路径积分的内在维数：数据挖掘量子临界性和涌现简单性
Intrinsic dimension of path integrals: data mining quantum criticality and emergent simplicity
摘要：
量子多体系统的特征是相关模式，当被解释为数据结构时，这些相关模式定义了高度非平凡的流形。相位和相变的物理性质通常通过与可观测响应函数相关的简单关联函数来检索。最近的实验已经证明了通过波函数快照完全表征量子多体系统的能力，为分析量子现象开辟了新的可能性。本文介绍了一种通过量子配分函数的路径积分（或等价的随机级数展开）流形来挖掘其关联结构的方法。我们利用固有维数（ID）和最近邻距离（NN）的方差来表征通过最先进的量子蒙特卡罗方法生成的路径积分流形：前者与数据集的复杂性有关，而后者能够诊断配置空间中点的连通性特征。我们展示了这些属性如何在量子临界附近的通用模式，揭示了如何数据结构{ \it简化}系统地在量子相变。NN距离的ID和方差在二阶和Berezinskiii-Kosterlitz-Thouless临界点附近表现出普遍的标度行为，这一事实进一步反映了这一点。最后，我们展示了非阿贝尔对称性如何极大地影响量子数据集，这是由于量子情况下（非对易）保守电荷的性质。神经网络表示的补充，我们的方法代表了第一个，基本的一步，系统的表征路径积分流形之前，采取任何降维，这是信息的普遍行为和复杂性，并可以立即应用到实验和蒙特卡罗模拟。
英文摘要：
Quantum many-body systems are characterized by patterns of correlations that define highly-non trivial manifolds when interpreted as data structures. Physical properties of phases and phase transitions are typically retrieved via simple correlation functions, that are related to observable response functions. Recent experiments have demonstrated capabilities to fully characterize quantum many-body systems via wave-function snapshots, opening new possibilities to analyze quantum phenomena. Here, we introduce a method to data mine the correlation structure of quantum partition functions via their path integral (or equivalently, stochastic series expansion) manifold. We characterize path-integral manifolds generated via state-of-the-art Quantum Monte Carlo methods utilizing the intrinsic dimension (ID) and the variance of distances from nearest neighbors (NN): the former is related to dataset complexity, while the latter is able to diagnose connectivity features of points in configuration space. We show how these properties feature universal patterns in the vicinity of quantum criticality, that reveal how data structures {\it simplify} systematically at quantum phase transitions. This is further reflected by the fact that both ID and variance of NN-distances exhibit universal scaling behavior in the vicinity of second-order and Berezinskii-Kosterlitz-Thouless critical points. Finally, we show how non-Abelian symmetries dramatically influence quantum data sets, due to the nature of (non-commuting) conserved charges in the quantum case. Complementary to neural network representations, our approach represents a first, elementary step towards a systematic characterization of path integral manifolds before any dimensional reduction is taken, that is informative about universal behavior and complexity, and can find immediate application to both experiments and Monte Carlo simulations.
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