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 inasymptotically Anti-de Sitter (AdS) space that have well-defined Coulomb-likepotential in three dimensions. The gauge field source to the Einstein equationis a power-Maxwell nonlinear electrodynamics with traceless energy-momentumtensor. The coupled Einstein-power-Maxwell-scalar gravity system, which carriesthe coupling $f(\phi)$ between the gauge and scalar fields, is analyzed, andhairy charged black hole solutions are found analytically. We consider threedifferent profiles of the coupling functions: (i) $f(\phi)=1$, corresponding tono direct coupling between the gauge and scalar fields, (ii)$f(\phi)=e^{\phi}$, and (iii) $f(\phi)=e^{\phi^2/2}$, corresponding to theirnon-minimal coupling. For all these cases, the scalar field, gauge fields, andcurvature scalars are regular and well-behaved everywhere outside the horizon.We further study the thermodynamics of the obtained hairy black hole in thecanonical and grand-canonical ensembles and find significant changes in itsthermodynamic structure due to the scalar field. In particular, for allconsidered coupling functions, the hairy parameter has a critical value abovewhich the hairy black hole undergoes the Hawking/Page phase transition, whereasbelow which no such phase transition appears.