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. This demonstrates that the dyonic sector exhibits a qualitatively different domain of existence from the purely electric nonlinear scalarization case. This behavior can be traced to the fact that the presence of both electromagnetic charges introduces a factor Δ (φ ) Δ ( ϕ ) in the source term of scalar field equations that vanishes when the coupling function f(φ ) f ( ϕ ) equals the ratio of the charges for some value of the scalar field φ c ϕ c . The scalar field of extremal black holes assumes precisely this value at the horizon, φ H=φ c ϕ H = ϕ c . We demonstrate the plunge for the coupling function f(φ )=exp (α φ ³) f ( ϕ ) = exp ( α ϕ 3 ) which provides a minimal representative of nonlinear scalarization models.