From ergodic to non-ergodic chaos in Rosenzweig-Porter model

The Rosenzweig-Porter model is a one-parameter family of random matrices with three different phases: ergodic, extended non-ergodic and localized. We characterize numerically each of these phases and the transitions between them. We focus on several quantities that exhibit non-analytical behaviour and show that they obey a single-parameter scaling. Based on this, we argue that the non-ergodic chaotic and the ergodic regimes are separated by a continuous phase transition, similarly to the transition between non-ergodic chaotic and localized phases.