Universality in the Vibrational Spectra of Single-Component Amorphous Clusters

We have numerically investigated the vibrational spectra of amorphous single-component clusters for several types of interactions among the particles. For all the potentials we have studied, we find that the density of states can be described, except at the two ends of the spectrum, by the same functional form to a very good approximation, and that the fluctuation properties of the spectra in this central region converge to those of the Gaussian orthogonal ensemble of random matrices with increasing system size. We conjecture that this scenario is true for a broad class of potentials.

Figure 1

Filled circles: Eigenvalue number vs eigenvalue ($\lambda$) for a typical spectrum (LJ potential with $N=2000$). For clarity some data points have been removed. Continuous line: Best fit curve obtained after removing the lowest 10% and the highest 20% of the spectrum. Inset: Histogram of eigenvalues, with bin width of 1.89, for the full spectrum. The dip in the count for the first bin is caused by DOS going to zero. Histogram for frequency ($\omega =\sqrt{\lambda }$) looks similar to those in Fig. 1 of Ref. 18.

Figure 2

Filled circles: Normalized nearest neighbor spacing ($s$) vs probability density $p(s)$ (LJ potential with $N=1000$). Crosses: Wigner’s surmise for GOE. Continuous line: Exact prediction for the GOE.

Figure 3

Variance of the number of levels in intervals of length $r$ shown as a function of $r$ for LJ potential. The dotted lines from top to bottom are for $N=200$, 500, 1000, and 2000. The continuous line is the prediction for the GOE.

Figure 4

Filled circles: Skewness and excess parameters of the distribution of $n(r)$, the number of levels in interval of length $r$, shown as a function of $r$ (LJ potential with $N=1000$). Prediction for the GOE is also superposed (dashed lines).