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A theory of vibrational excitations based on power-law spatial correlationsin the elastic constants (or equivalently in the internal stress) is derived,in order to determine the vibrational density of states $D(\omega)$ ofdisordered solids. The results provide the first prediction of a boson peak inamorphous materials where spatial correlations in the internal stresses (orelastic constants) are of power-law form, as is often the case in experimentalsystems, leading to logarithmic enhancement of (Rayleigh) phonon attenuation. Alogarithmic correction of the form $\sim -\omega^{2}\ln\omega$ is predicted tooccur in the plot of the reduced excess DOS for frequencies around the bosonpeak in 3D. Moreover, the theory provides scaling laws of the density of statesin the low-frequency region, including a $\sim\omega^{4}$ regime in 3D, andprovides information about how the boson peak intensity depends on the strengthof power-law decay of fluctuations in elastic constants or internal stress.Analytical expressions are also derived for the dynamic structure factor forlongitudinal excitations, which include a logarithmic correction factor, andnumerical calculations are presented supporting the assumptions used in thetheory.

Vibrational density of states of amorphous solids with long-ranged power-law correlated disorder in elasticity