Scalarization of asymptotically anti–de Sitter black holes with applications to holographic phase transitions

Abstract

We study the spontaneous scalarization of spherically symmetric, static and asymptotically anti--de Sitter (aAdS) black holes in a scalar-tensor gravity model with nonminimal coupling of the form ${\ensuremath{\phi}}^{2}(\ensuremath{\alpha}\mathcal{R}+\ensuremath{\gamma}\mathcal{G})$, where $\ensuremath{\alpha}$ and $\ensuremath{\gamma}$ are constants, while $\mathcal{R}$ and $\mathcal{G}$ are the Ricci scalar and Gauss-Bonnet term, respectively. Since these terms act as an effective ``mass'' for the scalar field, nontrivial values of the scalar field in the black hole space-time are possible for a priori vanishing scalar field mass. In particular, we demonstrate that the scalarization of an aAdS black hole requires the curvature invariant $\ensuremath{-}(\ensuremath{\alpha}\mathcal{R}+\ensuremath{\gamma}\mathcal{G})$ to drop below the Breitenlohner-Freedman bound close to the black hole horizon, while it asymptotes to a value well above the bound. The dimension of the dual operator on the AdS boundary depends on the parameters $\ensuremath{\alpha}$ and $\ensuremath{\gamma}$ and we demonstrate that---for fixed operator dimension---the expectation value of this dual operator increases with decreasing temperature of the black hole, i.e., of the dual field theory. When taking backreaction of the space-time into account, we find that the scalarization of the black hole is the dual description of a phase transition in a strongly coupled quantum system, i.e., corresponds to a holographic phase transition. A possible application are liquid-gas quantum phase transitions, e.g., in $^{4}\mathrm{He}$. Finally, we demonstrate that extremal black holes with ${\mathrm{AdS}}_{2}\ifmmode\times\else\texttimes\fi{}{S}^{2}$ near-horizon geometry cannot support regular scalar fields on the horizon in the scalar-tensor model studied here.

Topics & Concepts