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

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.