Unified theory of phonon in solids with phase diagram of non-Debye anomalies

The classical Debye model successfully predicts phononic contribution to the specific heat of solids in the continuum limit. However, as the phonon wavenumber increases, their vibrational density of states gradually deviates from the Debye prediction and eventually manifests as Van Hove singularities for crystals and a boson peak for glasses. So far, there is still much controversy over whether these two non-Debye anomalies are equivalent or not. Here we propose a unified model and demonstrate that it describes the vibrational density of states in both crystals and glasses. We achieve this by treating the vibrational excitation of solids as the elastic phonons resonating with local modes. Our modelling enables the construction of a phase diagram of non-Debye phonon anomalies. We clarify that the Van Hove singularity and boson peak can evolve as two variants of the same entity when the dispersion displays continuous softening; otherwise, they emerge separately due to resonance-induced extra acoustic softening, further proving by their coexistence. The model is supported by a comparison with experimental heat capacity data over a wide range of real solids, including 143 crystalline and glassy substances. These findings provide a unified picture of the Van Hove singularity and boson peak, and deepen our fundamental understanding of the continuum elasticity of real solids. Van Hove singularities in crystals and the boson peak in glasses are both deviations from the Debye theory of phonons. A unified model for the vibrational density of states now shows when these two phenomena have distinct phononic origins.