Charged black holes with axionic-type couplings: classes of solutions and dynamical scalarisation

We consider an augmented Einstein-Maxwell-scalar model including an axionic-type coupling between the scalar and electromagnetic field. We study dyonic black hole solutions in this model. For the canonical axionic coupling emerging from high energy physics, all charged black holes have axion hair. We present their domain of existence and investigate some physical properties. For other axionic-type couplings, two classes of black hole solutions may co-exist in the model: scalar-free Reissner-Nordstr\"om black holes and scalarised black holes. We show that in some region of the parameter space, the scalar-free solutions are unstable. Then, there is non-uniqueness since new scalarised black hole solutions with the same global charges also exist, which are entropically preferred over the scalar-free solutions and, moreover, emerge dynamically from the instability of the former.