非厄米随机矩阵的通用硬边统计特性
Universal hard-edge statistics of non-Hermitian random matrices
AI总结：
该研究将非厄米随机矩阵谱统计分类扩展至38个对称类，识别出28类独特硬边统计并给出量化度量，通过数值验证开放量子系统符合该统计规律，为检测开放系统量子混沌提供工具。
摘要：
随机矩阵理论是理解量子混沌系统固有谱关联的有力工具。尽管非厄米随机矩阵理论有着多样的应用，但其对称性的作用仍有待充分确立。本文全面研究了对称性对谱原点附近能级统计（即硬边统计）的影响，并将谱统计的分类扩展至涵盖非厄米随机矩阵的全部38个对称类。在该分类体系中，我们识别出28个具有与谱体区内能级统计不同的独特硬边统计特性的对称类，这些对称类进一步被划分为两组，即Altland-Zirnbauer₀分类及超出该分类的类别。我们引入并阐明了用于刻画所有对称类通用硬边统计特性的量化度量。此外，通过大量数值计算，我们研究了不同对称类下的多种开放量子系统，包括二次型和多体Lindbladian（林德布拉德算符），以及非厄米哈密顿量。研究表明，这些系统展现出与随机矩阵相同的硬边统计特性，且其在原点附近的系综平均谱分布呈现出符合随机矩阵行为的涌现对称性。本文的结果建立了对非厄米随机矩阵理论的全面理解，可用于检测开放量子系统中量子混沌的存在与否。
英文摘要：
Random matrix theory is a powerful tool for understanding spectral correlations inherent in quantum chaotic systems. Despite diverse applications of non-Hermitian random matrix theory, the role of symmetry remains to be fully established. Here, we comprehensively investigate the impact of symmetry on the level statistics around the spectral origin -- hard-edge statistics -- and expand the classification of spectral statistics to encompass all the 38 symmetry classes of non-Hermitian random matrices. Within this classification, we discern 28 symmetry classes characterized by distinct hard-edge statistics from the level statistics in the bulk of spectra, which are further categorized into two groups, namely the Altland-Zirnbauer$_0$ classification and beyond. We introduce and elucidate quantitative measures capturing the universal hard-edge statistics for all the symmetry classes. Furthermore, through extensive numerical calculations, we study various open quantum systems in different symmetry classes, including quadratic and many-body Lindbladians, as well as non-Hermitian Hamiltonians. We show that these systems manifest the same hard-edge statistics as random matrices and that their ensemble-average spectral distributions around the origin exhibit emergent symmetry conforming to the random-matrix behavior. Our results establish a comprehensive understanding of non-Hermitian random matrix theory and are useful in detecting quantum chaos or its absence in open quantum systems.
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