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 -limit of the energy in a regime which detects chirality transitions on one-dimensional interfaces between the two admissible chirality phases.
Highlights
Ordering problems in magnetism have been extensively studied by both the physics and the mathematics communities
Frustration in the context of spin systems refers to the situation where spins cannot find an orientation that simultaneously minimizes all the pairwise exchange interactions
Often frustration occurs in those systems where spins are subject to conflicting short range ferromagnetic and long range antiferromagnetic interactions, as when modulated phases appear
Summary
Ordering problems in magnetism have been extensively studied by both the physics and the mathematics communities. Due to the S1-symmetry, the energy at this regime cannot distinguish ground states with the same chirality, so that the relevant order parameter of the model is, not the spin field but its chirality: in Proposition 3.1 we prove that a sequence (uε) satisfying Fε(uε) ≤ C admits a subsequence (not relabeled) such that χ (uε) → χ strongly in L1( ) for some χ ∈ BV ( ; {−1, 1}), i.e., the admissible chiralities in the continuum limit are −1 and 1 and the chirality phases {χ = −1} and {χ = 1} have finite perimeter in. To conclude the argument one still needs to prove that the winding number of the spin field in the low-energy frame can be properly controlled (Step 3 of Proposition 4.2)
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