Euclidean random matrix theory: low-frequency non-analyticities and Rayleigh scattering

Abstract

By calculating all terms of the high-density expansion of Euclidean random matrix theory (up to second-order in the inverse density) for the vibrational spectrum of a topologically disordered system, we show that the low-frequency behavior of the self-energy is given by Σ(k, z) ∝ k ² z ^d/2 and not Σ(k, z) ∝ k ² z ^(d−2)/2, as claimed previously. This implies the presence of Rayleigh scattering and long-time tails of the velocity autocorrelation function of the analogous diffusion problem of the form Z(t) ∝ t ^(d+2)/2.

Keywords:

• topological disorder
• Euclidean random matrix
• Rayleigh scattering
• diffusion
• long-time tails
• velocity autocorrelation function
• low-frequency non-analyticity

Notes

Notes

1. The real-space transition rate will always carry the letter r in the argument. On all other occasions, it will refer to the Fourier transform.

2. For better readability, the z-dependence of G ₀ will be suppressed in the following.

3. In the following we shall not explicitly mention the q ² from the integration.