| Abstract: |
| We study the energy per particle of a ferromagnetic-anti-ferromagnetic frustrated spin chain with nearest and next-to-nearest interactions close to the Landau-Lifschitz point (where the helimagnetic-ferromagnetic transition occurs) , as the number of particles diverges. We rigorously prove the emergence of chiral ground states and we compute, by performing the $\Gamma$-limits of proper renormalizations and scalings, the energy for a chirality transition, if spins take value in the unit sphere $S^1$. Such a result does not hold if spins are $S^2$ valued, as in this case , as it is well established that in this case chirality transitions may emerge with vanishing energy. Inspired by recent work on the
N-clock model, we consider a spin model where spins are constrained to a diverging number $k_n$ of copies of $S^1$ covering $S^2$. We identify a critical energy-scaling regime and a threshold for the divergence rate of $k_n$, 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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