Topological Defects and Flows in BECs and Active Matter

Abstract

We study the nucleation and dynamics of topological defects in two-dimensional superfluid Bose-Einstein condensates and active liquid crystals. Using the property that these are emergent states of matter with broken rotational symmetry, we formulate a generic mathematical framework that we use to describe the properties of the corresponding topological defects. The active liquid crystals consist of micro-organisms that have an intrinsic activity which is injecting energy into the system. When the intrinsic energy production is large enough, it will result in the spontaneous creation of topological defects. These defects are localized sources of long-range elastic distortions which generate large-scale flows. We are able to solve the flow equations for isolated defects in the limit of point-like defects with an idealized far-field structure that is subject to both friction and viscous dissipation. The induced flow feeds back into the evolution equation for the order parameter of the liquid crystal and has effects on the motion of the defects by making them self-propelled and by mediating effective interactions between them. In contrast, the Bose-Einstein condensate is a passive system where energy is injected by externally applied potentials. One way to create defects is by stirring the condensate with a moving potential. Quantum vortices are then nucleated in pairs and shed from the stirring potential. We show how the defect nucleation and motion are determined by the evolution of the superfluid wave function. In this thesis, we demonstrate that even though the energy is injected and transported differently in these two systems, there are similarities in the fundamental mechanisms for the nucleation of topological defects and in the correspondence between defect kinematics and the evolution of the order parameter.