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Abstract

Abstract We study the instability of a charged scalar field in the near-horizon region of an extremal charged BTZ black hole. The extremal geometry contains an $$\textrm{AdS}_2\times S^1$$ AdS 2 × S 1 throat supported by a constant electric field, which lowers the effective scalar mass and can violate the $$\textrm{AdS}_2$$ AdS 2 Breitenlohner–Freedman bound. This identifies an infrared instability of the extremal normal state. We then construct a nearby finite-temperature branch of fully backreacted, source-free, node-free hairy black-hole solutions, thereby establishing the nonlinear continuation of the unstable mode near its onset. The scalar condensate modifies both the Maxwell field and the geometry, while stronger gravitational backreaction lowers the critical temperature and suppresses the condensate amplitude. Near the critical point, the condensate exhibits mean-field scaling. We also develop an effective near-horizon interpretation in terms of electric-flux screening. The exact zero-temperature endpoint of the hairy branch is not determined here and remains an open problem.