Self-similarity and scaling of wave function for binary quasiperiodic chains associated with quadratic irrationals
By establishing the correspondence between the substitution rule (a→anb; b→a) and the transformation on the value of ω=tan φ byω→1/(n+ω) in the cut-and-project (CP) method, it is proved that the necessary and sufficient conditions for a binary quasiperiodic (QP) sequence made by the CP method to be self-similar is that ω is a quadratic irrational (QI) number. And, vice versa, the necessary condition for a binary self-similar sequence generated by the substitution rule to be obtainable by the CP method is that the corresponding substitution rule can be rewritten as a simple composition of transformations with the type (a→anb; b→a). To illustrate some physical properties of the self-similar QP chains associated with QI numbers, we analyze the scaling behaviour of the wave function atE=0 for an off-diagnonal tight-binding model. It is shown that the wave function atE=0 grows at most by a power-law for the QP lattices, characterized by a special class of QI numbers. For the QP chains associated with general QI numbers, with the same logic, the typical scaling behaviour of the wave function is conjectured to be the same.
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