Inserted: 18 jul 2022
Last Updated: 18 jul 2022
We study the energy of a ferromagneticantiferromagnetic frustrated spin system with values on two disjoint circumferences of the 3-dimensional unit sphere in a one-dimensional and two-dimensional domain. It consists on the sum of a term that depends on the nearest and next-to-nearest interactions and a penalizing term that counts the spin's magnetic anisotropy transitions. We analyze the asymptotic behaviour of the energy, that is when the system is close to the helimagnetferromagnet transition point as the number of particles diverges. In the one-dimensional setting we compute the $\Gamma$-limit of renormalizations 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 the renormalization of the energy at second order, we we prove the emergence and study the geometric rigidity of chirality transitions.