Emergent percolation length and localization in random elastic networks

TL;DR

This paper investigates how vibrational modes in disordered elastic networks become localized or delocalized, revealing a percolation-driven transition and providing insights into phonon behavior in amorphous solids.

Contribution

It introduces a theoretical and numerical analysis of phonon localization in disordered elastic networks, highlighting a percolation-based transition and the role of a divergent percolation length.

Findings

Localization transition exists at a critical frequency above two dimensions.

Delocalized phonons exhibit a Debye spectrum at low frequencies.

Critical frequency for localization decreases exponentially with disorder.

Abstract

We study, theoretically and numerically, a minimal model for phonons in a disordered system. For sufficient disorder, the vibrational modes of this classical system can become Anderson localized, yet this problem has received significantly less attention than its electronic counterpart. We find rich behavior in the localization properties of the phonons as a function of the density, frequency and the spatial dimension. We use a percolation analysis to argue for a Debye spectrum at low frequencies for dimensions higher than one, and for a localization/delocalization transition (at a critical frequency) above two dimensions. We show that in contrast to the behavior in electronic systems, the transition exists for arbitrarily large disorder, albeit with an exponentially small critical frequency. The structure of the modes reflects a divergent percolation length that arises from the disorder in the springs without being explicitly present in the definition of our model. Within the percolation approach we calculate the speed-of-sound of the delocalized modes (phonons), which we corroborate with numerics. We find the critical frequency of the localization transition at a given density, and find good agreement of these predictions with numerical results using a recursive Green function method adapted for this problem. The connection of our results to recent experiments on amorphous solids are discussed.