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

摘要

The nearest-neighbor Aubry–André quasiperiodic localization model is generalized to include power-law translation-invariant hoppings Tl ∝ t/la or power-law Fourier coefficients Wm ∝ w/mb in the quasiperiodic potential. The Aubry–André duality between Tl and Wm 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 t of the hoppings yields that the eigenstates remain power-law localized in real space for a > 1 and are critical for ac = 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 quasiperiodic potential yields that the eigenstates remain delocalized in real space (power-law localized in Fourier space) for b > 1 and are critical for bc = 1 where they follow the weak multifractality Gaussian spectrum in real space (or strong multifractality linear spectrum in the Fourier basis). This critical case bc = 1 for the Fourier coefficients Wm corresponds to a periodic function with discontinuities, instead of the cosinus function of the standard self-dual Aubry–André model.