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)Z(2) as lambda(n)-> 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 -> infinity. We identify a critical energy-scaling regime and a threshold on the divergence rate of k(n) -> +infinity, 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 -> 1.