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-Nordstr"{o}m 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 $\mathrm{\Lambda }$ as well as the vacuum expectation value $v$ and the coupling constant $\lambda$ of the Higgs field. Fixing $\lambda$ small, we find there are two critical values of the cosmological constant $\Lambda_{\rm cr (1)}(v)$and$\Lambda_{\rm cr(2)}(v)$,whichdependon$v$\mathrm{.}If$\Lambda_{\rm cr(1)}(v)<\Lambda (<0)$,wefindthefirstordertransition,whileif$\Lambda_{\rm cr(2)}(v)<\Lambda<\Lambda_{\rm cr(1)}(v)$, the transition becomes second order. For the case of ${\mathrm{\Lambda }}_b(v)<\mathrm{\Lambda }<{\mathrm{\Lambda }}_{(2)}(v)$, we again find the first order irreversible transition from the monopole black hole to the extreme Reissner-Nordstr"{o}m black hole. Beyond ${\mathrm{\Lambda }}_b(v)$, no monopole black hole exists. We also discuss thermodynamical properties of the monopole black hole in a thermal bath system.