Two-Dimensional Ising Model with Competing Interactions as a Model for Interacting π-Rings

We investigate the Ising model on a square lattice with antiferromagnetic exchange between nearest and next-nearest neighbors and show that at low temperatures stripe-like and droplet-like superstructures appear. We show that the competing interactions introduce a strong frustration that could plausibly describe the systems with dipole-dipole type interactions and pay particular attention to arrays of interacting Josephson π-rings.