Spectral form factor and energy correlations in banded random matrices

TL;DR

This paper investigates the spectral properties of banded random matrices, revealing persistent long-range energy correlations in the nonergodic phase and identifying key timescales and spectral features that distinguish ergodic from nonergodic phases.

Contribution

It provides analytical and numerical insights into long-range energy correlations and spectral form factor behavior in banded random matrices, especially in the nonergodic phase.

Findings

Long-range energy correlations persist in the nonergodic phase.

Timescales for spectral correlation onset decrease with reduced bandwidth.

High-frequency power spectrum distinguishes ergodic and nonergodic phases.

Abstract

Banded random matrices were introduced as a more realistic alternative to full random matrices for describing the spectral statistics of heavy nuclei. Initially considered by Wigner, they have since become a paradigmatic model for investigating level statistics and the localization-delocalization transition in disordered quantum systems. In this work, we demonstrate that, despite the absence of short-range energy correlations, weak long-range energy correlations persist in the nonergodic phase of banded random matrices. This result is supported by our numerical and analytical studies of quantities that probe both short- and long-range energy correlations, namely, the spectral form factor, level number variance, and power spectrum. We derive the timescales for the onset of spectral correlations (ramp) and for the saturation (plateau) of the spectral form factor. Unexpectedly, we find that in the nonergodic phase, these timescales decrease as the bandwidth of the matrices is reduced. We also show that the high-frequency behavior of the power spectrum of energy fluctuations can distinguish between the nonergodic and ergodic phases of the banded random matrices.