Fracton dynamics of percolating elastic networks: Energy spectrum and localized nature

We have performed computer experiments on the dynamics of two- and three-dimensional (2D and 3D) percolating elastic networks with the site number N${>10\mathrm{}}^5$. The densities of vibrational states (DOS’s) for these large percolating networks as well as the mode patterns of fractons are obtained. It is confirmed, for both 2D and 3D networks, that the DOS is proportional to ${\omega }^{d̃\mathrm{-}1}$ with d̃ close to (4/3 in the regime above the characteristic frequency ${\omega }_c$. We could not find the notable steepness or hump in the DOS at the vicinity of a phonon-fracton crossover frequency ${\omega }_c$, i.e., our results were incompatible with the prediction by the effective-medium theory. Specific realizations of mode patterns show that the core of fracton excitations falls off sharply at their edges, and that the localized nature is quite unique.