The antiferromagnetic XY model on the triangular lattice: chirality transitions at the surface scaling

Abstract

We study the discrete-to-continuum variational limit of the antiferromagnetic XY model on the two-dimensional triangular lattice. The system is fully frustrated and displays two families of ground states distinguished by the chirality of the spin field. We compute the Γ-limit of the energy in a regime which detects chirality transitions on one-dimensional interfaces between the two admissible chirality phases.

The antiferromagnetic XY model on the triangular lattice: topological singularities

“…In particular, this implies that χ(u ε ) > 0, as (u ε (εi), u ε (εj), u ε (εk)) are in a counterclockwise order (see [9,Remark 2.3]). By Lemma 2.8, by Remark 2.5, and since…”

“…It aims at the first mathematically rigorous derivation of the coarse grained energy of the AFXY system as the lattice spacing vanishes and the energy scaling allows the formation of finitely many spin vortices. This is a further step towards a complete understanding of the AFXY model, whose variational analysis has been initiated in [9] at a different scaling, which leads to interfacial-type energies, as we recall below. It is worth mentioning that interfacial energies often result from different frustration mechanisms in the variational analysis of spin systems, e.g., those induced by the competition of ferromagnetic (favoring alignment) and antiferromagnetic interactions [2,23,14,44,19,24].…”

“…In [9] we analyze the energetic regime at which the two families of ground states coexist and the energy of the system concentrates at the interface between the two chiral phases {χ = 1} and {χ = −1}. This happens assuming that, as ε → 0, sequences of spin fields u ε : L ε → S 1 can deviate from ground states under the constraint E ε (u ε , Ω) ≤ Cε.…”

The $N$-clock model: Variational analysis for fast and slow divergence rates of $N$

“…To this end, given θ ε → 0 , we find the relevant scaling κ ε and we study the Γ -limit of 1 κε E ε . The limit strongly depends on the rate of convergence θ ε → 0 and can be characterized by interfacial-type singularities [23,2,21,27,8,17,20,24,12] (see also [18,5]) or vortex-like singularities [38,4,6,7,19,14,3], possibly coexisting. In this paper we are interested in the following regimes: ε| log ε| θ ε , θ ε ∼ ε| log ε|, and θ ε ε .…”