Solitonic vortices and black holes with vortex hair in AdS3

We study soliton and black hole solutions with scalar hair in AdS3 in a theory with a Maxwell field and a charged scalar field with double trace boundary conditions, which can trigger the dual boundary theory to become a superconductor. We investigate the phase diagram as a function of the temperature T and of the double trace coupling kappa and we find a rich pattern of phase transitions, which can be of the Hawking-Page kind or can be due to the condensation of the order parameter. We also find a transition between vortex solutions and the zero temperature limit of the black hole for a critical value of the double trace coupling kappa. The Little-Park periodicity is realized for the dual of the black hole solution with hair as a shift in the winding number and in the gauge field.