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 $d<p\le d+1$ 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 $0<p\le 1$ is also discussed). In $d\ge 2$ we prove, for $d<p\le d+1$, 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.