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
giuliani@princeton.edu

Joel L. Lebowitz,
Department of Mathematics and Physics, Rutgers University, Piscataway, NJ 08854 USA.
lebowitz@math.rutgers.edu

Elliott H. Lieb,
Department of Mathematics and Physics, Princeton University, Princeton 08544 NJ, USA
lieb@princeton.edu