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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 localization transitions as the parameter $γ$ of the model varies from 0 to $\infty$. One of them is the Anderson transition from the localized to the extended states that happens at $γ=2$. The other one at $γ=1$ is the transition from the extended non-ergodic (multifractal) states to the extended ergodic states similar to the eigenstates of the Gaussian Orthogonal Ensemble. We computed the two-level spectral correlation function, the spectrum of multifractality $f(α)$ and the wave function overlap which all show the transitions at $γ=1$ and $γ=2$.