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Models with correlated disorders are rather common in physics. In some ofthem, like the Aubry-Andr\'e (AA) model, the localization phase diagram can befound from the (self)duality with respect to the Fourier transform. In theothers, like the all-to-all translation-invariant Rosenzweig-Porter (TI RP)ensemble or the Hilbert-space structure of the many-body localization, oneneeds to develop more sophisticated and usually phenomenological methods tofind the localization transition. In addition, such models contain not onlylocalization but also the ergodicity-breaking transition, giving way to thenon-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 alower bound to the fractal dimensions $D_2$ and $D_\infty$, relevant forphysical observables. In order to verify this method, we apply it to the classof long-range (self-)dual models, interpolating between AA and TI RP ones viaboth 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-lawdecaying hopping terms leads to the emergence of the two types of fractalphases in an entire range of parameters, even without having anyquasiperiodicity of the AA potential. The analytical results of the abovemethod are in full agreement with the extensive numerical calculations.

Ergodicity-breaking phase diagram and fractal dimensions in long-range models with generically correlated disorder