In this study, we extend the application of the Lee-Yang phase transition theorem to the realm of anti–de Sitter (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.

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• Classical black holes
• Quantum aspects of black holes
• Thermodynamics