Universal spectral correlations in the chaotic wave function and the development of quantum chaos

We investigate the appearance of quantum chaos in a single many-body wave function by analyzing the statistical properties of the eigenvalues of its reduced density matrix $\rho_A$ of a spatial subsystem A. We find that the spectrum of $\rho_A$ is described by a so-called Wishart random matrix, which exhibits universal spectral correlations between eigenvalues separated by distances ranging from one up to many mean level spacings. We use these universal spectral characteristics of $\rho_A$ as a definition of chaos in the wave function. A simple and precise characterization of such correlations is a segment of linear growth at sufficiently long times, recently called the "ramp", of the spectral form factor. Specifically, numerical results for the spectral form factor of $\rho_A$ of generic non-integrable systems, such as one-dimensional quantum Ising and Floquet spin models, are found to exhibit an universal ramp identical to that appearing for a random pure state, in which $\rho_A$ is the Wishart random matrix. In addition, we study the development of chaos in the wave function by letting an initial direct product state evolve under the unitary time evolution. We find that the universal spectral correlations as manifested by the ramp set in as soon as the entanglement entropy begins to grow, and first develop for the eigenvalues at the top of the spectrum of $\rho_A$, subsequently spreading over the entire spectrum at later times. Finally, we study a prethermalized regime described by a generalized Gibbs ensemble, which develops in a rapidly driven Floquet model at intermediate times. We find that the prethermalized regime exhibits no chaos, as evidenced by the absence of a ramp in the spectral form factor of $\rho_A$, while the universal spectral correlations start to develop when the prethermalized regime finally relaxes at late times to the fully thermalized chaotic regime.