Nonminimally coupled gravitating vortex and critical coupling in AdS_3

We consider the Nielsen-Olesen vortex nonminimally coupled to Einstein gravity with cosmological constant $\Lambda$. The nonminimal coupling term $\xi\,R\,|\phi|^2$ where $\xi$ is a dimensionless coupling constant and $R$ is the Ricci scalar, plays a dual role: it contributes to the potential of the scalar field $\phi$ and to the Einstein-Hilbert term for gravity. 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 the Ginzburg-Landau (GL) mean-field theory of second-order phase transitions. The critical coupling exists only in an AdS$_3$ background; it does not exist in asymptotically flat spacetime (topologically a cone). 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.