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

Models with correlated disorder are rather common in physics. In some of them, like the Aubry-André (AA) model, the localization phase diagram can be found from the (self)duality with respect to the Fourier transform. In others, like the all-to-all translation-invariant Rosenzweig-Porter (TI RP) ensemble or the many-body localization (MBL), one needs to develop more sophisticated and usually phenomenological methods to obtain the localization transition. In addition, such models contain not only localization, but also the ergodicity-breaking transition in the Hilbert-space eigenstate structure, giving way to the nonergodic (i.e., MBL) phase of states with nontrivial fractal dimensions <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:msub><a:mi>D</a:mi><a:mi>q</a:mi></a:msub><a:mo>&gt;</a:mo><a:mn>0</a:mn></a:mrow></a:math>. In this work, we suggest a method to calculate both the above transitions and a lower bound to the fractal dimensions <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:msub><b:mi>D</b:mi><b:mn>2</b:mn></b:msub></b:math> and <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:msub><c:mi>D</c:mi><c:mi>∞</c:mi></c:msub></c:math>, relevant for the physical observables. To verify this method, we apply it to the class of long-range (self-)dual models, interpolating between a tight-binding ballistic model and TI RP via both power-law dependencies of the hopping terms and the on-site disorder correlations, 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 nonergodic phases in the 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.