Horizon, Null-Orbit, and Photon-Sphere Phase Structures in Logarithmic Nonlinear Electrodynamics
We investigate static, charged, asymptotically flat black holes in logarithmic nonlinear electrodynamics (LNED) within the $(Q,ε)$ parameter space. We distinguish three phase structures: the horizon geometry, the background circular null orbits, and the circular photon orbits of the effective optical geometry. All share a zero-radius boundary $Q_{nc}(ε;M)=γM^{2/3}ε^{1/6}$, but possess distinct critical structures: the extremal curve $Q_{ext}(ε;M)$ and the background merger curve $Q_{m}(ε;M)$ both terminate at the bifurcation point $(Q_{b},ε_{b})$, located at $r=0$. In contrast, the effective photon-orbit merger curve $Q_{m}^{ph}(ε;M)$ does not pass through this point. For $ε\geε_{b}$, approaching $Q_{nc}$ shrinks the horizon to zero radius, resulting in black points with non-zero, quantum-corrected entropy. Their Hawking temperature vanishes at the bifurcation point and diverges for $ε>ε_{b}$, except when $ηδ=1$, where the temperature at the bifurcation point also diverges. The background potential $V_{eff}^{bg}(r)$ develops an infinite barrier at $r=0$ without local extrema, acting as a perfect repeller. In contrast, the photon potential $V_{eff}^{ph}(r)$ is regularized with a maximum at a finite radius, acting as a partial reflector. Consequently, the black points retain photon spheres of finite radius. Away from $Q_{nc}$, for $ε<ε_{b}$, both geometries support horizonless structures above $Q_{ext}$; for $ε\geε_{b}$, only the optical geometry does so in the region $Q_{nc}<Q<Q_{m}^{ph}$. Non-radial geodesics are blocked from the singularity and can form metastable bound states, while radial geodesics either reach $r=0$ or escape to infinity. The critical impact parameter encodes both gravitational and optical-geometry effects, probing the distinct phase structures.