Année 2022-2023

A scaling law for a two-dimensional frustrated spin system

by Melanie Koser (Humboldt-Universität zu Berlin)


We are interested in pattern formation in two-dimensional magnetic compounds. We consider materials whose atoms are ordered in a regular crystalline structure and associate to each atom its so called spin, a unit vector in $\mathbb{S}^1$. Complex geometric structures in the spin field may be the result of the competition between anti- and ferromagnetic interactions. In ferromagnetic materials spins prefer to be aligned, whereas in antiferromagnetic compounds one cannot observe a global orientation of the spins. The competition between these two interactions leads to frustration mechanisms in the system. We consider the lattice energy of certain materials, in which antiferromagnetic (AF) and ferromagnetic (F) interactions coexist, and are modeled by the $J_1 $-$J_3$ F-AF model on a square lattice. In this talk we present our current research results. These include a scaling law for the optimal energy, which also describes arising patterns in a minimal spin field. Further, we discuss a $\Gamma$-convergence result which in a certain parameter regime relates the discrete model with a suitable continuous counterpart.

Based on a joint work with Janusz Ginster and Barbara Zwicknagl.

Organized by

Elise Bonhomme