We study the energy of a ferromagnetic/antiferromagnetic frustrated spin system where the spin takes values on two disjoint circles of the 2-dimensional unit sphere. This analysis will be carried out first on a one-dimensional lattice and then on a two-dimensional lattice. The energy consists of the sum of a term that depends on nearest and next-to-nearest interactions and a penalizing term related to the spins' magnetic anisotropy transitions. We analyze the asymptotic behaviour of the energy, that is when the system is close to the helimagnet/ferromagnet transition point as the number of particles diverges. In the one-dimensional setting we compute the $ \Gamma $-limit of scalings of the energy at first and second order. As a result, it is shown how much energy the system spends for any magnetic anistropy transition and chirality transition. In the two-dimensional setting, by computing the $ \Gamma $-limit of a scaling of the energy, we study the geometric rigidity of chirality transitions.
Citation: Andrea Kubin, Lorenzo Lamberti. Variational analysis in one and two dimensions of a frustrated spin system: chirality and magnetic anisotropy transitions[J]. Mathematics in Engineering, 2023, 5(6): 1-37. doi: 10.3934/mine.2023094
We study the energy of a ferromagnetic/antiferromagnetic frustrated spin system where the spin takes values on two disjoint circles of the 2-dimensional unit sphere. This analysis will be carried out first on a one-dimensional lattice and then on a two-dimensional lattice. The energy consists of the sum of a term that depends on nearest and next-to-nearest interactions and a penalizing term related to the spins' magnetic anisotropy transitions. We analyze the asymptotic behaviour of the energy, that is when the system is close to the helimagnet/ferromagnet transition point as the number of particles diverges. In the one-dimensional setting we compute the $ \Gamma $-limit of scalings of the energy at first and second order. As a result, it is shown how much energy the system spends for any magnetic anistropy transition and chirality transition. In the two-dimensional setting, by computing the $ \Gamma $-limit of a scaling of the energy, we study the geometric rigidity of chirality transitions.
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Andrea Kubin, Lorenzo Lamberti. Variational analysis in one and two dimensions of a frustrated spin system: chirality and magnetic anisotropy transitions[J]. Mathematics in Engineering, 2023, 5(6): 1-37. doi: 10.3934/mine.2023094
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