Spontaneous Scalarization of Brane Black Holes: Quantum-Enhanced Tachyonic Instabilities

We investigate the spontaneous scalarization of brane-localized charged black holes, focusing on the role of quantum-enhanced tachyonic instabilities. By solving the coupled bulk-brane equations numerically within the Einstein-DeTurck formulation, we construct fully backreacted static black hole solutions and map the system from the semi-classical limit to the strongly coupled quantum-dominated regime. We demonstrate that the holographic quantum effects, parameterized by $\kappa$, significantly modify the background geometry. Stability analysis, conducted via both effective potential diagnostics and quasinormal mode calculations, confirms that this geometric deformation deepens the negative well of the effective potential, thereby triggering a tachyonic instability. Integrating these results, we establish the global phase diagram of spontaneous scalarization in the $T_h/\mu-\kappa$ parameter space. The results reveal that these quantum-enhanced instabilities promote scalarization by substantially raising the critical temperature, providing a clear quantitative signature of quantum effects on black hole hair formation.