Localization and ergodicity breaking in long-range self-dual models with correlated disorder

Self-dual Aubry-Andr'e model provides an example of a system with the fully correlated quasiperiodic disorder potential, which demonstrates the Anderson transition already in a one-dimensional system. Its self-dual cousin with uncorrelated disorder and all-to-all translation-invariant (TI) coupling, known as a TI Rosenzweig-Porter ensemble, carries along with the ergodic and localized phases also a fractal one. In this paper, we consider an interpolation between the above two models, characterized by both the power-law correlated diagonal elements and the TI off-diagonal elements, power-law decaying with a distance from the diagonal. We show that the interplay of the partially correlated disorder and the power-law decay hopping terms may lead to the emergence of the two types of the fractal phases in an entire range of parameters, even without having any quasiperiodicity of the Aubry-Andr'e potential.