Weyl's theorem and the theoretical origin of Boson peak in two-dimensional scalar spring network

The origin of the anomalous low-frequency vibrational density of states (VDOS) universally observed in amorphous solids has been debated for decades. In this work, we model amorphous solids as a zero-temperature scalar spring network, the simplest model that captures both the amorphous structure and the features of the anomalous VDOS. Using this scalar network, we mathematically demonstrate that disorder can lead to a shift of the VDOS toward low frequencies, thereby accumulating the anomalous part there. Disorder is defined through the topological connectivity of the network. The validity and universality of the theoretical framework are general and valuable for real-world scenarios because it rests only on Weyl’s theorem for Hermitian matrices and is free of perturbation and phenomenological assumptions. This work strips away complex factors and reveals, at the level of mathematical principles, a profound and clean mechanism for the origin of the anomalous VDOS in amorphous solids.