Euler–Heisenberg BH solution in nonlinear quantum electrodynamics: Thermodynamic behaviors

The thermodynamic and geometric properties of the Euler–Heisenberg BH in nonlinear quantum electrodynamics (QED) are investigated under the incorporation of Barrow entropy, which is assumed to encode quantum-gravity-induced fractal deformations of the event horizon. A richer thermodynamic phase structure than the classical Reissner–Nordström case is found, as the combined effects of strong-field QED vacuum polarization and Barrow fractal entropy corrections are considered. The heat capacity is observed to undergo a sequence of transitions, changing from negative to positive values, diverging at a critical point, and becoming positive again for large horizon radii, which indicates the presence of a second-order phase transition. The Hawking temperature is shown to develop both a minimum and a maximum, the free-energy–temperature curve is demonstrated to exhibit double points signaling metastability and phase coexistence, and a van der Waals-like behavior is seen in the pressure–temperature plane. Meanwhile, curvature diagnostics reveal that the Kretschmann scalar remains finite, whereas the Ricci scalar displays a divergence, establishing a correspondence between thermodynamic and geometric critical behavior. Thus, a physically consistent framework to probe horizon microstructure under quantum-gravity inspired corrections is provided through Barrow entropy, and the Euler–Heisenberg model is shown to serve as a viable setting to explore the interplay between nonlinear electrodynamics and fractal horizon geometry.