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In the presence of quasiperiodic potentials, the celebrated Kitaev chainpresents an intriguing phase diagram with ergodic, localized and andmultifractal states. In this work, we generalize these results by studying thelocalization properties of the Aubry-Andr\'e-Harper model in the presence oflong-range hopping and superconducting pairing amplitudes. These amplitudesdecay with power-law exponents $\xi$ and $\alpha$ respectively. To this end, wereview and compare a toolbox of global and local characterization methods inorder to investigate different types of transitions between ergodic, localizedand multifractal states. We report energy-dependent transitions from ergodic tomultifractal states for pairing terms with $\alpha<1$ and energy-dependenttransitions from ergodic to localized states with an intermediate multifractalregion for $\alpha>1$. The size of the intermediate multifractal region dependsnot only on the value of the superconducting pairing term $\Delta$, but also onthe energy band. The transitions are not described by a mobility edge, butinstead we report hybridization of bands with different types of localizationproperties. This leads to coexisting multifractal regimes where fractaldimensions follow different distributions.

Localization and multifractal properties of the long-range Kitaev chain in the presence of an Aubry-Andr\'e-Harper modulation