Introduction to topological defects: from liquid crystals to particle physics

Abstract

Liquid crystals are assemblies of rod-like molecules which self-organize to form mesophases, inbetween 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.

Revisiting thermodynamic topologies of black holes

“…Then using Duan's φ-mapping topological current theory [24], the local winding number of the topological defects at the points τ = 1/T can be calculated, and the global topological number of the black hole solution is obtained by adding those winding number at each defect in [10]. We can see 1 We also have ∂F ∂S = 0, so we can also construct a vector field as…”

“…The topological defect is a universal pattern of nature and plays a fundamental role in providing stability information against perturbation for systems from liquid crystals to particle physics [1]. The study of topological properties in strong gravity systems beyond in quantum many-body systems can be traced back to [2][3][4], where the existence of at least one light ring around the four-dimensional stationary non-extremal rotating black hole was proved.…”