  Rosenzweig-Porter (RP) model has garnered much attention in the last decade,
as it is a simple analytically tractable model showing both ergodic--nonergodic
extended and Anderson localization transitions. Thus, it is a good toy model to
understand the Hilbert-space structure of many body localization phenomenon. In
our study, we present analytical evidence, supported by exact numerical
computations, that demonstrates the controllable tuning of the phase diagram in
the RP model by employing on-site potentials with a non-trivial fractal
dimension instead of the conventional random disorder. We demonstrate that
doing so extends the fractal phase and creates unusual dependence of fractal
dimensions of the eigenfunctions. Furthermore, we study the fate of level
statistics in such a system and analyze the return probability of a wave packet
localized at a single site to provide a dynamical test-bed for our theory.
