Quantifying quantum chaos through microcanonical distributions of entanglement

Abstract

A characteristic feature of "quantum chaotic" systems is that their eigenspectra and eigenstates display universal statistical properties described by random matrix theory (RMT). However, eigenstates of local systems also encode structure beyond RMT. To capture this, we introduce a quantitative metric for quantum chaos which utilizes the Kullback-Leibler divergence to compare the microcanonical distribution of entanglement entropy (EE) of midspectrum eigenstates with a reference RMT distribution generated by pure random states (with appropriate constraints). The metric compares not just the averages of the distributions, but also higher moments. The differences in moments are compared on a highly-resolved scale set by the standard deviation of the RMT distribution, which is exponentially small in system size. This distinguishes between chaotic and integrable behavior, and also quantifies the degree of chaos in systems assumed to be chaotic. We study this metric in local minimally structured Floquet random circuits, as well as a canonical family of many-body Hamiltonians, the mixed field Ising model (MFIM). For Hamiltonian systems, the reference random distribution must be constrained to incorporate the effect of energy conservation. The metric captures deviations from RMT across all models and parameters, including those that have been previously identified as strongly chaotic, and for which other diagnostics of chaos such as level spacing statistics look strongly thermal. In Floquet circuits, the dominant source of deviations is the second moment of the distribution, and this persists for all system sizes. For the MFIM, we find significant variation of the KL divergence in parameter space. Notably, we find a small region where deviations from RMT are minimized, suggesting that "maximally chaotic" Hamiltonians may exist in fine-tuned pockets of parameter space.