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Ohmic Dissipation of Tunnelling in Dielectric Glasses at Low Temperature

An additional interaction term of a two-level tunnelling system with the thermal lattice motion is derived, which is missed when expanding the Hamiltonian in terms of the elastic distortion. As a consequence, the bath spectral function comprises a part linear in frequency, besides the well-known contribution proportional to ω3. Whereas the latter is due to the derivative of the phonon field ∇u, the new term arises from coupling to the vibrational amplitude u itself. Since it requires an asymmetric atomic environment, this novel feature is intimately connected to the disordered configuration of atoms in amorphous solids; it leads to an Ohmic damping function which becomes predominant at very low temperature. The resulting linear variation of sound attenuation with temperature possibly resolves a manifest discrepancy between predictions of the tunnelling model and attenuation data below 100 millikelvins.