We study the structure and melting of a classical bilayer system of dipoles, in a setup where the dipoles are oriented perpendicular to the planes of the layers and the density of dipoles is the same in each layer. Due to the anisotropic character of
the dipole-dipole interactions, we find that the ground-state configuration is given by two hexagonal crystals positioned on top of each other, independent of the interlayer spacing and dipolar density. For large interlayer distances these crystals are independent, while in the opposite limit of small interlayer distances the system behaves as a two-dimensional crystal of paired dipoles. Within the harmonic approximation for the phonon excitations, the melting temperature of these crystalline configurations displays a non-monotonic dependence on the interlayer distance, which is associated with a re-entrant melting behavior in the form of solid-liquid-solid-liquid transitions at fixed temperature.
