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 show that the distribution of non-ergodic extended states features level repulsion at small energies and differs from the Wigner-Dyson distribution. This is characteristic of non-ergodic wave functions that exhibits a weak form of chaos, not strong enough to reproduce the full behavior of Gaussian ensembles. Then, we analyze the two transitions with the standard tools of critical phenomena. Our results show that a single parameter is needed to obtain finite-size scaling at both transitions. Based on this, we argue that a continuous phase transition occurs between non-ergodic chaotic and ergodic phases.