We introduce a complex-plane generalization of the consecutive level-spacing ratio distribution, used to distinguish regular from chaotic quantum spectra. Our approach features the distribution of complex-valued ratios between nearest- and next-to-nearest neighbor spacings. We show that this quantity can successfully detect the chaotic or regular nature of complex-valued spectra. This is done in two steps. First, we show that, if eigenvalues are uncorrelated, the distribution of complex spacing ratios is flat within the unit circle, whereas random matrices show a strong angular dependence in addition to the usual level repulsion. The universal fluctuations of Gaussian Unitary and Ginibre Unitary universality classes in the large-matrix-size limit are shown to be well described by Wigner-like surmises for small-size matrices with eigenvalues on the circle and on the two-torus, respectively. To study the latter case, we introduce the Toric Unitary Ensemble, characterized by a flat joint eigenvalue distribution on the two-torus. Second, we study different physical situations where nonhermitian matrices arise: dissipative quantum systems described by a Lindbladian, non-unitary quantum dynamics described by nonhermitian Hamiltonians, and classical stochastic processes. We show that known integrable models have a flat distribution of complex spacing ratios whereas generic cases, expected to be chaotic, conform to Random Matrix Theory predictions. Specifically, we are able to clearly distinguish chaotic from integrable dynamics in boundary-driven dissipative spin-chain Liouvillians and in the classical asymmetric simple exclusion process and to differentiate localized from delocalized regimes in a nonhermitian disordered many-body system.

中文翻译：

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复数间距比：耗散量子混沌的标志

我们介绍了连续平面间距比分布的复平面概括，用于区分规则量子和混沌量子光谱。我们的方法的特点是最接近和最接近的邻居间距之间的复数值比率分布。我们表明，该数量可以成功地检测复数值谱的混沌或规则性质。这分两个步骤完成。首先，我们表明，如果特征值不相关，则复数间距比率的分布在单位圆内是平坦的，而随机矩阵除了通常的水平斥力外还表现出较强的角度依赖性。高斯Unit性和吉尼伯Unit一性类在大矩阵极限中的普遍涨落已被Wigner样假设很好地描述了，小矩阵的特征值分别在圆和双环上。为了研究后一种情况，我们引入了Toric ary合奏，其特征是在两个花托上的联合特征值分布平坦。其次，我们研究了非厄密矩阵出现的不同物理情况：由Lindbladian描述的耗散量子系统，由非赫密尔顿哈密顿量描述的非-量子动力学和经典随机过程。我们表明，已知的可积模型具有复杂间距比的平面分布，而一般情况下，预期是混乱的，符合随机矩阵理论的预测。特别，