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, $D_{L}(ω)\simω^α$, that establishes this link for systems at zero temperature under quasi-static loading. It predicts two regimes depending on the density of shear transformations $P(x)\sim x^θ$ (with $x$ the additional stress needed to trigger a shear transformation). If $θ>1/4$, $α=4$ and a finite fraction of quasi-localised modes form shear transformations, whose amplitudes vanish at low frequencies. If $θ<1/4$, $α=3+ 4 θ$ and all quasi-localised modes form shear transformations with a finite amplitude at vanishing frequencies. We confirm our predictions numerically.