Phase transition at a critical coupling for a vortex nonminimally coupled in AdS3

We consider a Nielsen-Olesen vortex nonminimally coupled to Einstein gravity with negative cosmological constant (AdS3 background). The nonminimal coupling term plays a dual role: it contributes to the potential of the scalar field and to the Einstein-Hilbert term for gravity. This leads to a novel feature: there exists a critical coupling where the vacuum expectation value (VEV) is zero but becomes non-zero when the coupling crosses below the critical value. The critical coupling exists only in an AdS3 background and not in a Minkowski background. The VEV near the critical coupling has a power law behaviour with exponent of 1/2. The critical coupling can therefore be viewed as the analog of the critical temperature in the Ginzburg-Landau mean-field theory of second-order phase transitions.