Numerical study of the localization transition of Aubry-André type models

We use tools based on the modern theory of polarization for a numerical study of the localization transition of the Aubry-Andr\'e model. In this model, the spatial modulation of the potential, $\ensuremath{\alpha}$, 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-Andr\'e 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.