From discrete to continuum in the helical $XY$ model: emergence of chirality transitions in the $S^{1}$ to $S^{2}$ limit

We study the discrete-to-continuum limit of a frustrated ferromagnetic/anti-ferromagnetic S^{2} -valued spin system on the lattice \lambda_{n} \mathbb{Z}^{2} as \lambda_{n} \to 0 . For S^{2} spin systems near the Landau–Lifschitz point (where the helimagnetic/ferromagnetic transition occurs) it is well known that chirality transitions emerge with vanishing energy. Inspired by recent advances on the N -clock model, we consider a spin system in which the spins are constrained to k_{n} copies of S^{1} covering S^{2} as n \to \infty . We identify a critical energy-scaling regime and a threshold on the divergence rate of k_{n} \to +\infty , below which the \Gamma -limit of the discrete energies captures chirality transitions while preserving an S^{2} -valued energy description in the continuum limit. To achieve this, we establish a connection with the variational analysis of a discrete approximation of a vector-valued Modica–Mortola-type functional, where the k_{n} disconnected wells converge in the Hausdorff sense to a connected set as n \to \infty .