Spectral density correlations and eigenfunction fluctuations in one-dimensional quasi-periodic systems

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Abstract
The authors study the scaling and correlation-fluctuation properties for the spectra and wavefunctions of a simple one-dimensional quasi-periodic system that displays an Anderson metal-insulator transition. This system and its extensions are models for studying localization phenomena, and their usefulness for the description of the Anderson transition as well as insulating and metallic phases in disordered systems is exploited. They present numerical work on the critical behaviour of the spectrum and wavefunctions, which display multifractal fluctuations. Appropriate probability densities are studied, and it is demonstrated that: (i) the subband energy width statistics, which may express spectral correlations, is consistent with linear level repulsion at the critical point and a Poisson distribution in the insulating regime: and (ii) the critical wavefunction probability amplitude distributions approach a universal function as the system size increases. It is concluded that certain critical properties of quasi-periodic models are similar to what is expected for electronic states in weakly disordered metals, and others show a striking similarity to mobility edge behaviour in three dimensions.