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

TL;DR

This paper investigates the transition from discrete to continuum models in a frustrated spin system on a lattice, revealing conditions under which chirality transitions emerge in the continuum limit, especially near the Landau-Lifschitz point.

Contribution

It introduces a novel spin model constrained to multiple copies of S^1 covering S^2, identifying a critical energy-scaling regime for capturing chirality transitions in the continuum limit.

Findings

Identifies a critical energy-scaling regime for the model.

Determines a threshold for the divergence rate of the covering copies.

Shows when the continuum limit captures chirality transitions with S^2-valued energies.

Abstract

We analyze the discrete-to-continuum limit of a frustrated ferromagnetic/anti-ferromagnetic $\mathbb{S}^2$-valued spin system on the lattice $\lambda_n\mathbb{Z}^2$ as $\lambda_n\to 0$. For $\mathbb{S}^2$ spin systems close to the Landau-Lifschitz point (where the helimagnetic/ferromagnetic transition occurs), it is well established that for chirality transitions emerge with vanishing energy. Inspired by recent work on the $N$-clock model, we consider a spin model where spins are constrained to $k_n$ copies of $\mathbb{S}^1$ covering $\mathbb{S}^2$ as $n\to\infty$. We identify a critical energy-scaling regime and a threshold for the divergence rate of $k_n\to+\infty$, below which the $\Gamma$-limit of the discrete energies capture chirality transitions while retaining an $\mathbb{S}^2$-valued energy description in the continuum limit.