Black holes with mathfrak{s}mathfrak{u}(N) gauge field hair and superconducting horizons

Abstract

We present new planar dyonic black hole solutions of the mathfrak{s}mathfrak{u}(N) Einstein-Yang-Mills equations in asymptotically anti-de Sitter space-time, focussing on mathfrak{s}mathfrak{u}(2) and mathfrak{s}mathfrak{u}(3) gauge groups. The magnetic part of the gauge field forms a condensate close to the planar event horizon. We compare the free energy of a non-Abelian hairy black hole with that of an embedded Reissner-Nordström-anti-de Sitter (RN-AdS) black hole having the same Hawking temperature and electric charge. We find that the hairy black holes have lower free energy. We present evidence that there is a phase transition at a critical temperature, above which the only solutions are embedded RN-AdS black holes. At the critical temperature, an RN-AdS black hole can decay into a hairy black hole, and it is thermodynamically favourable to do so. Working in the probe limit, we compute the frequency-dependent conductivity, and find that enlarging the gauge group from mathfrak{s}mathfrak{u}(2) to mathfrak{s}mathfrak{u}(3) eliminates a divergence in the conductivity at nonzero frequency.