Hovering Black Holes from Charged Defects

TL;DR

We study localized electrically charged defects in a strongly coupled 2+1D CFT via a spatially varying boundary chemical potential $\mu(r)$ and construct the holographic duals in $AdS_4$ Einstein–Maxwell theory. The analysis yields an analytic marginal defect with an $AdS_2\times S^1$ IR geometry and a delta-function defect charge, and shows that sufficiently localized defects with large amplitude $a$ nucleate hovering extremal charged black holes—a second-order quantum phase transition with a universal defect entropy scaling. Numerical solutions confirm the IR behavior and reveal hovering BH regimes across multiple boundary profiles, with near-horizon physics matching extremal RN–AdS and a RN–AdS entropy–charge relation. The results illuminate mid-infrared physics around defects in strongly coupled CFTs and motivate future work on defect lattices, finite-temperature phases, and possible analytic connections to charged C-metric solutions.

Abstract

We construct the holographic dual of an electrically charged, localised defect in a conformal field theory at strong coupling, by applying a spatially dependent chemical potential. We find that the IR behaviour of the spacetime depends on the spatial falloff of the potential. Moreover, for sufficiently localized defects with large amplitude, we find that a new gravitational phenomenon occurs: a spherical extremal charged black hole nucleates in the bulk: a hovering black hole. This is a second order quantum phase transition. We construct this new phase with several profiles for the chemical potential and study its properties. We find an apparently universal behaviour for the entropy of the defect as a function of its amplitude. We comment on the possible field theory implications of our results.

Figures (18)

• Figure 1: Entropy of the hovering black hole as a function of $a/a_{\star}$ for several boundary profiles. The different symbols, which are labeled on the right, indicate the various profiles we have considered. One of the profiles has an additional parameter which here is $b=0.075$. (Here and in the remainder of the paper, we make plots in units of the AdS length $L=1$.)
• Figure 2: Sketches for the two coordinate systems (left) Eq. (\ref{['eq:poincare']}) and (right) Eq. (\ref{['eq:polar']}).
• Figure 3: Charge $Q$ on defect as a function of applied electric field $a_{{\lambda}}$. The lower branch of solutions is continuously connected to vacuum $AdS_4$ with zero charge and applied field.
• Figure 4: Regulated defect entropy for the two branches of the point-charge solution as a function of defect charge $Q$. Note the existence of a maximum charge.
• Figure 5: Domain of integration for hovering black hole solutions.