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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2412.15994 |
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| _version_ | 1866929642610360320 |
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| author | Cicalese, Marco Reggiani, Dario Solombrino, Francesco |
| author_facet | Cicalese, Marco Reggiani, Dario Solombrino, Francesco |
| contents | We analyze the discrete-to-continuum limit of a frustrated ferromagnetic/anti-ferromagnetic $\mathbb{S}^2$-valued spin system on the lattice $λ_n\mathbb{Z}^2$ as $λ_n\to 0$. 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 $Γ$-limit of the discrete energies capture chirality transitions while retaining an $\mathbb{S}^2$-valued energy description in the continuum limit. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_15994 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | From discrete to continuum in the helical XY-model: emergence of chirality transitions in the $S^1$ to $S^2$ limit Cicalese, Marco Reggiani, Dario Solombrino, Francesco Analysis of PDEs We analyze the discrete-to-continuum limit of a frustrated ferromagnetic/anti-ferromagnetic $\mathbb{S}^2$-valued spin system on the lattice $λ_n\mathbb{Z}^2$ as $λ_n\to 0$. 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 $Γ$-limit of the discrete energies capture chirality transitions while retaining an $\mathbb{S}^2$-valued energy description in the continuum limit. |
| title | From discrete to continuum in the helical XY-model: emergence of chirality transitions in the $S^1$ to $S^2$ limit |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2412.15994 |