06-137 Alessandro Giuliani, Joel L. Lebowitz, Elliott H. Lieb
Ising models with long--range dipolar and short range ferromagnetic interactions (94K, LaTeX) Apr 28, 06
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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.

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