We investigate the vibrational properties of topologically disordered materials by analytically studying particles that harmonically oscillate around random positions. Exploiting classical field theory in the thermodynamic limit at $T=0$, we build up a self-consistent model by analyzing the Hessian utilizing Euclidean random matrix theory. In accordance with earlier findings [T. S. Grigera et al.J. Stat. Mech. (2011) P02015.], we take nonplanar diagrams into account to correctly address multiple local scattering events. By doing so, we end up with a first principles theory that can predict the main anomalies of athermal disordered materials, including the boson peak, sound softening, and Rayleigh damping of sound. In the vibrational density of states, the sound modes lead to Debye’s law for small frequencies. Additionally, an excess appears in the density of states starting as ${\omega }^4$ in the low frequency limit, which is attributed to (quasi-) localized modes.

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• Classical statistical mechanics
• Mechanical & acoustical properties
• Disordered systems
• Glasses
• Diagrammatic methods
• Elastic wave theory
• Green's function methods
• Statistical methods