The simplest model of a scalarized black hole in the Einstein-Klein-Gordon theory

We investigate scalarized black holes in the Einstein-minimally coupled scalar theory with a negative potential $V(\phi)=-\alpha^2\phi^6$. The tachyonic instability is absent from analyzing the linearized scalar equation, which could not allow for spontaneous scalarization. However, we obtain the black hole solutions with scalar hair by solving three full equations because this scalar potential violates the weak energy condition. This shows clearly that scalarized black holes can be obtained without introducing a non-minimal scalar coupling term. We perform the stability analysis for scalarized black holes by adopting radial perturbations, implying that all scalarized black holes belonging to a single branch are unstable.