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

TL;DR

This paper investigates the ground states of a d-dimensional Ising model with competing long-range dipolar and short-range ferromagnetic interactions, revealing phase transitions and the structure of ground states depending on interaction parameters.

Contribution

It provides rigorous analysis of the ground state configurations and energy asymptotics for the model with varying interaction decay and strength, including phase transition insights.

Findings

For p > d+1 and large J, the ground state is ferromagnetic.

For d < p ≤ d+1, the ground state is not ferromagnetic regardless of J.

In 1D, the ground state transitions from antiferromagnetic to periodic states as J increases.

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.