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(ϕ) 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(ϕ)=1, corresponding to no direct coupling between the gauge and scalar fields, (ii) f(ϕ)=e^ϕ, and (iii) f(ϕ)=e^ϕ^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.