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

TL;DR

This paper introduces new black hole solutions with non-Abelian gauge fields in anti-de Sitter space, showing they are thermodynamically favored over Reissner-Nordström-AdS black holes and exhibit phase transitions and modified conductivity properties.

Contribution

It provides novel ${ m su}(N)$ Einstein-Yang-Mills black hole solutions with gauge hair and analyzes their thermodynamics and conductivity, extending previous models to larger gauge groups.

Findings

Hairy black holes have lower free energy than RN-AdS black holes.

A phase transition occurs at a critical temperature, favoring hairy black holes.

Increasing gauge group size removes divergence in conductivity at nonzero frequency.

Abstract

We present new planar dyonic black hole solutions of the ${\mathfrak {su}}(N)$ Einstein-Yang-Mills equations in asymptotically anti-de Sitter space-time, focussing on ${\mathfrak {su}}(2)$ and ${\mathfrak {su}}(3)$ 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-Nordstr\"om-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 ${\mathfrak {su}}(2)$ to ${\mathfrak {su}}(3)$ eliminates a divergence in the conductivity at nonzero frequency.