MULTIFRACTALITY IN THE GENERALIZED AUBRY–ANDRÉ QUASIPERIODIC LOCALIZATION MODEL WITH POWER-LAW HOPPINGS OR POWER-LAW FOURIER COEFFICIENTS

Abstract

The nearest-neighbor Aubry–André quasiperiodic localization model is generalized to include power-law translation-invariant hoppings [Formula: see text] or power-law Fourier coefficients [Formula: see text] in the quasiperiodic potential. The Aubry–André duality between [Formula: see text] and [Formula: see text] manifests when the Hamiltonian is written in the real-space basis and in the Fourier basis on a finite ring. The perturbative analysis in the amplitude [Formula: see text] of the hoppings yields that the eigenstates remain power-law localized in real space for [Formula: see text] and are critical for [Formula: see text] where they follow the strong multifractality linear spectrum, as in the equivalent model with random disorder. The perturbative analysis in the amplitude [Formula: see text] of the quasiperiodic potential yields that the eigenstates remain delocalized in real space (power-law localized in Fourier space) for [Formula: see text] and are critical for [Formula: see text] where they follow the weak multifractality Gaussian spectrum in real space (or strong multifractality linear spectrum in the Fourier basis). This critical case [Formula: see text] for the Fourier coefficients [Formula: see text] corresponds to a periodic function with discontinuities, instead of the cosinus function of the standard self-dual Aubry–André model.