Abstract
The interest in the precise nature of critical states and their role in the physics of aperiodic systems has witnessed a renewed interest in the last few years. In this work we present a review on the notion of critical wave functions and, in the light of the obtained results, we suggest the convenience of some conceptual revisions in order to properly describe the relationship between the transport properties and the wave functions distribution amplitudes for eigen functions belonging to singular continuous spectra related to both fractal and quasiperiodic distribution of atoms through the space.
Highlights
The interest in the precise nature of critical states and their role in the physics of aperiodic systems has witnessed a renewed interest in the last few years
In this work we present a review on the notion of critical wave functions and, in the light of the obtained results, we suggest the convenience of some conceptual revisions in order to properly describe the relationship between the transport properties and the wave functions distribution amplitudes for eigen functions belonging to singular continuous spectra related to both fractal and quasiperiodic distribution of atoms through the space
It was soon realized that quasicrystals (QCs), exhibiting long-range order along with orientational symmetries not compatible with periodic translations, represented a new order style, which should be properly interpreted as a natural extension of the notion of a crystal to structures with quasiperiodic (QP), instead of periodic, arrangements of atoms [3]
Summary
Numerical studies suggested that some electronic states are localized in these lattices, in such a way that the rate of spatial decay of the wave functions is intermediate between power and exponential laws [23,24,25] These results clearly illustrate that there is not any simple relation between the spectral nature of the Hamiltonian describing the dynamics of elementary excitations propagating through an aperiodic lattice and the spatial structure of the lattice potential. The Aubry-Andremodel provides an illustrative example of a QPS which can exhibit extended, localized, or critical wave functions depending on the value of a control parameter which measures the potential strength This parameter can be regarded as an order parameter controlling the existence of a metal-insulator transition in the system. By checking that finer discretization produces almost the same results one can be confident enough of the reliability of the obtained results [44]
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