Abnormal diffusion of a single vortex in the two dimensional XY model

We study thermal diffusion dynamics of a single vortex in two dimensional XY model. By numerical simulations we find an abnormal diffusion such that the mobility decreases with time $t$ as $1/\ln t$. In addition we construct a one dimensional diffusion-like equation to model the dynamics and confirm that it conserves quantitative property of the abnormal diffusion. By analyzing the reduced model, we find that the radius of the collectively moving region with the vortex core grows as $R(t) \propto t^{1/2}$. This suggests that the mobility of the vortex is described by dynamical correlation length as $1/\ln R(t)$.