  We consider the Ising model in a transverse field with long-range
antiferromagnetic interactions that decay as a power law with their distance.
We study both the phase diagram and the entanglement properties as a function
of the exponent of the interaction. The phase diagram can be used as a guide
for future experiments with trapped ions. We find two gapped phases, one
dominated by the transverse field, exhibiting quasi long range order, and one
dominated by the long range interaction, with long range N\'eel ordered ground
states. We determine the location of the quantum critical points separating
those two phases. We determine their critical exponents and central-charges. In
the phase with quasi long range order the ground states exhibit exotic
corrections to the area law for the entanglement entropy coexisting with gapped
entanglement spectra.
