We propose a characterization of quantum many-body chaos: given a collection of simple operators, the set of all possible pair-correlations between these operators can be organized into a matrix with random-matrix-like spectrum. This approach is particularly useful for locally interacting systems, which do not generically show exponential Lyapunov growth of out-of-time-ordered correlators. We demonstrate the validity of this characterization by numerically studying the Sachdev-Ye-Kitaev model and a one-dimensional spin chain with random magnetic field (XXZ model).

1 aGharibyan, Hrant1 aHanada, Masanori1 aSwingle, Brian1 aTezuka, Masaki uhttps://arxiv.org/abs/1902.1108601364nas a2200157 4500008004100000245003000041210003000071260001500101490000800116520096500124100002101089700002101110700001901131700001901150856003701169 2019 eng d00aQuantum Lyapunov Spectrum0 aQuantum Lyapunov Spectrum c04/10/20190 v0823 aWe introduce a simple quantum generalization of the spectrum of classical Lyapunov exponents. We apply it to the SYK and XXZ models, and study the Lyapunov growth and entropy production. Our numerical results suggest that a black hole is not just the fastest scrambler, but also the fastest entropy generator. We also study the statistical features of the quantum Lyapunov spectrum and find universal random matrix behavior, which resembles the recently-found universality in classical chaos. The random matrix behavior is lost when the system is deformed away from chaos, towards integrability or a many-body localized phase. We propose that quantum systems holographically dual to gravity satisfy this universality in a strong form. We further argue that the quantum Lyapunov spectrum contains important additional information beyond the largest Lyapunov exponent and hence provides us with a better characterization of chaos in quantum systems.

1 aGharibyan, Hrant1 aHanada, Masanori1 aSwingle, Brian1 aTezuka, Masaki uhttps://arxiv.org/abs/1809.01671