Long-range level correlations in quantum systems with finite Hilbert space dimension

TL;DR

This paper investigates spectral correlations in finite-dimensional quantum systems, revealing that eigenlevels cannot be entirely uncorrelated due to the unfolding process, and introduces models to explain observed spectral statistics.

Contribution

It proves a theorem on level correlations in finite quantum systems, derives an analytic power spectrum for a specific intermediate statistics model, and tests models on various quantum systems.

Findings

Eigenlevels in finite systems are inherently correlated due to unfolding.

Unfolding procedure affects the spectral statistics, spoiling certain expected results.

Models accurately describe spectral correlations in disordered and integrable quantum systems.

Abstract

We study the spectral statistics of quantum systems with finite Hilbert spaces. We derive a theorem showing that eigenlevels in such systems cannot be globally uncorrelated, even in the case of fully integrable dynamics, as a consequence of the unfolding procedure. We provide an analytic expression for the power spectrum of the $\delta_n$ statistic for a model of intermediate statistics with level repulsion but independent spacings, and we show both numerically and analytically that the result is spoiled by the unfolding procedure. Then, we provide a simple model to account for this phenomenon, and test it by means of numerics on the disordered XXZ chain, the paradigmatic model of many-body localization, and the rational Gaudin-Richardson model, a prototypical model for quantum integrability.