A random matrix model with localization and ergodic transitions

Abstract

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 $\gamma$ 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 $\gamma =2$. The other one at $\gamma =1$ 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 $f(\alpha )$ and the wave function overlap which all show the transitions at $\gamma =1$ and $\gamma =2$.

8 pages, 9 figures (main text) + 7 pages, 4 figures (supplementary materials)

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• Localization of interacting fermions at high temperature
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• Dynamical scaling for critical states: is Chalker's ansatz valid for strong fractality?
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