AdS monopole black hole and phase transition

We study the Einstein-SO(3)Yang-Mills-Higgs system with a negative cosmological constant and find the monopole black hole solutions as well as the trivial Reissner-Nordström black hole. We discuss thermodynamical stability of the monopole black hole in an isolated system. We expect a phase transition between those two black holes when the mass of a black hole increases or decreases. The type of phase transition depends on the cosmological constant Λ, as well as the vacuum expectation value v and the coupling constant λ of the Higgs field. Keeping λ small, we find there are two critical values of the cosmological constant, Λcr(1)(v) and Λcr(2)(v), which depend on v. If Λcr(1)(v)<Λ(<0), we find the first order transition, whereas if Λcr(2)(v)<Λ<Λcr(1)(v), the transition becomes second order. For the case of Λb(v)<Λ<Λ(2)(v), we again find the first order irreversible transition from the monopole black hole to the extreme Reissner-Nordström black hole. Beyond Λb(v), no monopole black hole exists. We also discuss thermodynamical properties of the monopole black hole in a thermal bath system.