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Quasinormal Modes of Charged Fermions and Phase Transition of Black Holes

We study the quasinormal modes of massless charged fermions in a Reissner-Nordstrom-anti-de Sitter black hole spacetime. In the probe limit, we find that the imaginary part of quasinormal frequency will become positive when the temperature of the black hole is below a critical value. This indicates an instability of the black hole occurs and a phase transition happens. In the AdS/CFT correspondence, this transition can be viewed as a superconducting phase transition and the bulk fermion is regarded as the order parameter. When the coupling of the fermions and the background electric field becomes stronger, the critical temperature of the phase transition becomes higher. If the interaction between the fermion and the electric field can be ignored, namely in the case of a neutral fermion, the imaginary part of the quasinormal modes is always negative, which indicates that the black hole is stable and no phase transition occurs.