Charged Gauss-Bonnet black holes with curvature induced scalarization in the extended scalar-tensor theories

Recently new scalarized black hole solutions were constructed in the extended scalar-tensor-Gauss-Bonnet gravity, where the scalar field is sourced by the curvature of the spacetime via the Gauss-Bonnet invariant. A natural extension of these results is to consider the case of nonzero black hole charge. In addition we have explored a large set of coupling functions between the Gauss-Bonnet invariant and the scalar field, that was not done until now even in the uncharged case, in order to understand better the behavior of the solutions and the deviations from pure general relativity. The results show that in the case of nonzero black hole charge two bifurcation points can exist - one at larger masses where the scalarized solutions bifurcated from the Reissner-Nordstrom one, and one at smaller masses where the scalar charge of the solutions decreases again to zero and the branch merges again with the GR one. All of the constructed scalarized branches do not reach an extremal limit. We have examined the entropy of the black holes with nontrivial scalar field and it turns out, that similar to the uncharged case, the fundamental branch which possesses scalar field with no nodes is thermodynamically favorable over the Reissner-Nordstrom one for the considered coupling functions, while the rest of the branches possessing scalar field with one or more zeros have lower entropy compared to the GR case and they are supposed to be unstable.