We numerically construct asymptotically AdS4 solutions to Einstein-Maxwell-dilaton theory. These have a dipolar electrostatic potential turned on at the conformal boundary S^2\times \mathbb{R}_t. We find two classes of geometries: AdS soliton solutions that encode the full backreaction of the electric field on the AdS geometry without a horizon, and neutral black holes that are "polarised" by the dipolar potential. For a certain range of the electric field \mathcal{E}, we find two distinct branches of the AdS soliton that exist for the same value of \mathcal{E}. For the black hole, we find either two or four branches depending on the value of the electric field and horizon temperature. These branches meet at critical values of the electric field and impose a maximum value of \mathcal{E} that should be reflected in the dual field theory. For both the soliton and black hole geometries, we study boundary data such as the stress tensor. For the black hole, we also consider horizon observables such as the entropy. At finite temperature, we consider the Gibbs free energy for both phases and determine the phase transition between them. We find that the AdS soliton dominates at low temperature for an electric field up to the maximum value. Using the gauge/gravity duality, we propose that these solutions are dual to deformed ABJM theory and compute the corresponding weak coupling phase diagram.
