In this paper, we suggest an extension of the Rosenzweig-Porter (RP) model, the LN-RP model, in which the off-diagonal matrix elements have a wide, log-normal distribution. We argue that this model is more suitable to describe a generic many-body localization problem. In contrast to RP model, in LN-RP model, a fragile weakly ergodic phase appears that is characterized by broken basis-rotation symmetry which the fully ergodic phase, also present in this model, strictly respects in the thermodynamic limit. Therefore, in addition to the localization and ergodic transitions in LN-RP model, there exists also the transition between the two ergodic phases (FWE transition). We suggest new criteria of stability of the nonergodic phases that give the points of localization and ergodic transitions and prove that the Anderson localization transition in LN-RP model involves a jump in the fractal dimension of the egenfunction support set. We also formulate the criterion of FWE transition and obtain the full phase diagram of the model. We show that truncation of the log-normal tail shrinks the region of weakly ergodic phase and restores the multifractal and the fully ergodic phases.

• Received 3 August 2020
• Accepted 19 November 2020

DOI:https://doi.org/10.1103/PhysRevResearch.2.043346

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by the Max Planck Society.

Published by the American Physical Society