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.

中文翻译：

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持久同源相变的定量和可解释顺序参数

我们将计算拓扑学中的现代方法应用于发现和表征相变的任务。作为说明，我们将我们的方法应用于四个二维晶格自旋模型：Ising、方冰、XY 和完全受挫的 XY 模型。特别是，我们使用持久同源性，随着粗粒度尺度或亚级阈值的增加，计算单个拓扑特征的出生和死亡，以总结自旋配置中的多尺度和高点相关性。我们使用称为持久性图像的这种信息的矢量表示来制定和执行区分阶段的统计任务。对于我们考虑的模型，对这些图像的简单逻辑回归足以识别相变。然后从回归的权重中读取可解释的顺序参数。这种方法足以将磁化、挫折和涡旋-反涡旋结构识别为我们模型中相变的相关特征。我们还定义了“持久性”临界指数，并研究它们与通常考虑的临界指数之间的关系。