Preprint

Inserted: 20 dec 2024
Last Updated: 20 dec 2024

Year: 2024

Abstract:

We analyze 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 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 $S^1$ covering $S^2$ as $n\to\infty$. We identify a critical energy-scaling regime and a threshold for the divergence rate of $k_n\to+\infty$, below which the $\Gamma$-limit of the discrete energies capture chirality transitions while retaining an $S^2$-valued energy description in the continuum limit.

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