Sparse phonon modes of a limit-periodic structure

Abstract

Limit-periodic structures are well ordered but nonperiodic, and hence have nontrivial vibrational modes. We study a ball and spring model with a limit-periodic pattern of spring stiffnesses and identify a set of extended modes with arbitrarily low participation ratios, a situation that appears to be unique to limit-periodic systems. The balls that oscillate with large amplitude in these modes live on periodic nets with arbitrarily large lattice constants. By studying periodic approximants to the limit-periodic structure, we present numerical evidence for the existence of such modes, and we give a heuristic explanation of their structure.

Spectral Classification of One‐Dimensional Binary Aperiodic Crystals: An Algebraic Approach

“…[20,26] The point-group symmetry of the di¤raction spectrum of the limit-periodic structures belonging to the III spectral class is compatible with periodicity (unlike QCs) and their overall atomic structure can be described in terms of a union of periodic substructures with ever increasing lattice constants, forming a sequence of successive sublattices. [27,28] Analogously, the di¤raction spectrum of the limit-quasiperiodic class representatives can be generated by a superposition of spectra of an in…nite number of quasiperiodic patterns. In fact, geometrically, a limit-quasiperiodic structure can be regarded as a section of a limit-periodic lattice in a higher dimension space, just as quasiperiodic structures can be obtained as sections of periodic lattices in high dimensions.…”