Excess modes in the vibrational spectrum of disordered systems and the boson peak

We study a disordered vibrational model system, where the spring constants k are chosen from a distribution P(k)∝1/k above a cut-off value kₘᵢₙ>0. We can motivate this distribution by the presence of free volume in glassy materials. We show that the model system reproduces several important features of the boson peak in real glasses: (i) a low-frequency excess contribution to the Debye density of states, (ii) the hump of the specific heat cV(T) including the power-law relation between height and position of the hump, and (iii) the transition to localized modes well above the boson peak frequency.