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. 1. Basic Notions 1.1. Orderings of Matter The notion of order is one of the most fundamental ones. In fact, order inspires the best human civilization achievements in politics, ethics, arts, and sciences [1]. Order pervades also most workings of Nature as the universe unfolds creating symmetric patterns and stable structures. Among them, solid matter arrangements were initially categorized in a dichotomist way, namely, as either ordered or disordered matter forms. In this way, ordered matter was identified with periodic arrays of atoms through the three-dimensional space, while disordered matter was related to random atomic distributions instead. Thus, the notions of crystalline matter and spatial periodicity were born interwoven from the very beginning, just as amorphous matter was conceptually related to randomness in a natural way (Figure 1(a)). Figure 1: In 1992 the notion of crystal was widened beyond mere periodicity. This conceptual diagram presents the position of aperiodic crystals, no longer based on the notion of periodic translation symmetry, among the different orderings of matter. The diverse aperiodic crystal families are arranged according to the dimension (see ( 2)) of their embedding hyperspaces (numerical labels). Nevertheless, the unexpected finding of incommensurate phases during the 1960s and 1970s, followed by the discovery of quasicrystalline alloys in 1982, opened up a discussion forum on the very crystal notion in the crystallographic, condensed matters physics and materials science communities. Indeed, initially it was thought that quasicrystals (short for quasiperiodic crystals) corresponded to a somewhat intermediate order form between that of crystals and amorphous materials [2]. However, it was soon realized that quasicrystals (QCs), exhibiting long-range order along with orientational symmetries not compatible with periodic translations, actually represented a new order style, which should be properly interpreted as a natural
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The theory of these functions was developed by Harald Bohr (1887–1951), brother of the well-known physicist Niels Bohr [93].
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As an illustrative counter example we can consider the function: which is AP but not QP, since its Fourier spectrum does not have a finite basis.
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Though exponentially localized decaying states are considered as representative of most localization processes in solid state physics one must keep in mind that other possible mathematical functions are also possible to this end.
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The tight-binding approximation considers that both the potential and the electronic wave functions are sharply peaked on the atomic sites.
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We note that (9) is invariant under the simultaneous transformation: This property shows that the spectra corresponding to opposite values of the potential are mirror images of each other with respect to the center of the spectrum.
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For the sake of simplicity one can adopt units such that without loss of generality.
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Note that all of the transfer integrals take on the same value.
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The existence of exponentially localized wave functions has been recently reported from numerical studies of (10) with the bichromatic potential with rational corresponding to certain specific energy values ( ). This system then suggests the possible presence of a pure-point spectral component not restricted to low energy values in certain periodic systems [44].
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This equation is also referred to as Harper's equation (see Section 1.6) or the almost Mathieu operator, which results from the discretization of the classical Mathieu equation: where and are real constants.
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The details of the crossover between localized and extended regimes have been studied in detail for some special values of irrational values only, so that one may ask whether these particular values are representative enough.
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That is, , where and are coprime integers.
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By comparing (14) and (20) we realize that in Harper's equation the potential strength maximum value is constrained to the value from the onset, whereas in the Aubry-André equation this potential strength maximum is one of the free parameters of the model.
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Note that this discontinuous character excludes the possibility of an absolutely continuous spectral measure.
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Quite interestingly, this description written in the pre-fractal era uses the term “clustering” instead of “splitting” in order to describe the energy spectrum fragmented pattern. This semantics seems to be more appropriate when considering the physical origin of the hierarchical patterns in terms of resonance effects involving a series of neighboring building blocks interspersed through the lattice.
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The choice of binary lattices composed of just two different types of atoms is just a matter of mathematical simplicity, though one should keep in mind that all of the thermodynamically stable QCs synthesized during the period 1987–2000 were ternary compounds. In 2000 the discovery of the first binary QC was reported (belonging to the (Cd, Ca)Yb system), and its atomic structure was solved in 2007 [7].
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To the best of our knowledge, this state was first studied by Kumar [50], who apparently confused it with the eigenstate of the Fibonacci standard transfer model shown in Figure 13(d).
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In an analogous way, the condition reduces the global transfer matrix of the original triadic Cantor lattice to that corresponding to a periodic structure with a unit cell (plus an isolated at the right end). On the contrary, the condition transforms the original global transfer matrix into that corresponding to the nonhomogeneous fractal lattice .
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