Abstract
We consider a one-dimensional structure obtained by stringing two types of “beads” (short and long bonds) on a line according to a quasiperiodic rule. This model exhibits a new kind of order, intermediate between quasiperiodic and random, with a singular continuous Fourier transform (i.e., neither Dirac peaks nor a smooth structure factor). By means of an exact renormalization transformation acting on the two-parameter family of circle maps that defines the model, we study in a quantitative way the local scaling properties of its Fourier spectrum. We show that it exhibits power-law singularities around a dense set of wavevectorsq, with a local exponentγ(q) varying continuously with the ratio of both bond lengths. Our construction also sheds some new light on the interplay between three characteristic properties of deterministic structures, namely: (1) a bounded fluctuation of the atomic positions with respect to their average lattice; (2) a quasiperiodic Fourier transform, i.e., made of Dirac peaks; and (3) for sequences generated by a substitution, the number-theoretic properties of the eigenvalue spectrum of the substitution.
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Aubry, S., Godrèche, C. & Luck, J.M. Scaling properties of a structure intermediate between quasiperiodic and random. J Stat Phys 51, 1033–1075 (1988). https://doi.org/10.1007/BF01014899
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DOI: https://doi.org/10.1007/BF01014899