  Analyzing the magnetic structure factor of a field demagnetized artificial
square ice, qualitative deviations from what would predict the square ice model
are observed. Combining micromagnetic and Monte Carlo simulations, we
demonstrate that these deviations signal the presence of interactions between
nanomagnets that extend beyond nearest neighbors. Including further neighbor,
dipolar-like couplings in the square ice model, we find that the first seven or
eight coupling strengths are needed to reproduce semi-quantitatively the main
features of the magnetic structure factor measured experimentally. An
alternative, more realistic numerical scenario is also proposed in which the
ice condition is slightly detuned. In that case as well, the features evidenced
in the experimental magnetic structure factor are only well-described when
further neighbor couplings are taken into account. Our results show that long
range dipolar interactions are not totally washed out in a field demagnetized
artificial square ice, and cannot be neglected as they impact the magnetic
correlations within or at the vicinity of the ice manifold.
