Moment-Preserving Theory of Vibrational Dynamics of Topologically Disordered Systems - Folli, Viola; Ruocco, Giancarlo; Schirmacher, Walter (Frontiers in Physics, 2017)
We investigate a class of simple mass-spring models for the vibrational dynamics of topologically disordered solids. The dynamical matrix of these systems corresponds to the Euclidean-RandomMatrix (ERM) scheme. We show that the self-consistent ERM approximation introduced by Ganter and Schirmacher, Phys. Rev. B82, 094205 (2010) preserves the ﬁrst two nontrivial moments of the level density exactly. We further establish a link between these approximations and the ﬂuctuating elasticity approaches. Using this correspondence we derive and solve a new, simpliﬁed mean-ﬁeld theory for calculating the vibrational spectrum of disordered mass-spring models with topological disorder. We calculate and discuss the level density and the spectral moments for a model in which the force constants obey a Gaussian site-separation dependence. We ﬁnd fair agreement between the results of the new theory and a numerical simulation of the model. For systems with ﬁnite size we ﬁnd that the moments strongly depend on the number of sites, which poses a caveat for extrapolating ﬁnite-system simulations to the inﬁnite-size limit.