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We present new planar dyonic black hole solutions of the ${\mathfrak{su}}(N)$ Einstein-Yang-Mills equations in asymptotically anti-de Sitterspace-time, focussing on ${\mathfrak {su}}(2)$ and ${\mathfrak {su}}(3)$ gaugegroups. The magnetic part of the gauge field forms a condensate close to theplanar event horizon. We compare the free energy of a non-Abelian hairy blackhole with that of an embedded Reissner-Nordstr\"om-anti-de Sitter (RN-AdS)black hole having the same Hawking temperature and electric charge. We findthat the hairy black holes have lower free energy. We present evidence thatthere is a phase transition at a critical temperature, above which the onlysolutions are embedded RN-AdS black holes. At the critical temperature, anRN-AdS black hole can decay into a hairy black hole, and it isthermodynamically favourable to do so. Working in the probe limit, we computethe frequency-dependent conductivity, and find that enlarging the gauge groupfrom ${\mathfrak {su}}(2)$ to ${\mathfrak {su}}(3)$ eliminates a divergence inthe conductivity at nonzero frequency.

Black holes with ${\mathfrak {su}}(N)$ gauge field hair and superconducting horizons