Dyonic Einstein-Maxwell-scalar black holes: the cold, the hot and the plunge

We investigate dyonic nonlinearly scalarized black holes in Einstein-Maxwell-scalar theory. The domain of existence of scalarized dyonic black holes consists of three branches. The cold branch and the hot branch bifurcate at a minimal value of the charge, analogous to the purely electrically charged scalarized black holes. However, the presence of both charges allows for regular extremal black holes, leading to a third branch featuring a sudden plunge in Hawking temperature. In fact, the presence of both electromagnetic charges introduces a factor $\Delta(\phi)$ in the source term of scalar field equations that vanishes when the coupling function $f(\phi)$ equals the ratio of the charges for some value of the scalar field $\phi_c$. The scalar field of extremal black holes assumes precisely this value at the horizon, $\phi_H=\phi_c$. We demonstrate the plunge for the coupling function $f(\phi)=\exp(\alpha \phi^3)$.