Self-consistent Euclidean-random-matrix theory

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

By calculating all terms of the high-density expansion of the 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 $\Sigma(k,z)\propto k^2z^{d/2}$ and not $\Sigma(k,z)\propto k^2z^{(d-2)/2}$, 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 $Z(t)\propto t^{(d+2)/2}$.

Model calculations for vibrational properties of disordered solids and the “boson peak”

Abstract It is demonstrated that a disordered system of coupled classical harmonic oscillators with a continuous distribution of coupling parameters exhibits generally a low-frequency enhancement (“boson peak”) of the density of states, as compared with the Debye law. This phenomenon is most pronounced if the system is close to an instability. This is shown by means of a scalar model on a simple cubic lattice. The force constants are assumed to fluctuate from bond to bond according to a Gaussian distribution which is truncated at its lower end. The model is solved for the density of states and the one-phonon dynamic structure factor S(q, ω) by applying the two-site coherent potential approx…

Harmonic Vibrational Excitations in Disordered Solids and the "Boson Peak"

We consider a system of coupled classical harmonic oscillators with spatially fluctuating nearest-neighbor force constants on a simple cubic lattice. The model is solved both by numerically diagonalizing the Hamiltonian and by applying the single-bond coherent potential approximation. The results for the density of states $g(\omega)$ are in excellent agreement with each other. As the degree of disorder is increased the system becomes unstable due to the presence of negative force constants. If the system is near the borderline of stability a low-frequency peak appears in the reduced density of states $g(\omega)/\omega^2$ as a precursor of the instability. We argue that this peak is the anal…