0.0173$, as expected. (Right): $P$--$v$ diagram for the standard AdS-Schwarzschild black hole ($\alpha = 0$) at three different temperatures with $\beta = 1$, illustrating the absence of van der Waals phase transition.
• Figure 4: Free energy versus temperature plots for two cases: (Left): $\alpha_{c_{1}}=11.4998, \,\,\alpha_{c_{2}}=14.8394,\,\,\alpha = 13.5$, $l = 15$, $\beta = 1$, showing a van der Waals-like first-order phase transition. (Right): $\alpha = 18$, $l = 15$, $\beta = 1$, where no such phase transition is observed. These plots correspond to the red and magenta curves shown in Fig. \ref{['T-rh-new']} (Left).
• Figure 5: Plot of the heat capacity as a function of $\tilde{r}_{h}$, evaluated at the same parameter values used in Fig. \ref{['tilde-F-t']}. The left panel illustrates the van der Waals-like behaviour, while the right panel corresponds to a regime where the first-order phase transition is absent.