A numerical study of the localization transition of Aubry-Andr\'{e} 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, $\alpha$, 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 calculate 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. At any given finite system size we find that at many densities the transition occurs at or near $W=2t$ ($t$ denotes the hopping). This is where single particle states are known to localize. However, we also find "spikes", densites at which the transition occurs in the range $0<W<2t$. These spikes occur for densities at which there are no partially filled bands. As the system size (and both $F_n$ and $F_{n+1}$ in $\alpha$) is increased the spikes tend towards zero, but the density at which they occur also changes slightly: they approach irrational numbers which can be written as Fibonacci ratios or sums thereof. For densities which are fixed ratios for all system sizes, the transition occurs at $W=2t$. 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 which do not necessarily tend to zero, but to finite values of $W$, determined by the modifed gap structure.