  Models with correlated disorders are rather common in physics. In some of
them, like the Aubry-Andr\'e (AA) model, the localization phase diagram can be
found from the (self)duality with respect to the Fourier transform. In the
others, like the all-to-all translation-invariant Rosenzweig-Porter (TI RP)
ensemble or the Hilbert-space structure of the many-body localization, one
needs to develop more sophisticated and usually phenomenological methods to
find the localization transition. In addition, such models contain not only
localization but also the ergodicity-breaking transition, giving way to the
non-ergodic extended phase of states with non-trivial fractal dimensions $D_q$.
In this work, we suggest a method to calculate both the above transitions and a
lower bound to the fractal dimensions $D_2$ and $D_\infty$, relevant for
physical observables. In order to verify this method, we apply it to the class
of long-range (self-)dual models, interpolating between AA and TI RP ones via
both power-law dependences of on-site disorder correlations and hopping terms,
and, thus, being out of the validity range of the previously developed methods.
We show that the interplay of the correlated disorder and the power-law
decaying hopping terms leads to the emergence of the two types of fractal
phases in an entire range of parameters, even without having any
quasiperiodicity of the AA potential. The analytical results of the above
method are in full agreement with the extensive numerical calculations.
