Low-frequency non-phononic vibrations of amorphous solids.

The inherent disorder imparts amorphous solids with a range of anomalous yet universally observed mechanical and thermodynamic properties at low temperatures, which distinguish them from their crystalline counterparts. A comprehensive understanding of these low-temperature anomalies is imperative from all perspectives but still seems to be a long-lasting challenge. In particular, it has long been recognized that low-frequency vibrations play an indispensable role in understanding low-temperature properties of amorphous solids. One noteworthy aspect is that the past decade has witnessed a remarkable surge in numerical and theoretical investigations of the low-frequency non-phononic spectrum beyond the Debye prediction; however, despite great efforts and advancements, some debated problems remain unsolved. Therefore, the discussion of the low-frequency non-phononic spectrum constitutes the primary focus of this review. Additionally, insights provided by low-frequency non-phononic vibrations in comprehending other elusive issues, such as the glass transition, phonon attenuation, two-level systems, and soft spots, are discussed.