Charge-dependent scalarization of Einstein- Euler-Heisenberg black holes

Charge-dependent scalarization of the Einstein-Euler-Heisenberg (EEH) black hole is carried out in the EEH-scalar theory by introducing an exponential scalar coupling with $\alpha$ coupling constant to the Maxwell and nonlinear electrodynamic terms. The bald black hole (EEHBH) is described by mass $M$ and arbitrary magnetic charge $q$ and has a single horizon when choosing the action parameter $\mu=0.3$. The spontaneous scalarization ($\alpha^+$) of this black hole is available for charge $0<q< q_c=1.115$ and positive $\alpha$, whereas its new scalarization ($\alpha^-$) occurs for $q> q_c$ and negative $\alpha$. The former case of $q=0.5$ implies infinite branches of scalarized EEHBHs but its fundamental branch ($n=0$) is stable against radial perturbations, while the latter cases of $q=2,20$ show two stable single branches of scalarized EEHBHs.