Introduction to topological defects: from liquid crystals to particle physics

Liquid crystals are assemblies of rod-like molecules which self-organize to form mesophases, in-between ordinary liquids and anisotropic crystals. At each point, the molecules collectively orient themselves along a privileged direction, which locally defines an orientational order. Sometimes, this order is broken and singularities appear in the form of topological defects. This tutorial article is dedicated to the geometry, topology and physics of these defects. We introduce the main models used to describe the nematic phase and discuss the isotropic-nematic phase transition. Then, we present the different families of defects in nematics and examine some of their physical outcomes. Finally, we show that topological defects are universal patterns of nature, appearing not only in soft matter, but also in biology, cosmology, geology and even particle physics.