Replica field theory for anharmonic sound attenuation in glasses

A saddle-point treatment of interacting phonons in a disordered environment is developed. In contrast to crystalline solids, anharmonic attenuation of density fluctuations becomes important in the hydrodynamic regime, due to a broken momentum conservation. The variance of the shear modulus Δ2 turns out to be the strength of the disorder enhanced phonon–phonon interaction. In the low-frequency regime (below the boson peak frequency) we obtain an Akhiezer-like sound attenuation law Γ ∝ Τω2. Together with the usual Rayleigh scattering mechanism this yields a crossover of the Brillouin linewidth from a ω2 to a ω4 regime. The crossover frequency ωc is fully determined by the boson peak frequency and the temperature. For network glasses like SiO2 at room temperature this crossover is predicted to be situated one order of magnitude below the boson peak frequency.