发表机构
Institute for Solid State Physics, University of Tokyo（东京大学固体物理研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文从精确联合特征值概率密度出发，推导了非厄米随机矩阵高斯系综AI†和AII†中复间距比的精确分布，给出AII†类任意N的代数表达式及AI†类N=3的积分表示和渐近行为，并经数值验证。
AI 中文摘要
复间距比定义为从参考特征值到其最近邻的复位移除以其到次近邻的相应位移。其统计量可为开放量子系统中的谱关联和不可积性提供有用的诊断。本文从精确的联合特征值概率密度出发，推导了类AI†和AII†高斯系综非厄米随机矩阵的有限N复间距比分布，这些系综分别由复对称和复自对偶随机矩阵实现。在类AII†中，我们获得了任意N的精确代数表达式，并明确计算了N=3,4,5,6时的分布及代表性矩。在类AI†中，尽管联合密度保留了非酉特征向量自由度上的非紧致积分，我们解析地推导了N=3时归一化的一维积分表示，并确定了渐近行为，包括对立方级排斥的对数修正以及对角密度的非解析贡献。我们进一步通过非厄米随机矩阵的直接数值对角化验证了这些解析结果。
英文摘要
The complex spacing ratio is the complex displacement from a reference eigenvalue to its nearest neighbor divided by the corresponding displacement to its next-to-nearest neighbor. Its statistics provide a useful diagnostic of spectral correlations and nonintegrability in open quantum systems. Here, starting from the exact joint eigenvalue probability densities, we derive finite-$N$ complex-spacing-ratio distributions for the Gaussian ensembles of non-Hermitian random matrices in classes AI$^†$ and AII$^†$, realized by complex symmetric and complex self-dual random matrices, respectively. In class AII$^†$, we obtain an exact algebraic expression for arbitrary $N$ and explicitly evaluate the distributions and representative moments for $N=3, 4, 5, 6$. In class AI$^†$, although the joint density retains a noncompact integral over nonunitary eigenvector degrees of freedom, we analytically derive a normalized one-dimensional integral representation for $N=3$ and determine the asymptotic behavior, including a logarithmic correction to the cubic level repulsion and a nonanalytic contribution to the angular density. We further confirm these analytical results through direct numerical diagonalization of non-Hermitian random matrices.
Comments16 pages, 4 figures, 3 tables
