The Abrikosov vortex in curved space

Abstract

We study the self-gravitating Abrikosov vortex in curved space with and with-out a (negative) cosmological constant, considering both singular and non-singular solutions with an eye to hairy black holes. In the asymptotically flat case, we find that non-singular vortices round off the singularity of the point particle’s metric in 3 dimensions, whereas singular solutions consist of vortices holding a conical singularity at their core. There are no black hole vortex solutions. In the asymptotically AdS case, in addition to these solutions there exist singular solutions containing a BTZ black hole, but they are always hairless. So we find that in contrast with 4-dimensional ’t Hooft-Polyakov monopoles, which can be regarded as their higher-dimensional analogues, Abrikosov vortices cannot hold a black hole at their core. We also describe the implications of these results in the context of AdS/CFT and propose an interpretation for their CFT dual along the lines of the holographic superconductor.

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

“…Later, work was then extended to include singular vortex solutions besides non-singular ones [4]. Vortices with conical singularities were obtained in flat backgrounds and BTZ black hole solutions were obtained in curved backgrounds, though it was found that the vortex cannot ultimately hold a black hole at its core [4]. Our present paper introduces the non-minimal coupling term which is missing in all previous studies of gravitating vortices.…”

“…We work in natural units where = c = 1. Though our parameters and quantities such as the radius, mass and magnetic field are quoted as numbers, they should be thought of as having a unit attached to them (except for the winding number n which is a pure number) 4 . As we have seen, a negative Λ automatically ensures that the asymptotic cosmological constant Λ ef f will be negative.…”

“…We now place this paper in context, with a focus on previous studies of gravitating vortices that we referred to earlier [1][2][3][4]. It was recognized a long time ago that Einstein gravity in 2+1 dimensions yields a locally flat spacetime outside localized sources, albeit with the topology of a cone [7].…”

“…In this work, we consider the Nielsen-Olesen vortex, a 2 + 1-dimensional abelian Higgs model, non-minimally coupled to Einstein gravity with and without cosmological constant. Compared to previous work on the effects of gravity on vortices [1][2][3][4] the new ingredient in the action is the non-minimal coupling term ξ R |φ| 2 where R is the Ricci scalar, ξ is a dimensionless coupling constant and φ is a complex scalar field. When gravity is present, it is perfectly fitting to add this term to the action as it preserves the local U (1) gauge invariance of the vortex.…”

“…The mass of the vortex increased as the magnitude of the cosmological constant increased and led to a slightly smaller core for the vortex. Later, work was then extended to include singular vortex solutions besides non-singular ones [4]. Vortices with conical singularities were obtained in flat backgrounds and BTZ black hole solutions were obtained in curved backgrounds, though it was found that the vortex cannot ultimately hold a black hole at its core [4].…”