Non-ergodic extended regime in random matrix ensembles: insights from eigenvalue spectra

The non-ergodic extended (NEE) regime in physical and random matrix (RM) models has attracted a lot of attention in recent years. Formally, NEE regime is characterized by its fractal wavefunctions and long-range spectral correlations such as number variance or spectral form factor. More recently, it's proposed that this regime can be conveniently revealed through the eigenvalue spectra by means of singular-value-decomposition (SVD), whose results display a super-Poissonian behavior that reflects the minibands structure of NEE regime. In this work, we employ SVD to a number of RM models, and show it not only qualitatively reveals the NEE regime, but also quantitatively locates the ergodic-NEE transition point. With SVD, we further suggest the NEE regime in a new RM model-the sparse RM model.

Automated summary

The non-ergodic extended (NEE) regime has garnered attention in the study of physical and random matrix models. By using singular-value-decomposition (SVD) to analyze the eigenvalue spectra of these models, a super-Poissonian behavior is observed, revealing the minibands structure of the NEE regime. This study applies SVD to various random matrix models, qualitatively demonstrating the NEE regime and quantitatively determining the ergodic-NEE transition point. The NEE regime is further discovered in a new type of random matrix model, the sparse RM model.