The statistics of gap ratios between consecutive energy levels is a widely used tool—in particular, in the context of many-body physics—to distinguish between chaotic and integrable systems, described, respectively, by Gaussian ensembles of random matrices and Poisson statistics. In this work, we extend the study of the gap ratio distribution $P(r)$ to the case where discrete symmetries are present. This is important since in certain situations it may be very impractical, or impossible, to split the model into symmetry sectors, let alone in cases where the symmetry is not known in the first place. Starting from the known expressions for surmises in the Gaussian ensembles, we derive analytical surmises for random matrices comprised of several independent blocks. We check our formulas against simulations from large random matrices, showing excellent agreement. We then present a large set of applications in many-body physics, ranging from quantum clock models and anyonic chains to periodically driven spin systems. In all these models, the existence of a (sometimes hidden) symmetry can be diagnosed through the study of the spectral gap ratios, and our approach furnishes an efficient way to characterize the number and size of independent symmetry subspaces. We finally discuss the relevance of our analysis for existing results in the literature, as well as its practical usefulness, and point out possible future applications and extensions.

中文翻译：

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通过间隙比统计探索量子多体系统的对称性

连续能级之间的间隙比统计是一种广泛使用的工具 - 特别是在多体物理的背景下 - 用于区分混沌和可积系统，分别由随机矩阵的高斯集合和泊松统计描述。在这项工作中，我们扩展了对间隙比率分布的研究$磷(r)$对于存在离散对称性的情况。这很重要，因为在某些情况下，将模型拆分为对称扇区可能非常不切实际或不可能，更不用说首先不知道对称性的情况了。从已知的高斯系综中推测的表达式开始，我们推导出由几个独立块组成的随机矩阵的解析推测。我们根据大型随机矩阵的模拟检查我们的公式，显示出非常好的一致性。然后，我们展示了多体物理学中的大量应用，从量子钟模型和任意子链到周期性驱动的自旋系统。在所有这些模型中，可以通过研究光谱间隙比来诊断（有时是隐藏的）对称性的存在，我们的方法提供了一种有效的方法来表征独立对称子空间的数量和大小。最后，我们讨论了我们的分析与文献中现有结果的相关性，以及它的实际用途，并指出了未来可能的应用和扩展。