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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. Fixing $λ$ small, we find there are two critical values of the cosmological constant $Λ_{\rm cr (1)}(v)$ and $Λ_{\rm cr(2)}(v)$, which depend on $v$. If $Λ_{\rm cr(1)}(v)<Λ(<0)$, we find the first order transition, while if $Λ_{\rm cr(2)}(v)<Λ<Λ_{\rm cr(1)}(v)$, the transition becomes second order. For the case of $Λ_{b}(v)<Λ<Λ_{\rm (2)}(v)$, we again find the first order irreversible transition from the monopole black hole to the extreme Reissner-Nordström black hole. Beyond $Λ_{b}(v)$, no monopole black hole exists. We also discuss thermodynamical properties of the monopole black hole in a thermal bath system.