二维XY模型中涡旋的拓扑动力学和动力学标度行为
Topological dynamics and dynamical scaling behavior of vortices in a two-dimensional XY model
摘要：
利用拓扑流理论研究二维XY模型中涡旋的内部拓扑结构，得到与序参量场有关的拓扑流。通过拓扑流理论引入标量场$ψ$。通过求解标量场，重新讨论了二维XY模型中涡旋的相互作用能。研究了涡旋的动力学演化，给出了涡旋产生、湮灭、交叉、分裂和合并的分支条件。在二维XY模型中，给定动力学指数$z=2$，在极限点附近得到了相应长度的动力学标度律$ξ(t)\propto(t-t^*)^{1/z}$.这种动力学标度行为与重整化群理论、数值模拟和实验结果是一致的。此外，还发现在涡旋的交叉、分裂和合并过程中，相应长度的动力学标度律为$ξ(t)\propto(t-t^*)$。然而，如果旋涡在分裂或合并过程中处于静止状态，则相应长度的动力学标度律是常数。
英文摘要：
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, $ψ$, 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, $ξ(t)\propto(t-t^*)^{1/z}$, 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 $ξ(t)\propto(t-t^*)$. However, if vortices are at rest during splitting or merging, the dynamical scaling law of relevant length is a constat.
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