Self-averaging characteristics of spectral fluctuations

TL;DR

This paper investigates the self-averaging properties of spectral fluctuations in quantum systems, showing that smoothing techniques reveal universal behavior and that noise impacts the ability to distinguish between different random matrix ensembles.

Contribution

The study demonstrates that smoothing spectral correlators induces self-averaging behavior and provides analytical and numerical evidence for the variance of the smoothed correlator in CUE.

Findings

Smoothing turns spectral correlators into self-averaging quantities.

Noise diminishes as the Hilbert space dimension increases.

Self-averaging is lost in certain spectral windows, affecting ensemble distinction.

Abstract

The spectral form factor as well as the two-point correlator of the density of (quasi-)energy levels of individual quantum dynamics are not self-averaging. Only suitable smoothing turns them into useful characteristics of spectra. We present numerical data for a fully chaotic kicked top, employing two types of smoothing: one involves primitives of the spectral correlator, the second a small imaginary part of the quasi-energy. Self-averaging universal (like the CUE average) behavior is found for the smoothed correlator, apart from noise which shrinks like $1\over\sqrt N$ as the dimension $N$ of the quantum Hilbert space grows. There are periodically repeated quasi-energy windows of correlation decay and revival wherein the smoothed correlation remains finite as $N\to\infty$ such that the noise is negligible. In between those windows (where the CUE averaged correlator takes on values of the order ${1\over N^2}$) the noise becomes dominant and self-averaging is lost. We conclude that the noise forbids distinction of CUE and GUE type behavior. Surprisingly, the underlying smoothed generating function does not enjoy any self-averaging outside the range of its variables relevant for determining the two-point correlator (and certain higher-order ones). --- We corroborate our numerical findings for the noise by analytically determining the CUE variance of the smoothed single-matrix correlator.