Ising models with long-range dipolar and short range ferromagnetic interactions

We study the ground state of a d-dimensional Ising model with both long-range (dipole-like) and nearest-neighbor ferromagnetic (FM) interactions. The long-range interaction is equal to r−p, p>d, while the FM interaction has strength J. If p>d+1 and J is large enough the ground state is FM, while if d<p⩽d+1 the FM state is not the ground state for any choice of J. In d=1 we show that for any p>1 the ground state has a series of transitions from an antiferromagnetic state of period 2 to 2h-periodic states of blocks of sizes h with alternating sign, the size h growing when the FM interaction strength J is increased (a generalization of this result to the case 0<p⩽1 is also discussed). In d⩾2 we prove, for d<p⩽d+1, that the dominant asymptotic behavior of the ground-state energy agrees for large J with that obtained from a periodic striped state conjectured to be the true ground state. The geometry of contours in the ground state is discussed.