Newly scalarization of the Einstein-Euler-Heisenberg black hole

Th spontaneous scalarization of the Einstein-Euler-Heisenberg (EEH) black hole is performed in the EEH-scalar theory by introducing an exponential scalar coupling (with $\alpha$ coupling constant) to the Maxwell term.Here, the EEH black hole as a blad black hole is described by mass $M$ and magnetic charge $q$ with an action parameter $\mu$. A choice of $\mu =0.3$ gurantees a single horizon with unrestricted magnetic charge $q$. The onset scalarization of this black hole appears for a positive coupling $\alpha$ with an unlimited magnetic charge $q$. However, there exists a difference between $q\le 1$ and $q>1$ onset scalarizations. We notify the presence of infinite branches labeled by the number of $n=0,1,2,\cdots$ of scalarized charged black holes by taking into account the scalar seeds around the EEH black hole. We find that the $n=0$ fundamental branch of all scalarized black holes is stable against the radial perturbations, while the $n=1$ excited branch is unstable.