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 ρ_A. We find that ρ_A is described by a so-called Wishart random matrix, which exhibits universal spectral correlations between eigenvalues. A simple and precise characterization of such correlations is a segment of linear growth at long times, recently called the ramp, of the spectral form factor. Numerical results for a generic non-integrable systems are found to exhibit an universal ramp identical to that appearing for a random pure state. In addition, we study the development of chaos in the wave function by letting an initial product state evolve under the unitary time evolution. We find that the ramp sets in as soon as the entanglement entropy begins to grow, and first develops at the top of the spectrum of ρ_A, subsequently spreads over the entire spectrum. Finally, we study a prethermalized regime described by a generalized Gibbs ensemble. We find that the prethermalized regime exhibits no chaos, as evidenced by the absence of a ramp , while the spectral correlations start to develop when the prethermalized regime finally relaxes at late times to the fully thermalized chaotic regime.