Black holes of $(2+1)$-dimensional $f(R)$ gravity coupled to a scalar field

Abstract: We consider $f(R)$ gravity theories in the presence of a scalar field minimally coupled to gravity with a self-interacting potential in $(2+1)$-dimensions. Without specifying the form of the $f(R)$ function, we first obtain an exact black hole solution dressed with scalar hair with the scalar charge to appear in the $f(R)$ function and we discuss its thermodynamics. This solution at large distances gives a hairy BTZ black hole, and it reduces to the BTZ black hole when the scalar field decouples. In a pure $f(R)$ gravity supported by the scalar field, we find an exact hairy black hole similar to the BTZ black hole with phantom hair and an analytic $f(R)$ form and discuss its thermodynamics.