The Nielsen-Olesen vortex nonminimally coupled to Einstein gravity in AdS₃: phase transition at a critical coupling ξ_c

A Nielsen-Olesen vortex (Abelian-Higgs model) is nonminimally coupled to Einstein gravity with negative cosmological constant (i.e. AdS₃ background). The nonminimal coupling term ξ R Φ² plays a dual role: it contributes to the potential of the scalar field Φ and to the Einstein-Hilbert term for gravity. This leads to a novel feature: the existence of a critical coupling ξ_c . Beloew this critical value, the VEV becomes nonzero leading to a phase transition where the gauge symmetry is spontaneously broken. We show that the VEV near the critical coupling has a power law behaviour with an exponent of ½. The critical coupling can be viewed as the analog of the critical temperature in the Ginzburg-Landau mean-field theory of second-order phase transitions. In a Minkowski background, there is no critical coupling; it exists only in an AdS₃ background. However, in an asymptotically flat spacetime (topologically a cone), the deficit angle depends on the coupling ξ and is no longer determined solely by the mass; remarkably, a higher mass does not necessarily yield a higher deficit angle.