  We investigate the ground states of the Abelian Higgs model in AdS_4 with
various choices of parameters, and with no deformations in the ultraviolet
other than a chemical potential for the electric charge under the Abelian gauge
field. For W-shaped potentials with symmetry-breaking minima, an analysis of
infrared asymptotics suggests that the ground state has emergent conformal
symmetry in the infrared when the charge of the complex scalar is large enough.
But when this charge is too small, the likeliest ground state has Lifshitz-like
scaling in the infrared. For positive mass quadratic potentials, Lifshitz-like
scaling is the only possible infrared behavior for constant nonzero values of
the scalar. The approach to Lifshitz-like scaling is shown in many cases to be
oscillatory.
