Phonon interpretation of the 'boson peak' in supercooled liquids

Glasses^1,2 are amorphous solids, in the sense that they display elastic behaviour. In crystalline solids, elasticity is associated with phonons, which are quantized vibrational excitations. Phonon-like excitations also exist in glasses at very high (terahertz; 10¹² Hz) frequencies; surprisingly, these persist in the supercooled liquids³. A universal feature of such amorphous systems is the boson peak: the vibrational density of states has an excess compared to the Debye squared-frequency law. Here we investigate the origin of this feature by studying the spectra of inherent structures⁴ (local minima of the potential energy) in a realistic glass model. We claim that the peak is the signature of a phase transition in the space of the stationary points of the energy, from a minima-dominated phase (with phonons) at low energy to a saddle-point-dominated phase^5-7 (without phonons). The boson peak moves to lower frequencies on approaching the phonon-saddle transition, and its height diverges at the critical point. Our numerical results agree with the predictions of euclidean random matrix theory⁸ on the existence of a sharp phase transition⁹ between an amorphous elastic phase and a phonon-free one.