AdS Monopole Black Hole and Phase Transition

Abstract

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 $\Lambda$ 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 ${\Lambda }_{\mathrm{cr}(1)}(v)$ and ${\Lambda }_{\mathrm{cr}(2)}(v)$, which depend on $v$. If ${\Lambda }_{\mathrm{cr}(1)}(v)<\Lambda (<0)$, we find the first order transition, while if ${\Lambda }_{\mathrm{cr}(2)}(v)<\Lambda <{\Lambda }_{\mathrm{cr}(1)}(v)$, the transition becomes second order. For the case of ${\Lambda }_b(v)<\Lambda <{\Lambda }_{(2)}(v)$, we again find the first order irreversible transition from the monopole black hole to the extreme Reissner-Nordström black hole. Beyond ${\Lambda }_b(v)$, no monopole black hole exists. We also discuss thermodynamical properties of the monopole black hole in a thermal bath system.

arXiv comment16 pages, 34 figures

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