  Motivated by a series of recent works, an interest in multifractal phases has
risen as they are believed to be present in the Many-Body Localized (MBL) phase
and are of high demand in quantum annealing and machine learning. Inspired by
the success of the RosenzweigPorter (RP) model with Gaussian-distributed
hopping elements, several RP-like ensembles with the fat-tailed distributed
hopping terms have been proposed, with claims that they host the desired
multifractal phase. In the present work, we develop a general (graphical)
approach allowing a self-consistent analytical calculation of fractal
dimensions for a generic RP model and investigate what features of the RP
Hamiltonians can be responsible for the multifractal phase emergence. We
conclude that the only feature contributing to a genuine multifractality is the
on-site energies' distribution, meaning that no random matrix model with a
statistically homogeneous distribution of diagonal disorder and uncorrelated
off-diagonal terms can host a multifractal phase.
