Theory for the density of interacting quasi-localised modes in amorphous solids

TL;DR

This paper develops a theoretical framework linking the frequency density of quasi-localised vibrational modes in amorphous solids to shear transformation densities, predicting two regimes based on the shear transformation distribution, and confirms these predictions numerically.

Contribution

It provides the first formal theory connecting the density of quasi-localised modes to shear transformation statistics in amorphous solids at zero temperature.

Findings

For $ heta>1/4$, $ ext{D}_L( extomega) extasciimath=' extasciimath{ extomega}^4$ with vanishing amplitudes at low frequencies.

For $ heta<1/4$, $ ext{D}_L( extomega) extasciimath=' extasciimath{ extomega}^{3+4 heta}$ with finite amplitudes at low frequencies.

Numerical confirmation of the theoretical predictions.

Abstract

Quasi-localised modes appear in the vibrational spectrum of amorphous solids at low-frequency. Though never formalised, these modes are believed to have a close relationship with other important local excitations, including shear transformations and two-level systems. We provide a theory for their frequency density, $D_{L}(\omega)\sim\omega^{\alpha}$, that establishes this link for systems at zero temperature under quasi-static loading. It predicts two regimes depending on the density of shear transformations $P(x)\sim x^{\theta}$ (with $x$ the additional stress needed to trigger a shear transformation). If $\theta>1/4$, $\alpha=4$ and a finite fraction of quasi-localised modes form shear transformations, whose amplitudes vanish at low frequencies. If $\theta<1/4$, $\alpha=3+ 4 \theta$ and all quasi-localised modes form shear transformations with a finite amplitude at vanishing frequencies. We confirm our predictions numerically.