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 $$\Gamma $$ Γ -limit of the energy in a regime which detects chirality transitions on one-dimensional interfaces between the two admissible chirality phases.

Renormalized energy between fractional vortices with topologically induced free discontinuities on 2-dimensional Riemannian manifolds

“…Definition 3 (Approximate limit). Let u ∈ L 1 loc (T M ) and let T x be the transport map from x ∈ M defined in (9). We say that u has an approximate limit z ∈ T…”

“…Functionals of this kind have been rigorously investigated from a mathematical point of view in connection with models of ripple phases coexistence in biological systems in [25]. Slightly different energy functionals leading to point and line defects have been recently investigated in the context of discrete systems to model chirality phase separations in geometrically frustrated spin systems [9,10], the dependence of the energy concentration phenomenon on the rate of divergence of n in the n-clock model [20,21,22,23] or the formation of partial vortices and line defects in modified xy models [11]. The latter analysis is also connected to the investigation of orientability issues of the director field of some liquid crystals model as first discussed in [12,13].…”

Variational analysis of the $J_1$-$J_2$-$J_3$ model: a non-linear lattice version of the Aviles-Giga functional

“…Nevertheless, since the approximation in [26, Lemma 2.5] can be achieved with entropies whose supports don't contain 0 , [26, Lemma 2.6] remains true under our more restrictive notion of entropy. 9 We recall that for any separable and reflexive Banach space X , strong compactness of (v * n )n ⊂ X * is equivalent to v * n , vn → 0 for every sequence (vn)n ⊂ X with vn 0 weakly in X .…”

“…In the recent years, several examples of frustrated spin models have been investigated from a variational perspective, cf. [1,29,30,21,31,15,17,23,9,10]. As these examples show, the presence of frustration in a lattice spin system depends on both the topological properties of the lattice and the symmetry properties of the interaction potentials.…”

Existence results in large-strain magnetoelasticity

“…Among the many contributions, we single out the seminal works [43,49] (see also [37] and the references therein). We refer to [1] and the references therein for a study of effective theories and chirality transitions in the discrete-to-continuous setting. We also mention [30] for recent results on the numerics of chiral magnets, as well as [11] and [12] and the references therein, for periodic homogenization of chiral magnetic materials and for the static, dynamic, and numerical study of reduced models obtained starting from three-dimensional theories including asymmetric exchange, respectively.…”