From discrete to continuum in the helical $XY$ model: emergence of chirality transitions in the $S^1$ to $S^2$ limit

Abstract

We study the discrete-to-continuum limit of a frustrated ferromagnetic/anti-ferromagnetic $S^2$-valued spin system on the lattice ${\lambda }_n{\mathbb{Z}}^2$ as ${\lambda }_n\to 0$. For $S^2$ spin systems near the Landau–Lifschitz point (where the helimagnetic/ferromagnetic transition occurs) it is well known that chirality transitions emerge with vanishing energy. Inspired by recent advances on the $N$-clock model, we consider a spin system in which the spins are constrained to $k_n$ copies of $S^1$ covering $S^2$ as $n\to \infty$. We identify a critical energy-scaling regime and a threshold on the divergence rate of $k_n\to +\infty$, below which the $\Gamma$-limit of the discrete energies captures chirality transitions while preserving an $S^2$-valued energy description in the continuum limit. To achieve this, we establish a connection with the variational analysis of a discrete approximation of a vector-valued Modica–Mortola-type functional, where the $k_n$ disconnected wells converge in the Hausdorff sense to a connected set as $n\to \infty$.