Marcoux, C; Socolar, JES, *Sparse phonon modes of a limit-periodic structure*,
Physical Review B - Condensed Matter and Materials Physics, vol. 93 no. 17
(May, 2016), American Physical Society [PhysRevB.93.174102], [doi] .
**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.*