Final fate of instability of Reissner-Nordström-anti-de Sitter black holes by charged complex scalar fields

We investigate instability of four-dimensional Reissner-Nordström- anti-de Sitter (RN-AdS4) black holes with various topologies by charged scalar field perturbations. We numerically find that the RN-AdS4 black holes become unstable against the linear perturbations below a critical temperature. It is analytically shown that charge extraction from the black holes occurs during the unstable evolution. To explore the end state of the instability, we perturbatively construct static black hole solutions with the scalar hair near the critical temperature. It is numerically found that the entropy of the hairy black hole is always larger than the one of the unstable RN-AdS4 black hole in the microcanonical ensemble. Our results support the speculation that the black hole with charged scalar hair always appears as the final fate of the instability of the RN-AdS4 black hole.