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We investigate the appearance of quantum chaos in a single many-body wavefunction by analyzing the statistical properties of the eigenvalues of itsreduced density matrix $\rho_A$ of a spatial subsystem A. We find that thespectrum of $\rho_A$ is described by a so-called Wishart random matrix, whichexhibits universal spectral correlations between eigenvalues separated bydistances ranging from one up to many mean level spacings. We use theseuniversal spectral characteristics of $\rho_A$ as a definition of chaos in thewave function. A simple and precise characterization of such correlations is asegment of linear growth at sufficiently long times, recently called the"ramp", of the spectral form factor. Specifically, numerical results for thespectral form factor of $\rho_A$ of generic non-integrable systems, such asone-dimensional quantum Ising and Floquet spin models, are found to exhibit anuniversal 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 ofchaos in the wave function by letting an initial direct product state evolveunder the unitary time evolution. We find that the universal spectralcorrelations as manifested by the ramp set in as soon as the entanglemententropy begins to grow, and first develop for the eigenvalues at the top of thespectrum of $\rho_A$, subsequently spreading over the entire spectrum at latertimes. Finally, we study a prethermalized regime described by a generalizedGibbs ensemble, which develops in a rapidly driven Floquet model atintermediate 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 theprethermalized regime finally relaxes at late times to the fully thermalizedchaotic regime.

Universal Spectral Correlations in the Chaotic Wave Function, and the Development of Quantum Chaos