Probing quantum chaos through singular-value correlations in the sparse non-Hermitian Sachdev-Ye-Kitaev model

Exploring the spectral properties of non-Hermitian systems presents a substantial theoretical challenge due to the presence of a complex eigenvalue spectrum. Singular values for such systems are inherently real and non-negative, and the techniques used for Hermitian systems can be used with ease. As a prototypical example of such systems, we investigate the singular-value spectrum of a non-Hermitian extension of the sparse Sachdev-Ye-Kitaev (SYK) model, a solvable toy model of quantum chaos and quantum gravity with significant interest in digital quantum simulation. Our findings reveal a congruence between the statistics of singular values and those of the analogous Hermitian Gaussian ensembles. An increase in sparsity results in the model deviating from its chaotic behavior, a phenomenon precisely captured by both short- and long-range correlations in the singular-value spectrum. These findings indicate the existence of a critical sparsity threshold, beyond which the chaotic nature of the model breaks down, suggesting a potential collapse of holographic duality in such systems.