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Full range spectral correlations and their spectral form factors in chaotic and integrable models

Ruth ShirPablo Martinez-AzconaAur\'elia Chenu
Nov 2023
Quantum chaotic systems are characterized by energy correlations in their spectral statistics, usually probed by the distribution of nearest-neighbor level spacings. Some signatures of chaos, like the spectral form factor (SFF), take all the correlations into account, while others sample only short-range or long-range correlations. Here, we characterize correlations between eigenenergies at all possible spectral distances. Specifically, we study the distribution of $k$-th neighbor level spacings ($k$nLS) and compute its associated $k$-th neighbor spectral form factor ($k$nSFF). This leads to two new full-range signatures of quantum chaos, the variance of the $k$nLS distribution and the minimum value of the $k$nSFF, which quantitatively characterize correlations between pairs of eigenenergies with any number of levels $k$ between them. We find exact and approximate expressions for these signatures in the three Gaussian ensembles of random matrix theory (GOE, GUE and GSE) and in integrable systems with completely uncorrelated spectra (the Poisson ensemble). We illustrate our findings in a XXZ spin chain with disorder, which interpolates between chaotic and integrable behavior. Our refined measures of chaos allow us to probe deviations from Poissonian and Random Matrix behavior in realistic systems. This illustrates how the measures we introduce bring a new light into studying many-body quantum systems, which lie in-between the fully chaotic or fully integrable models.
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