Compactified holographic conformal order

Abstract We study holographic conformal order compactified on S 3 . The corresponding boundary CFT4 has a thermal phase with a nonzero expectation value of a certain operator. The gravitational dual to the ordered phase is represented by a black hole in asymptotically A d S 5 that violates the no-hair theorem. While the compactification does not destroy the ordered phase, it does not cure its perturbative instability: we identify the scalar channel QNM of the hairy black hole with Im [ w ] > 0 . On the contrary, we argue that the disordered thermal phase of the boundary CFT is perturbatively stable in holographic models of Einstein gravity and scalars.