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

Quasilocalized modes appear in the vibrational spectrum of amorphous solids at low frequency. Though never formalized, 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) similar to omega(alpha), that establishes this link for systems at zero temperature under quasistatic loading. It predicts two regimes depending on the density of shear transformations P(x) similar to x(theta) (with x the additional stress needed to trigger a shear transformation). If theta > 1/4, then alpha = 4 and a finite fraction of quasilocalized modes form shear transformations, whose amplitudes vanish at low frequencies. If theta < 1/4, then alpha = 3 + 4 theta and all quasilocalized modes form shear transformations with a finite amplitude at vanishing frequencies. We confirm our predictions numerically.