Electronic states of a disordered two-dimensional quasiperiodic tiling: From critical states to Anderson localization

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We consider critical eigenstates in a two dimensional quasicrystal and their evolution as a function of disorder. By exact diagonalization of finite size systems we show that the evolution of properties of a typical wave-function is non-monotonic. That is, disorder leads to states delocalizing, until a certain crossover disorder strength is attained, after which they start to localize. Although this non-monotonic behavior is only present in finite-size systems and vanishes in the thermodynamic limit, the crossover disorder strength decreases logarithmically slowly with system size, and is quite large even for very large approximants. The non-monotonic evolution of spatial properties of eigenstates can be observed in the anomalous dimensions of the wave-function amplitudes, in their multifractal spectra, and in their dynamical properties. We compute the two-point correlation functions of wave-function amplitudes and show that these follow power laws in distance and energy, consistent with the idea that wave-functions retain their multifractal structure on a scale which depends on disorder strength. Dynamical properties are studied as a function of disorder. We find that the diffusion exponents do not reflect the non-monotonic wave-function evolution. Instead, they are essentially independent of disorder until disorder increases beyond the crossover value, after which they decrease rapidly, until the strong localization regime is reached. The differences between our results and earlier studies on geometrically disordered ``phason-flip'' models lead us to propose that the two models are in different universality classes. We conclude by discussing some implications of our results for transport and a proposal for a Mott hopping mechanism between power law localized wave-functions, in moderately disordered quasicrystals.