Scalarization of Bardeen spacetime

In this paper, we study the scalarization of the entire Bardeen spacetime which is constructed from a nonlinear magnetic monopole. We find that once the scalarization coupling parameter exceeds the scalarized threshold $a_t$, a scalarized Bardeen spacetime (SBS) solution exists for any magnetic charge $q$. However, the nature of the scalarization depends on the magnetic charge $q$. For $q$ less than a critical vaule $q_c$, when $a$ exceeds $a_t$, the scalar field emerges. As $a$ approaches $a_t$ from above (i.e., $a\to a_t^{+}$), the scalar field vanishes and the metric is reduced to a pure Bardeen spacetime. This behavior indicates that the solution exhibits the general spontaneous" scalarization phenomenon. Conversely, for $q \geq q_c$, in the limit $a \to a_t^+$, the scalar field is non-vanishing and the SBS approaches a frozen" SBS. Considering the importance of photon orbits in astronomical observations, we analyze the trajectories of photons around SBSs by analyzing null geodesics.