The non-minimally coupled gravitating vortex: phase transition at critical coupling $\xi_c$ in AdS$_3$

We consider the Nielsen-Olesen vortex non-minimally coupled to Einstein gravity with cosmological constant $\Lambda$. A non-minimal coupling term $\xi\,R\,|\phi|^2$ is natural to add to the vortex as it preserves gauge-invariance (here $R$ is the Ricci scalar and $\xi$ a dimensionless coupling constant). This term plays a dual role: it contributes to the potential of the scalar field and to the Einstein-Hilbert term for gravity. As a consequence, the vacuum expectation value (VEV) of the scalar field and the cosmological constant in the AdS$_3$ background depend on $\xi$. This leads to a novel feature: there is a critical coupling $\xi_c$ where the VEV is zero for $\xi\ge \xi_c$ but becomes non-zero when $\xi$ crosses below $\xi_c$ and the gauge symmetry is spontaneously broken. Moreover, we show that the VEV near the critical coupling has a power law behaviour proportional to $|\xi-\xi_c|^{1/2}$. Therefore $\xi_c$ can be viewed as the analog of the critical temperature $T_c$ in Ginzburg-Landau (GL) mean-field theory where a second-order phase transition occurs below $T_c$ and the order parameter has a similar power law behaviour $|T-T_c|^{1/2}$ near $T_c$. The critical coupling exists only in an AdS$_3$ background; it does not exist in asymptotically flat spacetime (topologically a cone) where the VEV remains at a fixed non-zero value independent of $\xi$. However, the deficit angle of the asymptotic conical spacetime depends on $\xi$ and is no longer determined solely by the mass; remarkably, a higher mass does not necessarily yield a higher deficit angle. The equations of motion are more complicated with the non-minimal coupling term present. However, via a convenient substitution one can reduce the number of equations and solve them numerically to obtain exact vortex solutions.