Variational Methods for Classical Spin Systems with Topological Defects

Date: 23 June 2025 - 27 June 2025

Abstract

This course focuses on the analysis of low-energy configurations of lattice spin systems, their complex structures, geometric features, and energetic properties. The multiscale character of spin systems leads to challenging mathematical questions, often requiring a combination of variational, stochastic, geometric, and combinatorial methods. The aim is to introduce a variational framework for the study of some energy-driven lattice models, with particular emphasis on energy scalings that give rise to topological singularities. After briefly reviewing key tools from Geometric Measure Theory and the Direct Methods in the Calculus of Variations, we will examine the large-scale behavior of ferromagnetic and antiferromagnetic XY systems of S^1-valued spins on (possibly geometrically frustrated) lattices. Time permitting, simple systems of S^2-valued spins leading to the emergence of baby skyrmions as energetically favorable topological defects will also be discussed.