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

Abstract

We consider the Nielsen-Olesen vortex non-minimally coupled to Einstein gravity with cosmological constant Λ. A non-minimal coupling term ξ R |φ| 2 is natural to add to the vortex as it preserves gauge-invariance (here R is the Ricci scalar and ξ 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 ξ. This leads to a novel feature: there is a critical coupling ξ c where the VEV is zero for ξ ≥ ξ c but becomes non-zero when ξ crosses below ξ 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 |ξ − ξ c | 1/2 . Therefore ξ 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 proportional to |T − T c | 1/2 near T c . The plot of the VEV as a function of ξ shows a clear discontinuity in the slope at ξ c and looks similar to plots of the order parameter versus temperature in GL theory. 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 ξ. However, the deficit angle of the asymptotic conical spacetime depends on ξ 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.

Non-BPS bubbling geometries in AdS3

“…The chain of rods is located at r = 0 and divided into different sections of θ delimited by the critical angles θ (i) c (5.8). The locus of each rod is given by 25 The metric can be written in the Weyl cylindrical coordinate system with (4.3), and the component along y can be written as…”

“…There are other interesting constructions as[24,25] in AdS3, but being constructed from Einstein gravity in three dimensions, they do not admit UV description within a quantum gravity theory.3 Note that some solutions in this paper require a KKm charge, k, such that the asymptotic isAdS3×S 3 /Z k ×T 4 .The holographic dictionary of such a system is much less understood than that of the D1-D5 system already at the level of BPS states 4. See[32][33][34][35][36][37][38] as a non-exhaustive list of techniques used to generate solutions in four-dimensional General Relativity in vacuum of with gauge fields.…”

Generalized Abelian Higgs model with analytical vortex solutions

“…We have obtained vortex solutions by solving equation ( 18) for r > 0 and using boundary conditions ( 10)-( 11) at r = 0. Alternatively, it is possible to extend (18) to the whole plane [22]: writing ϕ(r, θ) = e u(r) e inθ and substituting in (8), we can solve for the vector field out of the origin in the form…”

“…Out of the vortex centers, equation (8) gives the vector field components in terms of the modulus and phase of the scalar field:…”

“…The situation changes, however, when vortices are considered on a curved manifold, where the metric can possibly depend on the scalar field, instead of on the plane: in these cases several examples of integrable vortex equations have been found and classified, see [6] and references therein. From a different perspective, the Abelian Higgs model on a curved spacetime and with coupling to the gravitational field has been also thoroughly investigated, see [7] for a review or [8] for some recent new solutions obtained numerically.…”