Cloud Screening of extremal charged BTZ black hole

TL;DR

This work analyzes how an extremal charged BTZ black hole's near-horizon $AdS_2$ region with a constant electric field can drive a BF-bound violation for a charged scalar, initiating Schwinger pair production and forming a static scalar cloud. The authors develop a consistent 2D effective theory by dimensional reduction, impose BF-violating boundary conditions, and show that the cloud backreacts to partially screen the electric flux, yielding a self-consistent, hairy extremal solution. They connect the onset of condensation to mixed (double-trace) boundary conditions and perform a zero-mode quantization that reveals discretized cloud charge and conditions for full or partial screening. Overall, the paper provides a concrete electric analogue of magnetic hairy black holes, highlighting how near-horizon infrared dynamics can generate and stabilize electric screening in extremal geometries, with implications for defect CFTs and IR criticality.

Abstract

We study the dynamics of a charged scalar field in the near-horizon region of an extremal charged BTZ black hole. The near-horizon geometry contains an AdS2 throat with a constant electric field, which lowers the effective mass of the scalar and can trigger a violation of the AdS2 Breitenlohner-Freedman bound. We show that this instability is resolved by the formation of a static scalar cloud supported by Schwinger pair production. The condensate backreacts on the gauge field and partially screens the electric flux, leading to a self-consistent stationary configuration. The scalar profile is obtained analytically from the near-horizon equations and exhibits the characteristic behavior of a BF-violating mode in AdS2. We analyze the associated boundary conditions, the induced charge density, and the resulting modification of the electric field. The resulting configuration can be interpreted as an electric analogue of known magnetic hairy black hole solutions. Our results provide a concrete realization of electric screening in extremal charged black holes and clarify the role of near-horizon dynamics in shaping the infrared structure of the solution.

Figures (3)

• Figure 1: Effective potential $V_{\text{eff}}(x)$ as a function of the radial coordinate $x$.
• Figure 2: Radial profiles of the scalar mode $\phi(r)$ (blue) and the electric field $E(r)\, r^{2}$ (orange), obtained from the nonlinear system in the near-horizon $\mathrm{AdS}_{2}$ region of the extremal charged BTZ geometry. The horizontal axis is the Poincaré radius $r$, where $r \to 0$ corresponds to the $\mathrm{AdS}_{2}$ boundary. For visibility, the electric field is normalized by its value at a UV reference point $r_{\mathrm{UV}}$. The three panels show increasing values of the IR electric field $E_{0}$, which is proportional to the black-hole charge $Q$. Larger $E_{0}$ enhances the BF-violating instability: the scalar cloud develops a larger log-oscillatory envelope near the horizon, and the electric field exhibits stronger suppression as $r \to 0$, reflecting classical screening by the condensate. Full screening requires the one-loop quantization analysis of the defect, discussed in Sec. \ref{['seclev:6']}.
• Figure 3: Normalized fluctuation flux $r^{2}E_{\mathrm{fluc}}^{(\mathrm{nor})}(r)$ associated with the gauge-field zero mode. The left panel shows the profile on a linear radial scale, while the right panel uses a logarithmic scale to resolve the near-boundary region. The negative dip at small $r$ reflects suppression of the electric field by the scalar cloud in the near-horizon (IR) region. The apparent zero of the flux at $r=r_{\mathrm{IR}}$ is a boundary artifact resulting from the imposed condition $a_{t}'(r_{\mathrm{IR}})=0$ and must not be interpreted as full screening. Physical screening is instead determined by the integrated cloud charge $q_{\mathrm{eff}}$, which remains finite, implying partial screening of the background BTZ electric field.