Dynamical theory of topological defects I: the multivalued solution of the diffusion equation

Point-like topological defects are singular configurations that manifest in and out of various equilibrium systems with two-dimensional orientational order. Because they are associated with a nonzero circuitation condition, the presence of defects induces a long-range perturbation of the orientation landscape around them. The effective dynamics of defects is thus generally described in terms of quasi-particles interacting via the orientation field they produce, whose evolution in the simplest setting is governed by the diffusion equation. Because of the multivalued nature of the orientation field, its expression for a defect moving with an arbitrary trajectory cannot be determined straightforwardly and is often evaluated in the quasi-static approximation. Here, we instead derive the exact expression for the orientation created by multiple moving defects, which we find to depend on their past trajectories and thus to be nonlocal in time. Performing various expansions in relevant regimes, we demonstrate how improved approximations with respect to the quasi-static defect solution can be obtained. Moreover, our results lead to so far unnoticed structures in the orientation field of moving defects, which we discuss in light of existing experimental results.