Singularities in Hessian element distributions of amorphous media*

We show that the distribution of elements H in the Hessian matrices associated with amorphous materials exhibit singularities P(H) ∼ |H|^γ with an exponent γ < 0, as |H| → 0. We utilize the rotational invariance of amorphous structures to derive these exponents exactly for systems with particles interacting via radially symmetric, pairwise potentials. We show that γ depends only on the degree of smoothness at cutoff of the interaction potential, independent of other details of interaction, and is identical in both two and three dimensions. We verify our predictions with numerical simulations of models of structural glass formers. Finally, we show that such singularities affect the vibrational properties of amorphous solids, through the distributions of the minimum eigenvalue of the Hessian matrix. Crucially, short-ranged interaction models display a novel, universal, non-debye regime in the density of states.

Reference: https://arxiv.org/abs/2005.02180