Robustness of density of low-frequency states in amorphous solids

Low-frequency quasilocalized modes of amorphous glass appear to exhibit universal density of states, depending on the frequencies as $D\left(\omega \right)\sim {\omega }^4$. To date, various models of glass formers with short-range binary interaction and network glass with both binary and ternary interactions were shown to conform with this law. In this paper, we examine granular amorphous solids with long-range electrostatic interactions and find that they exhibit the same law. To rationalize this wide universality class, we return to a model proposed by Gurevich, Parshin, and Schober (GPS) [Phys. Rev. B 67, 094203 (2003)] and analyze its predictions for interaction laws with varying spatial decay, exploring this wider-than-expected universality class. Numerical and analytic results are provided for both the actual system with long-range interaction and for the GPS model.

Figure 1

The low (small-frequency) tail of the density of states of the two-dimensional model of charged disks discussed in Sec. 2 for four different system sizes, i.e., $N=600,1000,2000$, and 8000 in panels (a)–(d), respectively. In all cases, we employed 10 000 independent configurations. The lines are best fits to the small-frequency tail of the DOS. We conclude from this data that the density of states converges to the universal law Eq. (1).

Figure 2

The low (small-frequency) tail of the density of states of the three-dimensional model of charged spheres discussed in Sec. 2 for three different system sizes, i.e., $N=3600,4500$, and 5500 in panels a-c respectively. In all cases, we employed 5000 independent configurations. The lines are best fits to the small frequency tail of the DOS. We conclude from this data that the density of states converges to the universal law Eq. (1).

Figure 3

A representative example of an eigenmode in two dimensions of the Hessian matrix whose frequency lies in the range of the universal ${\omega }^4$ regime. The participation ratio of this mode is $5.9\times {10}^{-2}$. It is obviously quasilocalized.

Figure 4

A representative example of an eigenmode in three dimensions of the Hessian matrix whose frequency lies in the range of the universal ${\omega }^4$ regime. The participation ratio of this mode is $3\times {10}^{-2}$. It is obviously quasilocalized.

Figure 5

Probability distribution function of forces $P\left(f\right)$ for different system sizes, for the GPS model in three dimensions with $\alpha =3$ (a), $\alpha =2$ (b), and $\alpha =1$ (c). In all cases, we employed 10 000 independent realizations. As expected from the analysis in Sec. 3c, full convergence is observed in (a), marginal convergence in (b), and no convergence in (c).

Figure 6

The low-frequency regime of the density of states of the GPS model as found from the largest available system size for $\alpha =1$ (a), $\alpha =2$ (b), and $\alpha =3$ (c). In all cases, we employed 10 000 independent realizations. The continuous lines are best linear fits to the data. The dashed line in (c) has a slope of unity, revealing the regime of the first reconstruction, in agreement with Eq. (10).

Figure 7

The low-frequency regime of the density of states of the GPS model computed with a uniform distribution $g_0$ in the interval [0,1]. Here we employ 5000 realizations of a system with $N=$ 8000. We see a loss of the linear regime $g_1\propto \omega$, as expected, but we still get the universal behavior at low frequencies!