A random matrix model with localization and ergodic transitions

TL;DR

This paper introduces a generalized random matrix model exhibiting two localization transitions, one from localized to extended states at b3=2 and another from non-ergodic to ergodic extended states at b3=1, relevant to Many-Body Localization.

Contribution

It proposes a new random matrix model with two distinct localization transitions, extending previous models to better understand ergodic and non-ergodic phases.

Findings

Spectral correlation functions show transitions at b3=1 and b3=2.

Multifractality spectrum indicates a transition from multifractal to ergodic states.

Wave function overlaps confirm the existence of two localization 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 $\gamma=2$.