Topological dynamics and dynamical scaling behavior of vortices in a two-dimensional XY model

TL;DR

This paper investigates the topological structure and dynamical scaling behavior of vortices in a 2D XY model using topological current theory, revealing scaling laws during vortex evolution consistent with existing theories and experiments.

Contribution

It introduces a topological current framework to analyze vortex interactions and derives new dynamical scaling laws during vortex processes in the 2D XY model.

Findings

Dynamical scaling law during vortex growth or annihilation: $\xi(t)\propto(t-t^*)^{1/z}$ with $z=2$.

Scaling law during vortex crossing, splitting, and merging: $\xi(t)\propto(t-t^*)$.

Constant length scale when vortices are at rest during splitting or merging.

Abstract

By using topological current theory we study the inner topological structure of vortices a two-dimensional (2D) XY model and find the topological current relating to the order parameter field. A scalar field, $\psi$, is introduced through the topological current theory. By solving the scalar field, the interaction energy of vortices in a 2D XY model is revisited. We study the dynamical evolution of vortices and present the branch conditions for generating, annihilating, crossing, splitting and merging of vortices. During the growth or annihilation of vortices, the dynamical scaling law of relevant length in a 2D XY model, $\xi(t)\propto(t-t^*)^{1/z}$, is obtained in the neighborhood of the limit point, given the dynamic exponent $z=2$. This dynamical scaling behavior is consistent with renormalization group theory, numerical simulations, and experimental results. Furthermore, it is found that during the crossing, splitting and merging of vortices, the dynamical scaling law of relevant length is $\xi(t)\propto(t-t^*)$. However, if vortices are at rest during splitting or merging, the dynamical scaling law of relevant length is a constat.