Dynamic scaling of topological ordering in classical systems
Dynamic scaling of topological ordering in classical systems
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DOI：
10.1103/physrevb.97.024432
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发表时间：
2017-11
影响因子：
3.7
通讯作者：
Na Xu;C. Castelnovo;R. Melko;C. Chamon;A. Sandvik
中科院分区：
文献类型：
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作者：
Na Xu;C. Castelnovo;R. Melko;C. Chamon;A. Sandvik
我们分析了模拟退火的标度行为进行各种经典系统的拓扑顺序，获得适当的限制的环面代码在二维和三维。我们首先考虑三维$\mathbb{Z}_2$（Ising）格点规范模型，它在有限温度下表现出连续的拓扑相变.我们表明，广义Kibble-Zurek标度的anadministration适用于这种转变，尽管没有一个局部序参数。我们发现周长律标度的大小的非本地序参数（定义使用威尔逊循环）和一个动态指数z=2.70 \pm 0.03$，后者在很好的协议与以前的结果的平衡动力学（自相关）。然后，我们研究的系统（拓扑）秩序的形式，只有在零温度-伊辛链，二维$\mathbb{Z}_2$规范模型，和三维星星模型（另一个变种的$\mathbb{Z}_2$规范模型）。在这些系统中的相关长度发散指数，在一种方式，是非光滑的有限大小的系统接近零温度状态。我们表明，Kibble-Zurek理论不适用于任何这些系统。相反，动力学可以理解的扩散和湮灭的拓扑缺陷，我们用它来制定一个标度理论与我们的模拟结果很好地吻合。我们还讨论了开放边界的缺陷湮灭竞争的表面蒸发的更快的过程中的效果。
We analyze scaling behaviors of simulated annealing carried out on various classical systems with topological order, obtained as appropriate limits of the toric code in two and three dimensions. We first consider the three-dimensional $\mathbb{Z}_2$ (Ising) lattice gauge model, which exhibits a continuous topological phase transition at finite temperature. We show that a generalized Kibble-Zurek scaling ansatz applies to this transition, in spite of the absence of a local order parameter. We find perimeter-law scaling of the magnitude of a non-local order parameter (defined using Wilson loops) and a dynamic exponent $z=2.70 \pm 0.03$, the latter in good agreement with previous results for the equilibrium dynamics (autocorrelations). We then study systems where (topological) order forms only at zero temperature---the Ising chain, the two-dimensional $\mathbb{Z}_2$ gauge model, and a three-dimensional star model (another variant of the $\mathbb{Z}_2$ gauge model). In these systems the correlation length diverges exponentially, in a way that is non-smooth as a finite-size system approaches the zero temperature state. We show that the Kibble-Zurek theory does not apply in any of these systems. Instead, the dynamics can be understood in terms of diffusion and annihilation of topological defects, which we use to formulate a scaling theory in good agreement with our simulation results. We also discuss the effect of open boundaries where defect annihilation competes with a faster process of evaporation at the surface.