  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 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 were 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 phases in a nonhermitian disordered many-body
system.
