Global characteristics of all eigenstates of local many-body Hamiltonians: participation ratio and entanglement entropy

In the spectrum of many-body quantum systems, the low-energy eigenstates were the traditional focus of research. The interest in the statistical properties of the full eigenspectrum has grown more recently, in particular in the context of non-equilibrium questions. Wave functions of interacting lattice quantum systems can be characterized either by local observables, or by global properties such as the participation ratio (PR) in a many-body basis or the entanglement between various partitions. We present a study of the PR and of the entanglement entropy (EE) between two roughly equal spatial partitions of the system, in all the eigenfunctions of local Hamiltonians. Motivated by the similarity of the PR and EE - both are generically larger in the bulk and smaller near the edges of the spectrum - we quantitatively analyze the correlation between them. We elucidate the effect of (proximity to) integrability, showing how low-entanglement and low-PR states appear also in the middle of the spectrum as one approaches integrable points. We also determine the precise scaling behavior of the eigenstate-to-eigenstate fluctuations of the PR and EE with respect to system size, and characterize the statistical distribution of these quantities near the middle of the spectrum.