Variational analysis of the $J_1$-$J_2$-$J_3$ model: a non-linear lattice version of the Aviles-Giga functional

TL;DR

This paper analyzes the variational limit of a frustrated spin model on a square lattice, revealing its asymptotic behavior is governed by a non-linear perturbation of the Aviles-Giga functional through $ ext{Gamma}$-convergence.

Contribution

It introduces a novel $ ext{Gamma}$-convergence analysis of the $J_1$-$J_2$-$J_3$ model near a critical transition, linking it to a non-linear lattice version of the Aviles-Giga energy.

Findings

Characterizes the optimal cost of chirality transitions in $BV$.

Shows the system is asymptotically driven by a non-linear perturbation of the Aviles-Giga energy.

Provides a discrete $ ext{Gamma}$-convergence framework for the model.

Abstract

We study the variational limit of the frustrated $J_1$-$J_2$-$J_3$ spin model on the square lattice in the vicinity of the ferromagnet/helimagnet transition point as the lattice spacing vanishes. We carry out the $\Gamma$-convergence analysis of proper scalings of the energy and we characterize the optimal cost of a chirality transition in $BV$ proving that the system is asymptotically driven by a discrete version of a non-linear perturbation of the Aviles-Giga energy functional.