A complex network analysis on the eigenvalue spectra of random spin systems

Recent works have established a novel viewpoint that treats the eigenvalue spectra of disordered quantum systems as time-series, and corresponding algorithm such as singular-value-decomposition has proven its advantage in studying subtle physical quantities like Thouless energy and non-ergodic extended regime. On the other hand, algorithms from complex networks have long been known as a powerful tool to study highly nonlinear time-series. In this work, we combine these two ideas together. Using the particular algorithm called visibility graph (VG) that transforms the eigenvalue spectra of a random spin system into complex networks, it is shown that the degree distribution of the resulting network is capable of signaturing the eigenvalue evolution during the thermal to many-body localization transition, and the networks in the thermal phase have a small-world structure. We further show these results are robust even when the eigenvalues are incomplete with missing levels, which reveals the advantage of the VG algorithm.