Variational analysis in one and two dimensions of a frustrated spin system: chirality transitions and magnetic anisotropic transitions

TL;DR

This paper investigates the asymptotic behavior of a frustrated spin system's energy in one and two dimensions, focusing on transitions related to magnetic anisotropy and chirality, using Gamma-convergence analysis.

Contribution

It provides the first detailed Gamma-limit analysis of the energy for frustrated spin systems in both one and two dimensions, revealing the energetic costs of transitions.

Findings

Quantifies energy costs for magnetic anisotropy and chirality transitions.

Derives Gamma-limits for the energy in 1D and 2D settings.

Shows geometric rigidity of chirality transitions in 2D.

Abstract

We study the energy of a ferromagnetic/antiferromagnetic 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 helimagnet/ferromagnet 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.