Properties Of Normal Modes In A Modified Disordered Klein-Gordon Lattice: From Disorder To Order

We introduce a modified version of the disordered Klein-Gordon lattice model, having two parameters for controlling the disorder strength: D, which determines the range of the coefficients of the on-site potentials, and W, which defines the strength of the nearest-neighbor interactions. We fix W = 4 and investigate how the properties of the system's normal modes change as we approach its ordered version, i.e. D -> 0. We show that the probability density distribution of the normal modes' frequencies takes a 'U'-shaped profile as D decreases. Furthermore, we use two quantities for estimating the modes' spatial extent, the so-called localization volume V (which is related to the mode's second moment) and the mode's participation number P. We show that both quantities scale as proportional to D-2 when D approaches zero and we numerically verify a proportionality relation between them as V/P approximate to 2.6.