  In the present Letter we discuss the origin of the vertex population
inversion observed experimentally in the mediated Artificial Square Ice.
  An interaction modifier is a disc-shaped magnetic nanoisland (a dot) which is
placed at the center of a vertex to mediate the interaction between the nearby
islands. We show that the inversion is of entropic origin, and can be explained
via the renormalization of the vertex configuration energies due to local
interaction between the nanoislands and the dot. We show it in a model with
mixed Heisenberg and Ising spins (a spin-dot interaction) that as a function of
the island size, entropic effects become important. Because of the
renormalization of the energies, we observe a level crossing between
\textrm{Type I} and \textrm{Type II} vertices which is qualitatively similar to
the experimental results. We also discuss possible implications of spin-dot
interactions in the eight- and sixteen- vertex models phase diagrams, using
both the exact order parameter in the former case and the numerically inferred
one in the latter.
