Scalar hairs and exact vortex solutions in 3D AdS gravity

We investigate three-dimensional (3D) Anti-de Sitter (AdS) gravity coupled to a complex scalar field φwith self-interaction potential V(|φ|). We show that the mass of static, rotationally symmetric, AdS black hole with scalar hairs is determined algebraically by the scalar charges. We recast the field equations as a linear system of first order differential equations. Exact solutions, describing 3D AdS black holes with real spherical scalar hairs and vortex-black hole solutions are derived in closed form for the case of a scalar field saturating the Breitenlohner-Freedman (BF) bound and for a scalar field with asymptotic zero mass. The physical properties of these solutions are discussed. In particular, we show that the vortex solution interpolates between two different AdS_{3} vacua, corresponding respectively to a U(1)-symmetry-preserving maximum and to a symmetry-breaking minimum of the potential V.