  This paper contains three types of results:
  1. the construction of ground state solutions for a long-range Ising model
whose interfaces stay at a bounded distance from any given hyperplane,
  2. the construction of nonlocal minimal surfaces which stay at a bounded
distance from any given hyperplane,
  3. the reciprocal approximation of ground states for long-range Ising models
and nonlocal minimal surfaces.
  In particular, we establish the existence of ground state solutions for
long-range Ising models with planelike interfaces, which possess scale
invariant properties with respect to the periodicity size of the environment.
The range of interaction of the Hamiltonian is not necessarily assumed to be
finite and also polynomial tails are taken into account (i.e. particles can
interact even if they are very far apart the one from the other).
  In addition, we provide a rigorous bridge between the theory of long-range
Ising models and that of nonlocal minimal surfaces, via some precise limit
result.
