FM: 06-07, cond-mat/0604668

Author : A. Giuliani, E.H. Lieb, J.L. Lebowitz

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

Abstract: 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.

Keywords: ising model; long range interactions; periodic ground states.

Alessandro Giuliani,
Department of Physics, Princeton University, Princeton 08544 NJ, USA

Joel L. Lebowitz,
Department of Mathematics and Physics, Rutgers University, Piscataway, NJ 08854 USA.

Elliott H. Lieb,
Department of Mathematics and Physics, Princeton University, Princeton 08544 NJ, USA