Investigating the Vibrational Properties of Disordered Solids Using Random Matrix Models

Glasses and jammed packings exhibit interesting low-frequency vibrational behavior, including a region in the density of states, D(ω), that scales as ω⁴ with quasi-localized excitations important for flow and failure. However, to our knowledge, there are no constructive models that generate ω⁴ scaling and explain the mechanism for quasi-localization. Recent work indicates random matrix models can provide explanations for universal vibrational properties in glasses. Here we analyze the Laplacian Matrix of ring networks where disorder is controlled by the distribution of bond weights and adding bonds to distort the network. Simulating this model with high statistics, we find 2 crossover frequencies. The larger frequency, ω*, corresponds to the end of a plateau in D(ω) and scales with Δz, just as the boson peak in jammed systems. The second crossover frequency, ω_c, is dependent on the density of weak bonds. For a uniform bond distribution, we find ωc scales with Δz and D(ω) scales with ω³ below ω_c, and between ω_c and ω*, the spectrum scaling is consistent with ω⁴ independent of Δz and system size. This scaling disappears if we perturb away from a uniform distribution. These results suggest this simple model may be a powerful tool for understanding localization in structural glasses.