Thermodynamic supercriticality and complex phase diagram for charged Gauss-Bonnet AdS black holes (2511.10357v1)

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.

• The paper demonstrates that applying Lee-Yang zero theory rigorously delineates the supercritical regime and complex phase transitions in charged Gauss-Bonnet AdS black holes.
• The methodology leverages analytical root-finding of high-degree polynomials and scaling relations to pinpoint critical points and triple points across different dimensions.
• Results reveal distinct thermal behaviors with multiple Widom lines in six dimensions and a single Van der Waals-like critical point in five dimensions.

Thermodynamic Supercriticality and Complex Phase Diagram for Charged Gauss-Bonnet AdS Black Holes

Introduction

This paper presents a systematic paper of the thermodynamic supercritical regime and the detailed phase structure of charged Gauss-Bonnet anti-de Sitter (AdS) black holes, with emphasis on dimensional effects and critical phenomena. The key conceptual advance is the application of Lee-Yang zero theory to probe supercriticality, moving beyond previous analyses which focused on subcritical and critical regions. The work reveals how the presence of complex phase structures (notably triple points) in higher-dimensional black holes is directly reflected in the nature and partitioning of supercritical domains by multiple Widom lines.

Thermodynamics of Charged Gauss-Bonnet AdS Black Holes

The theoretical framework utilizes the Einstein-Gauss-Bonnet gravity action in $d \geq 5$ dimensions, incorporating electromagnetic fields and a negative cosmological constant interpreted as pressure in the extended thermodynamic phase space. The black holes are characterized by mass $M$ (interpreted as enthalpy), charge $Q$, temperature $T$, entropy $S$, and Gibbs free energy $G$, all expressed as functions of the horizon radius $r_h$, pressure $P$, and Gauss-Bonnet coupling $\alpha$. Essential scaling relations allow rescaling thermodynamic variables by $Q$, facilitating dimensionless analysis and separation of effects due to $\alpha$ and $d$.

The Gauss-Bonnet coupling $\alpha$ proves crucial for the emergence of complex phase behavior. In five dimensions, the system supports a single critical point reminiscent of Van der Waals fluids. In six dimensions, a sufficiently large $\alpha$ induces a triple point where small, intermediate, and large black hole phases coexist, parallel to classical triphasic thermodynamics. The normalization schemes employed (relative to critical or triple points) enable direct cross-comparison between different $d$.

Lee-Yang Zero Theory and Supercriticality

The analytic approach extends standard thermodynamic analysis into the complex domain using the Lee-Yang framework. Here, phase transitions are traced to the distribution of partition function zeros (singularities of $G$) in the complexified parameter space (e.g., horizon radius $z$). In the thermodynamic limit, these zeros approach the real axis, demarcating regions of nonanalyticity and thus locating critical phenomena.

Widom lines—loci of smooth but marked crossover in response functions—are rigorously redefined as projections onto the real parameter plane of the trajectories of the nearest complex Gibbs free energy singularities (rather than naive maxima of specific heats, which may be ambiguous for black holes). This approach is especially powerful for systems with multiple critical points or triple points, where several Widom lines may emerge.

Five-Dimensional Case: Single Critical Point

For $d=5$ and $\alpha=3.05$, there exists a single critical point. The thermodynamic phase diagram displays classic features:

• First-order phase transitions between small and large black holes, with the Gibbs free energy exhibiting a swallowtail structure below the critical pressure.
• Heat capacity diverges at two points, which coalesce at the critical point.
• Beyond the critical pressure, the swallowtail and the divergences disappear, replaced by smooth isobars and a local heat capacity maximum indicative of crossover behavior.

Application of the Lee-Yang framework demonstrates that in the supercritical regime, all singularities move off the real axis into the complex $z$ plane. The single Widom line defined by the trajectory of the closest complex singularity partitions the supercritical domain into small-black-hole-like and large-black-hole-like regions, consistent with previous analyses in four-dimensional Reissner-Nordström AdS black holes.

Six-Dimensional Case: Triple Point and Complex Supercritical Structure

For $d=6$ and $\alpha=3.05$, a triple point appears—a situation with three coexisting black hole phases (SBH, IBH, LBH). There are three critical points in the phase diagram, each associated with merging of different roots (singularities). The Gibbs free energy exhibits double swallowtail structure at sub-triple-point pressures, signifying two simultaneous first-order phase transitions.

Analysis of singularity distributions reveals that:

• Multiple roots on the real axis merge at the three distinct critical points.
• For $p > 1$, multiple sets of conjugate complex roots in the $z$-plane emerge.
• Lee-Yang theory gives rise to two distinct Widom lines, originating from the triple point and separating the supercritical domain into three sectors morphologically similar to SBH, IBH, and LBH.

This correspondence demonstrates that the number of Widom lines in the supercritical regime matches the number of coexisting phases at the triple point—a direct connection between phase structure complexity and supercritical thermodynamics. Crucially, local maxima in heat capacity do not correspond uniquely to Widom line positions, highlighting the necessity of the complex analytic approach.

Implementation, Analytical Methods, and Trade-Offs

The analytic and numerical implementation relies on:

• Derivation and root-finding for high-degree polynomial equations controlling singularity locations (heat capacity divergences as a function of $z$).
• Systematic exclusion of nonphysical roots (regions where $T<0$ or $|z| < z_{\text{min}}$).
• Construction of complex phase diagrams via mapping singularity data onto the $(Re(p), Re(t))$ plane, specifically extracting the trajectories for the first quadrant roots.

Trade-offs include computational complexity for higher $d$ (e.g., solving tenth-order polynomials for the six-dimensional case) and the necessity of careful parameter space scans to accurately identify phase transitions and Widom line structures. The choice of scaling (relative to critical or triple points) is essential for achieving universality and comparability of results.

Physical Implications and Future Directions

These results deepen the understanding of thermodynamic behavior in extended gravity theories and have broader implications:

• The existence and number of Widom lines in supercritical black hole thermodynamics are dictated by underlying phase structure complexity (triple points, multicriticality).
• Dimensionality and higher-curvature corrections (via $\alpha$) fundamentally shape the phase landscape, suggesting analogous phenomena in other modified gravity theories.
• The Lee-Yang methodology provides a robust framework for analyzing crossover and supercriticality, avoiding ambiguities in response function maxima.

Future work should systematically map out the evolution of Widom line multiplicity as $\alpha$ varies, explore similar behavior in Horndeski, Lovelock, or other higher-order gravity black holes, and investigate observable implications for AdS/CFT correspondence and the microstructure of quantum gravity.

Conclusion

The application of Lee-Yang zero theory to charged Gauss-Bonnet AdS black holes reveals a direct correspondence between the number of coexisting phases at critical and triple points and the partitioning of the supercritical domain by Widom lines. The five-dimensional system displays a single Widom line, while the six-dimensional system with a triple point is characterized by two, leading to three supercritical "phases." These results refine the understanding of black hole thermodynamics, demonstrate the necessity of complex analytic approaches for rigorous crossover definition, and suggest rich future avenues in probing supercriticality in gravitational systems with complex phase diagrams.