论文标题
持续同源性相变的定量和可解释的顺序参数
Quantitative and Interpretable Order Parameters for Phase Transitions from Persistent Homology
论文作者
论文摘要
我们将现代方法应用于计算拓扑中，以发现和表征相变的任务。作为插图，我们将方法应用于四个二维晶格自旋模型：Ising，Square Ice，XY和完全填充的XY模型。特别是，我们使用持续的同源性，该同源性将单个拓扑特征的出生和死亡计算为粗粒度量表或巨型阈值，以总结自旋配置中的多尺度和高点相关性。我们采用此信息的矢量表示称为持久图像，以制定和执行区分阶段的统计任务。对于我们考虑的模型，这些图像上的简单逻辑回归足以识别相变。然后从回归的权重读取可解释的顺序参数。这种方法足以确定磁化，挫败感和涡流 - 抗差结构作为我们模型中相变的相关特征。我们还定义了“持久性”关键指数，并研究了它们与通常考虑的关键指数的关系。
We apply modern methods in computational topology to the task of discovering and characterizing phase transitions. As illustrations, we apply our method to four two-dimensional lattice spin models: the Ising, square ice, XY, and fully-frustrated XY models. In particular, we use persistent homology, which computes the births and deaths of individual topological features as a coarse-graining scale or sublevel threshold is increased, to summarize multiscale and high-point correlations in a spin configuration. We employ vector representations of this information called persistence images to formulate and perform the statistical task of distinguishing phases. For the models we consider, a simple logistic regression on these images is sufficient to identify the phase transition. Interpretable order parameters are then read from the weights of the regression. This method suffices to identify magnetization, frustration, and vortex-antivortex structure as relevant features for phase transitions in our models. We also define "persistence" critical exponents and study how they are related to those critical exponents usually considered.
