Complex Spacing Ratios: A signature of Dissipative Quantum Chaos
The authors (doctoral student Lucas Sa and Pedro Ribeiro from the University of Lisbon and Prof. Dr. Tomaž Prosen from the Faculty of Mathematics and Physics of the University of Ljubljana) show that the distribution of the ratios between the spacings of a randomly selected own value of a non-Hermitian operator to the nearest and next-to-nearest neighbours is a very robust statistic that successfully diagnoses integrability (exact solvability), or, alternatively, chaos, in a wide panel of dissipative quantum and classical stochastic systems.
Quantum chaos is a field of theoretical and mathematical physics with a large number of possible applications, from condensed matter physics and atomic physics to high energy physics and the holographic theory of black holes. One of the main problems is quantum chaos diagnostics (or the absence of an exact solution/integrability), which is particularly problematic in multiparticle systems without any proper classical correspondence or in the absence of a small effective coupling constant. The only robust indication of chaos in such systems is matching the prediction of random-matrix theory. It is especially important to find an indicator that is robust and universal, i.e. independent of the non-universal properties of the model, such as the density of states.
In their scientific work, Sa, Ribeiro and Prosen introduced new universal statistic which is adapted to non-Hermitian or non-normal operators with a complex spectrum, and which is completely independent of the density of states or spectral density in the complex plane and of other details of the model. This statistic of the ratios between the spacings of a randomly selected own value to the nearest and next-to-nearest neighbours shows the universal behaviour in limit cases of the spectra with Poisson statistics which we expect in integrable models, and the spectra of the Ginibre ensemble of non-Hermitian Gaussian random matrices expected in the case of chaotic dissipative models. The new method was successfully tested in the detection of integrability in open quantum multiparticle systems and in classical stochastic (Markov) systems.
Source: Lucas Sa, Pedro Ribeiro in Tomaž Prosen, Complex Spacing Ratios: A Signature of Dissipative Quantum Chaos, Physical Review X 10, 021019 (2020)
Distribution of complex spacing ratios - “integrable” Poisson random complex spectrum (left) and the “chaotic” spectrum determined through the diagonalisation of non-symmetric random complex matrices with Gaussian distributed matrix elements (right)
Illustration: Lucas Sa, Pedro Ribeiro and Tomaž Prosen