Low participation ratio vibrational modes in a limit-periodic structure

Motivated by the demonstration that patterned colloidal particles may form a limit-periodic phase\footnote{C.\ Marcoux, T.\ W.\ Byington, Z.\ Qian, P.\ Charbonneau, and J.\ E.\ S.\ Socolar, {\it Phys. Rev. E} {\bf 90}, 012136 (2014).}, we study the nature of vibrational modes in a toy model based on the Taylor-Socolar tiling. We consider a triangular lattice of identical point masses with nearest neighbors connected by springs of two different strengths, where the pattern of spring constants reflects the limit-periodic structure of the tiling. Using calculations of the phonon spectra for crystalline approximants to the limit-periodic structure, we identify several hierarchies of modes shared by the full limit-periodic system that have arbitrarily low participation ratios. We present a heuristic explanation of the existence of such modes, which are robust in the presence of vacancies and small amounts of disorder in the spring constants.