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We study the spontaneous scalarization of spherically symmetric, static andasymptotically Anti-de Sitter (aAdS) black holes in a scalar-tensor gravitymodel with non-mininal coupling of the form $\phi^2\left(\alpha{\cal R} +\gamma {\cal G}\right)$, where $\alpha$ and $\gamma$ are constants, while${\cal R}$ and ${\cal G}$ are the Ricci scalar and Gauss-Bonnet term,respectively. Since these terms act as an effective ``mass'' for the scalarfield, non-trivial values of the scalar field in the black hole space-time arepossible for {\it a priori} vanishing scalar field mass. In particular, wedemonstrate that the scalarization of an aAdS black hole requires the curvatureinvariant $-\left(\alpha{\cal R} + \gamma {\cal G}\right)$ to drop below theBreitenlohner-Freedman bound close to the black hole horizon, while itasymptotes to a value well above the bound. The dimension of the dual operatoron the AdS boundary depends on the parameters $\alpha$ and $\gamma$ and wedemonstrate that -- for fixed operator dimension -- the expectation value ofthis dual operator increases with decreasing temperature of the black hole,i.e. of the dual field theory. When taking backreaction of the space-time intoaccount, we find that the scalarization of the black hole is the dualdescription of a phase transition in a strongly coupled quantum system, i.e.corresponds to a holographic phase transition. A possible application areliquid-gas quantum phase transitions, e.g. in $^4$He. Finally, we demonstratethat extremal black holes with $AdS_2\times S^2$ near-horizon geometry {\itcannot support regular scalar fields on the horizon} in the scalar-tensor modelstudied here.

Scalarization of asymptotically Anti-de Sitter black holes with applications to holographic phase transitions