Analytic three-dimensional primary hair charged black holes with Coulomb-like electrodynamics and their thermodynamics

We construct and discuss new solutions of primary hair charged black holes in asymptotically Anti-de Sitter (AdS) space that have well-defined Coulomb-like potential in three dimensions. The gauge field source to the Einstein equation is a power-Maxwell nonlinear electrodynamics with traceless energy-momentum tensor. The coupled Einstein-power-Maxwell-scalar gravity system, which carries the coupling $f(\phi)$ between the gauge and scalar fields, is analyzed, and hairy charged black hole solutions are found analytically. We consider three different profiles of the coupling functions: (i) $f(\phi)=1$, corresponding to no direct coupling between the gauge and scalar fields, (ii) $f(\phi)=e^{\phi}$, and (iii) $f(\phi)=e^{\phi^2/2}$, corresponding to their non-minimal coupling. For all these cases, the scalar field, gauge fields, and curvature scalars are regular and well-behaved everywhere outside the horizon. We further study the thermodynamics of the obtained hairy black hole in the canonical and grand-canonical ensembles and find significant changes in its thermodynamic structure due to the scalar field. In particular, for all considered coupling functions, the hairy parameter has a critical value above which the hairy black hole undergoes the Hawking/Page phase transition, whereas below which no such phase transition appears.