Thermodynamic supercriticality and complex phase diagram for charged Gauss-Bonnet AdS black holes

TL;DR

This work extends black hole thermodynamics into the supercritical regime by applying Lee–Yang theory to charged Gauss–Bonnet AdS black holes in 5d and 6d. Using Q=1 and fixed α=3.05, the authors construct reduced equations of state and analyze complex Gibbs free energy singularities, revealing a single Widom line in 5d (with one critical point) and two Widom lines in 6d (with a triple point and three coexisting phases SBH/IBH/LBH). The complex-phase-diagram analysis shows how real and complex singularities project onto the real plane to delineate phase-like sectors, highlighting dimensional and higher-curvature effects on supercritical behavior. These results substantiate the Lee–Yang framework in gravitational thermodynamics and suggest a general mechanism by which triple-point structure induces multiple Widom lines in the supercritical state, with implications for broader modified gravity contexts.

Abstract

Lee-Yang zero theory plays a crucial role in phase transition theory and is widely employed in the critical behavior of statistical thermodynamics. The supercritical regime of black hole thermodynamics remains a relatively unexplored area, and recent applications of this theory to charged anti-de Sitter (AdS) black holes have initiated probes into this regime, revealing a simple structure partitioned by a single Widom line. In this paper, we apply Lee-Yang theory to charged Gauss-Bonnet AdS black holes, which feature complex phase diagrams (e.g., triple points), to determine how such structures are reflected in the supercritical regime. Notably, we observe a key dimensional difference in that the five-dimensional (5d) system, which lacks a triple point, confirms the known single Widom line structure, while the six-dimensional (6d) system, which admits a triple point, generates two distinct Widom lines. These lines partition the supercritical domain into three sectors (small-, intermediate-, and large- black hole-like phases), corresponding to the three phases coexisting at the triple point. Our results reveal a direct correspondence between the number of coexisting phases at a triple point and the number of distinct supercritical sectors separated by Widom lines.

Figures (6)

• Figure 1: Thermodynamic behavior in 5d charged Gauss-Bonnet AdS black hole ($Q=1, \alpha=3.05$). Curves correspond to different reduced pressures: $p=0.50$ (black), $p=1.00$ (red dashed, critical point), $p=1.20$ (blue), $p=1.60$ (green), $p=2.00$ (purple). The vertical dashed line in (a) indicates the physical boundary $z_{\text{min}}(\text{5d}) \approx 0.176$, below which $t<0$. The critical point ($p=1$) corresponds to the critical radius $z_c=1$.
• Figure 2: The singularity distribution for the 5d charged Gauss-Bonnet AdS black hole. The distinct colors denote the four physical roots of Eq. (\ref{['fenbu1']}). The gray shaded region ($|z| < z_{\text{min}}(\text{5d})$) is non-physical, and roots within it are excluded. The vertical dashed line marks the critical point $z_c$ on the real axis.
• Figure 3: The complex phase diagram of the 5d charged Gauss-Bonnet AdS black hole and corresponding supercritical phenomena.
• Figure 4: Thermodynamic behavior in 6d charged Gauss-Bonnet AdS black hole ($Q=1, \alpha=3.05$). Curves correspond to different reduced pressures: $p=0.50$ (black), $p=1.00$ (red dashed, triple point), $p=1.20$ (blue), $p=1.60$ (green), $p=2.00$ (purple). The vertical dashed line in (a) indicates the physical boundary $z_{\text{min}}(\text{6d}) \approx 0.313$, below which $t<0$. The vertical dashed lines in (c) at $z_c^{(1)} \approx 0.769$, $z_c^{(2)} \approx 1.027$, and $z_c^{(3)} \approx 1.443$ mark the radii of the three critical points, respectively.
• Figure 5: The singularity distribution for the 6d charged Gauss-Bonnet AdS black hole. The distinct colors denote the four physical roots of Eq. (\ref{['fenbu2']}). The gray shaded region ($|z| < z_{\text{min}}(\text{6d})$) is non-physical, and roots within it are excluded. The vertical dashed line marks the critical point $z_c^{(1)}, z_c^{(2)}, z_c^{(3)}$ on the real axis.