  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 non-ergodic phases which 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
eigenfunction 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.
