A random matrix model with localization and ergodic transitions

Motivated by the problem of many-body localization and the recent numerical results for the level and eigenfunction statistics on the random regular graphs, a generalization of the Rosenzweig-Porter random matrix model is suggested that possesses two transitions. One of them is the Anderson localization transition from the localized to the extended states. The other one is the ergodic transition from the extended non-ergodic (multifractal) states to the extended ergodic states. We confirm the existence of both transitions by computing the two-level spectral correlation function, the spectrum of multifractality f (alpha) and the wave function overlap which consistently demonstrate these two transitions.