Anomalous phonon dispersion near yielding in athermal crystals

Vibrational properties of ordered athermal solids near yielding remain poorly understood. We show that yielding in a sheared crystal is governed not by a single localized instability but by directionally extended multimode softening that forms a cross-shaped low-frequency region in wave number space. Near yielding, the acoustic dispersion $\omega \sim k$ is replaced by $\omega \sim k^2$ along the soft direction, and the vibrational density of states crosses over from Debye to non-Debye scaling, with a diverging length scale. We analytically derive these scaling laws.