Thermodynamic supercriticality and complex phase diagram for the AdS black hole

TL;DR

The study develops a complex-analytic framework for supercritical black hole thermodynamics by applying Lee-Yang phase-transition theory to AdS black holes. It relates the Gibbs free energy to the Euclidean partition function through G = -T log Z and analyzes the distribution of zeros to identify phase boundaries. A complex phase diagram for a charged AdS black hole is obtained, with real zeros (p<1) signaling actual transitions and complex zeros (p>1) governing the supercritical regime; projecting these complex zeros onto the real plane defines a Widom line that partitions the supercritical region into small- and large-black-hole-like phases. This Widom-line construction yields a thermodynamic crossover without singularities and provides a novel, first-principles route to study black hole phase structure using complex-analytic methods, opening avenues for simulation and potential experimental analogs.

Abstract

In this study, we extend the application of the Lee-Yang phase transition theorem to the realm of AdS black hole thermodynamics, thereby deriving a comprehensive complex phase diagram for such systems. Our research augments extant studies on black hole thermodynamic phase diagrams, particularly in the regime above the critical point, by delineating the Widom line of AdS black holes. This boundary segregates the supercritical domain of the phase diagram into two disparate zones. As the system traverses the thermodynamic crossover within the supercritical region, it undergoes a transition from one supercritical phase to another, while maintaining the continuity of its thermodynamic state functions. This behavior is fundamentally different from that below the critical point, where crossing the coexistence line results in discontinuities of thermodynamic state functions. The Widom line enables a thermodynamic crossover between single-phase states without traversing the spinodal that emerges in the critical region.

Figures (5)

• Figure 1: A schematic picture of the phase diagram of a typical charged AdS black hole thermodynamic system. The left panel shows the phase diagram of a large black hole (LBH) and a small black hole (SBH) undergoing a phase transition (similar to a gas-liquid phase transition). The right panel shows the supercritical phenomenon of the charged AdS black hole, analogous to conventional supercritical fluids.
• Figure 2: The behaviors of dimensionless reduced Gibbs free energy $g$ and constant pressure heat capacity $c_p$, where black line for $p=0.50$, red dashed line for $p=1.00$ (critical point), blue line for $p=1.20$, orange line for $p=1.35$, green line for $p=1.60$, and purple line for $p=2.00$ for the charged AdS black hole.
• Figure 3: The singularity distribution of the Gibbs free energy in charged AdS black hole, corresponding to the Lee-Yang zeros, for different pressure values. The gray curve represents the unit circle, while distinct colors denote the four roots of Eq. (\ref{['fenbu']}) in the complex domain. Specifically, roots located on the real axis correspond to the critical region ($p<1$), whereas those in the complex plane (excluding the real axis) are associated with the supercritical region ($p>1$). There are also singularities in the second derivative of $g$ at $z=\pm 1$, on the unit circle.
• Figure 4: The complex phase diagram of the charged AdS black hole in a three-dimensional complex space (consisting of the positive real part of temperature $\text{Re} ~t$, the positive real part of pressure $\text{Re} ~p$, and the positive imaginary part of temperature $\text{Im} ~t$) and corresponding supercritical phenomena.
• Figure 5: The behaviors of moduli of the complex Gibbs free energy $g$ with respect to complex temperature $t$ for different pressure $p=1.0$ (upper-left), $p=1.1$ (upper-right), $p=1.2$ (lower-left), $p=1.5$ (lower-right) for the charged AdS black hole, where the red dot represents the Widom point (line) at this pressure, which is the boundary point (line) between the two phases in the supercritical region.