Effective Optical Geometry and Holographic Phase Structure of a Magnetic AdS Black Hole

Abstract

We study circular null orbits, thermodynamic topology, and the holographic thermodynamic description of a four-dimensional magnetically charged AdS black hole in nonlinear electrodynamics. The background orbit structure is classified using Duan s phi-mapping construction and compared with the extraordinary electromagnetic characteristics. The latter defines an effective optical metric with an additional radial factor that modifies the orbit condition and introduces explicit dependence on the AdS radius. For four representative configurations, the local indices of the five isolated orbits are unchanged while their radii shift by up to 5.187 %. The background mass thresholds vary smoothly across sigma = 1/4 when the total mass is used, whereas the fixed-m0 parametrization is singular there. Regularity at the center requires sigma > 1 and m0 = 0. The thermodynamic analysis separates local stability indices from equilibrium coexistence: stable Gibbs-free-energy branches cross only in model-dependent parameter intervals. The boundary canonical ensemble exhibits critical behavior consistent with mean-field exponents, and the examined grand-canonical ensemble has a Hawking-Page crossing. In the mixed ensemble, the free-energy branches have a finite endpoint discontinuity, but the sampled competing states do not satisfy the full local-stability criterion