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

We analyze the discrete-to-continuum limit of a frustrated ferromagnetic/anti-ferromagnetic ${\mathbb{S}}^2$-valued spin system on the lattice ${\lambda }_n{\mathbb{Z}}^2$ as ${\lambda }_n\to 0$. For ${\mathbb{S}}^2$ spin systems close to the Landau-Lifschitz point (where the helimagnetic/ferromagnetic transition occurs), it is well established that for chirality transitions emerge with vanishing energy. Inspired by recent work on the $N$-clock model, we consider a spin model where spins are constrained to $k_n$ copies of ${\mathbb{S}}^1$ covering ${\mathbb{S}}^2$ as $n\to \mathrm{\infty }$. We identify a critical energy-scaling regime and a threshold for the divergence rate of $k_n\to +\mathrm{\infty }$, below which the $\mathrm{\Gamma }$-limit of the discrete energies capture chirality transitions while retaining an ${\mathbb{S}}^2$-valued energy description in the continuum limit.