Multifractality in the generalized Aubry-Andre quasiperiodic localization model with power-law hoppings or power-law Fourier coefficients

The nearest-neighbor Aubry-Andr\'e quasiperiodic localization model is generalized to include power-law translation-invariant hoppings $T_l\propto t/l^a$ or power-law Fourier coefficients $W_m \propto w/m^b$ in the quasi-periodic potential. The Aubry-Andr\'e duality between $T_l$ and $W_m$ is manifest 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 $t$ of the hoppings yields that the eigenstates remain power-law localized in real space for $a>1$ and are critical for $a_c=1$ where they follow the Strong Multifractality linear spectrum, as in the equivalent model with random disorder. The perturbative analysis in the amplitude $w$ of the quasi-periodic potential yields that the eigenstates remain delocalized in real space (power-law localized in Fourier space) for $b>1$ and are critical for $b_c=1$ where they follow the Weak Multifractality gaussian spectrum in real space (or Strong Multifractality linear spectrum in the Fourier basis). This critical case $b_c=1$ for the Fourier coefficients $W_m$ corresponds to a periodic function with discontinuities, instead of the cosinus of the standard self-dual Aubry-Andr\'e model.