Multifractality in the generalized Aubry-Andre 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 $T_l\propto t/l^a$ or power-law Fourier coefficients $W_m\propto w/m^b$ in the quasi-periodic potential. The Aubry-André 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-André model.

16 pages

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• Many-body localization in the interpolating Aubry-André-Fibonacci model
• Anomalous multifractality in quantum chains with strongly correlated disorder
• Critical states and anomalous mobility edges in two-dimensional diagonal quasicrystals
• A numerical study of the localization transition of Aubry-André type models
• Entanglement scaling in fermion chains with a localization-delocalization transition and inhomogeneous modulations
• Localisation in quasiperiodic chains: a theory based on convergence of local propagators