From discrete to continuum in the helical XY-model: emergence of chirality transitions in the $S^1$ to $S^2$ limit
| Autor: | Cicalese, Marco, Reggiani, Dario, Solombrino, Francesco |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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 $n\to\infty$. 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 $\mathbb{S}^2$-valued energy description in the continuum limit. Comment: 33 pages, 2 figures |
| Databáze: | arXiv |
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