Negative potential-induced scalarization in the Einstein-Euler-Heisenberg black hole

We investigate a negative potential-induced scalarization of the Einstein-Euler-Heisenberg (EEH) black hole in the EEH-scalar (EEHS) theory, characterized by mass $M$, Euler-Heisenberg parameter $\mu$, and magnetic charge $q$. Within this framework, the charge $q$ can exceed the extremal bound $q/M > 1$, and a single event horizon is maintained provided the parameter $\mu$ exceeds the $\mu_{\text{max}} = 0.019$, with the ADM mass fixed at $M = 1/2$. We obtain a single branch of scalarized EEH (sEEH) black holes for $q > 0$ which is considered as the simplest model for scalarization of EEH black holes. We found that this class of hairy black holes is not thermodynamically favored, and their quasinormal modes indicate they are dynamically unstable. An interesting feature is that when $q < 1/2$, the scalar charge varies only slightly with $q$ for a fixed mass. In contrast, for $q>1/2$, the scalar charge increases more rapidly as $q$ increases. This distinct behavior suggests that the scalar charge exhibits the characteristics of a primary charge for $q < 1/2$, and of a secondary charge for $q > 1/2$. This finding reveals notable features of hairy black holes in EEH theory, specifically in the overcharging regime.