Modulated phases in Ising systems with long-range antiferromagnetic and short-range ferromagnetic interactions
Andrea Braides, Fabrizio Caragiulo
TL;DR
The paper analyzes a one-dimensional spin system with competing short-range ferromagnetic and long-range antiferromagnetic interactions under periodic boundary conditions. It uses $\Gamma$-convergence to derive a sharp limit description of near-ground-state configurations, showing that they decompose into a finite set of modulated phases translations of the absolute ground state with interfacial defects, and that the limiting energy $F_{\infty}$ is a sum of defect energies over the jump set of a limit map $r$. The defect energy density $\phi(j)$ is defined as a limit of renormalized minimal energies and is shown to satisfy $\phi(0)=0$, $\phi(j)>0$ for nonzero $j$ mod $2h^{\star}$, and subadditivity, enabling a rigorous $\Gamma$-limit $F_{\infty}^j(r)=\sum_{x\in J(r)} \phi(\Delta r(x))$ on $r:\mathbb{T}\to \mathbb{Z}/2h^{\star}\mathbb{Z}$. This framework bridges the discrete lattice problem with a continuum-like partition energy, clarifying how texture-forming modulated phases emerge and how interfacial costs govern the coarse-grained behavior, with potential extensions to higher dimensions.
Abstract
We consider large spin systems with short-range ferromagnetic interactions and long-range antiferromagnetic interactions subjected to periodic boundary conditions which have been proved by Giuliani, Lebowitz and Lieb to have minimizers that tend to alternate groups of $1$ and $-1$ of the same length $h^\star$. We consider states with energy of the same order as that of minimizers and show that they consist of a finite number of modulated phases of the same form as minimizers with some interfacial defects. The analysis is carried out using the notation of Gamma-convergence by exhibiting an interfacial energy that describes the minimal defect energy between different modulated phases.