Chirality transitions in frustrated $S^{2}$-valued spin systems

TL;DR

This paper investigates the transition behaviors of chiral states in a frustrated $S^{2}$-valued spin system, revealing conditions under which vectorial chirality transitions occur at zero energy cost and how modifications lead to complex chirality phenomena.

Contribution

It introduces a discrete-to-continuum analysis of a frustrated $S^{2}$-spin system, showing zero-cost vectorial chirality transitions and the emergence of complex chirality behaviors under energy penalization.

Findings

Vectorial chirality transitions have zero energy cost at certain scalings.

Modifying the energy to penalize deviations from $S^{1}$ copies allows complex chirality transitions.

The model connects $S^{2}$-spin systems with $S^{1}$-scalar chirality phenomena.

Abstract

We study the discrete-to-continuum limit of the helical XY $S^{2}$-spin system on the lattice $\mathbb{Z}^{2}$. We scale the interaction parameters in order to reduce the model to a spin chain in the vicinity of the Landau-Lifschitz point and we prove that at the same energy scaling under which the $S^{1}$-model presents scalar chirality transitions, the cost of every vectorial chirality transition is now zero. In addition we show that if the energy of the system is modified penalizing the distance of the $S^{2}$ field from a finite number of copies of $S^{1}$, it is still possible to prove the emergence of nontrivial (possibly trace dependent) chirality transitions.