We use tools based on the modern theory of polarization for a numerical study of the localization transition of the Aubry-André model. In this model, the spatial modulation of the potential, $\alpha$, is an irrational number, which we approximate as the ratio of Fibonacci numbers, $F_{n+1}/F_n$, where $F_n=L$ is also the system size. We determine the phase diagram as a function of particle density (filling) and potential strength $W$. We calculate the geometric Binder cumulant and also apply a renormalization approach. In accordance with an earlier study, we find that at certain fillings the transition occurs at $W=0$, while at others it occurs at $W=2t$ ($t$ is the hopping parameter). Using the Zeckendorf decomposition of natural numbers, we show that the $W=0$ fillings tend to irrational fillings given by the infinite limit of ratios of Fibonacci numbers. When a given filling is a “good” approximation to the irrational number to which it tends in the thermodynamic limit, the transition tends to $W=0$; if this is not the case, then $W=2t$. We also observe that “quasiperiodic” energy bands form, i.e., states cluster, and there are gaps between the clusters. We conjecture that the formation of these quasiperiodic bands is also related to how well a given filling approximates its irrational infinite limit. We also study an extension of the original Aubry-André model with second-nearest-neighbor hoppings. This model also exhibits a distorted phase diagram compared to the original one, with spikes that do not necessarily tend to zero, but to finite values of $W$, determined by the modified band structure.

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• Anderson localization
• Geometric & topological phases
• Localization
• Quantum phase transitions
• Quasicrystals
• Renormalization group