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

Understand

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.