Nonminimally coupled gravitating vortex: Phase transition at critical coupling<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>ξ</mml:mi><mml:mi>c</mml:mi></mml:msub></mml:math>in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow><mml:mi>AdS</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math>
Ariel Edery
Abstract
We consider the Nielsen-Olesen vortex nonminimally coupled to Einstein gravity with a cosmological constant $\mathrm{\ensuremath{\Lambda}}$. A nonminimal coupling term $\ensuremath{\xi}R|\ensuremath{\phi}{|}^{2}$ is natural to add to the vortex as it preserves gauge invariance (here $R$ is the Ricci scalar and $\ensuremath{\xi}$ 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 ${\mathrm{AdS}}_{3}$ background depend on $\ensuremath{\xi}$. This leads to a novel feature: There is a critical coupling ${\ensuremath{\xi}}_{c}$ where the VEV is zero for $\ensuremath{\xi}\ensuremath{\ge}{\ensuremath{\xi}}_{c}$ but becomes nonzero when $\ensuremath{\xi}$ crosses below ${\ensuremath{\xi}}_{c}$ and the gauge symmetry is spontaneously broken. Moreover, we show that the VEV near the critical coupling has a power-law behavior proportional to $|\ensuremath{\xi}\ensuremath{-}{\ensuremath{\xi}}_{c}{|}^{1/2}$. Therefore, ${\ensuremath{\xi}}_{c}$ 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 behavior proportional to $|T\ensuremath{-}{T}_{c}{|}^{1/2}$ near ${T}_{c}$. The plot of the VEV as a function of $\ensuremath{\xi}$ shows a clear discontinuity in the slope at ${\ensuremath{\xi}}_{c}$ and looks similar to plots of the order parameter versus temperature in GL theory. The critical coupling exists only in an ${\mathrm{AdS}}_{3}$ background; it does not exist in asymptotically flat spacetime (topologically a cone) where the VEV remains at a fixed nonzero value independent of $\ensuremath{\xi}$. However, the deficit angle of the asymptotic conical spacetime depends on $\ensuremath{\xi}$ 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 nonminimal coupling term present. However, via a convenient substitution, one can reduce the number of equations and solve them numerically to obtain exact vortex solutions.