Stringent Test on Power Spectrum of Quantum Integrable and Chaotic Systems

TL;DR

This paper rigorously tests the power spectrum of quantum integrable and chaotic systems, confirming the validity of previous approximations, and introduces a protocol to accurately analyze spectral correlations, especially in the crossover to many-body localization.

Contribution

It provides a rigorous validation of the power spectrum approximations for quantum systems and develops a protocol for exact analysis in the integrable limit, addressing long-range correlation issues.

Findings

Approximations for power spectra are valid under most conditions.

Huge statistical samples are needed to observe corrections in chaotic systems.

The protocol successfully describes the crossover to many-body localization.

Abstract

Quantum chaotic and integrable systems are known to exhibit a characteristic $1/f$ and $1/f^{2}$ noise, respectively, in the power spectrum associated to their spectral fluctuations. A recent work [R. Riser, V. A. Osipov, and E. Kanzieper, \textit{Power Spectrum of Long Eigenlevel Sequences in Quantum Chaotic Systems}, Phys. Rev. Lett. \textbf{118}, 204101 (2017)] calls into question the approximations used to derive these results from random matrix theory. In this paper we show that such approximations do remain valid under almost any circumstances. For the integrable limit, we devise a protocol to exactly recover the original results. As a corollary, we show that the theoretical predictions for other statistics are bound for failure regarding long-range correlations, due to unavoidable spurious effects emerging from the analysis. By means of a rigorous statistical test, we also show that the corrections for the chaotic case introduced in the aforementioned paper require huge statistics to become relevant ---averages over more than $1000$ realizations are mandatory. As an application, we study a paradigmatic model for the crossover from the thermal to the many-body localized phase. We show that our protocol succeeds in describing the crossover. Furthermore, it also succeeds in proving that the Gaussian $\beta$-ensemble fails to account for long-range correlations between the energy levels of this paradigmatic model.