Hairy AdS black holes with a toroidal horizon in 4D Einstein-nonlinear $\sigma $-model system

TL;DR

This paper constructs an exact asymptotically AdS black hole solution with scalar hair in a 4D Einstein-nonlinear sigma model, analyzing its properties and thermodynamic favorability over non-hairy black holes.

Contribution

It presents a novel exact hairy black hole solution in 4D Einstein-nonlinear sigma models with a flat horizon and analyzes its thermodynamics.

Findings

Hairy black hole is thermodynamically favored over non-hairy counterparts.

Solution depends on a discrete parameter without topological or Noether charge.

Inclusion of a U(1) gauge field is possible.

Abstract

An exact hairy asymptotically locally AdS black hole solution with a flat horizon in the Einstein-nonlinear sigma model system in (3+1) dimensions is constructed. The ansatz for the nonlinear $SU(2)$ field is regular everywhere and depends explicitly on Killing coordinates, but in such a way that its energy-momentum tensor is compatible with a metric with Killing fields. The solution is characterized by a discrete parameter which has neither topological nor Noether charge associated with it and therefore represents a hair. A $U(1)$ gauge field interacting with Einstein gravity can also be included. The thermodynamics is analyzed. Interestingly, the hairy black hole is always thermodynamically favored with respect to the corresponding black hole with vanishing Pionic field.