相变的拓扑持久机
Topological Persistence Machine of Phase Transitions
摘要：
使用数据驱动的方法研究相变是具有挑战性的，特别是当系统的先验知识很少时。拓扑数据分析是一个新兴的框架，用于表征数据的形状，最近在检测材料科学中的结构转变（如玻璃-液体转变）方面取得了成功。然而，从物理状态获得的数据可能不具有作为结构材料的明确形状。因此，我们提出了一个一般的框架，称为“拓扑持久性机”，以构建数据的形状从状态的相关性，这样我们就可以随后破译相变通过定性的形状变化。我们的框架，使一个有效的和统一的方法在相变分析。我们证明了该方法在检测经典XY模型中的Berezinskii-Kosterlitz-无相变和横向Ising和Bose-哈伯德模型中的量子相变方面的有效性。有趣的是，虽然这些相变已被证明是非常难以使用传统方法进行分析，但它们可以通过我们的框架进行表征，而无需事先了解相位。因此，我们的方法有望得到广泛的应用，并将为探索实验物理系统的阶段提供实用的见解。
英文摘要：
The study of phase transitions using data-driven approaches is challenging, especially when little prior knowledge of the system is available. Topological data analysis is an emerging framework for characterizing the shape of data and has recently achieved success in detecting structural transitions in material science, such as the glass--liquid transition. However, data obtained from physical states may not have explicit shapes as structural materials. We thus propose a general framework, termed "topological persistence machine," to construct the shape of data from correlations in states, so that we can subsequently decipher phase transitions via qualitative changes in the shape. Our framework enables an effective and unified approach in phase transition analysis. We demonstrate the efficacy of the approach in detecting the Berezinskii--Kosterlitz--Thouless phase transition in the classical XY model and quantum phase transitions in the transverse Ising and Bose--Hubbard models. Interestingly, while these phase transitions have proven to be notoriously difficult to analyze using traditional methods, they can be characterized through our framework without requiring prior knowledge of the phases. Our approach is thus expected to be widely applicable and will provide practical insights for exploring the phases of experimental physical systems.
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