Nonmonotonic crossover and scaling behavior in a disordered one-dimensional quasicrystal

We consider a noninteracting disordered 1D quasicrystal in the weak-disorder regime. We show that the critical states of the pure model approach strong localization in strikingly different ways, depending on their renormalization properties. A finite-size scaling analysis of the inverse participation ratios of states (IPR) of the quasicrystal shows that they are described by several kinds of scaling functions. While most states show a progressively increasing IPR as a function of the scaling variable, other states exhibit a nonmonotonic ``reentrant'' behavior wherein the IPR first decreases, and passes through a minimum, before increasing. This surprising behavior is explained in the framework of perturbation renormalization group treatment, where wave functions can be computed analytically as a function of the hopping amplitude ratio and the disorder, however it is not specific to this model. Our results should help to clarify results of recent studies of localization due to random and quasiperiodic potentials.