Complex Spacing Ratios: A Signature of Dissipative Quantum Chaos
Lucas S\'a, Pedro Ribeiro, Toma\v{z} Prosen

TL;DR
This paper introduces a complex-plane generalization of level-spacing ratios to distinguish regular from chaotic spectra in quantum and classical systems, revealing universal behaviors and aiding in identifying chaos.
Contribution
It presents a novel complex spacing ratio distribution, analyzes its universal properties across different ensembles, and applies it to various physical systems to detect chaos and phase transitions.
Findings
Complex spacing ratios distinguish regular from chaotic spectra.
Universal fluctuations are well described by Wigner-like surmises.
Application to physical models successfully identifies chaos and phase transitions.
Abstract
We introduce a complex-plane generalization of the consecutive level-spacing 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…
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