Periodic striped ground states in Ising models with competing interactions

TL;DR

This paper characterizes the ground states of 2D and 3D Ising models with competing short-range ferromagnetic and long-range antiferromagnetic interactions, showing that stripe and slab patterns are energetically favored near critical interaction ratios.

Contribution

It provides a rigorous proof that stripe and slab ground states are stable in certain parameter regimes for models with power-law decaying long-range interactions.

Findings

Stripe and slab states are ground states near the critical ratio J_c(p).

Ground states are characterized for decay exponent p > 2d.

The proof uses localization bounds and reflection positivity.

Abstract

We consider Ising models in two and three dimensions, with short range ferromagnetic and long range, power-law decaying, antiferromagnetic interactions. We let $J$ be the ratio between the strength of the ferromagnetic to antiferromagnetic interactions. The competition between these two kinds of interactions induces the system to form domains of minus spins in a background of plus spins, or vice versa. If the decay exponent $p$ of the long range interaction is larger than $d+1$, with $d$ the space dimension, this happens for all values of $J$ smaller than a critical value $J_c(p)$, beyond which the ground state is homogeneous. In this paper, we give a characterization of the infinite volume ground states of the system, for $p>2d$ and $J$ in a left neighborhood of $J_c(p)$. In particular, we prove that the quasi-one-dimensional states consisting of infinite stripes ($d=2$) or slabs ($d=3$), all of the same optimal width and orientation, and alternating magnetization, are infinite volume ground states. Our proof is based on localization bounds combined with reflection positivity.