Sparse phonon modes of a limit-periodic structure

TL;DR

This paper investigates vibrational modes in a limit-periodic structure, revealing unique extended phonon modes with low participation ratios and large-scale oscillations, supported by numerical analysis of periodic approximants.

Contribution

It identifies and characterizes a new class of extended vibrational modes specific to limit-periodic systems, supported by numerical evidence and heuristic explanations.

Findings

Existence of extended modes with low participation ratios

Oscillations on periodic nets with large lattice constants

Numerical evidence from periodic approximants

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.