Vibrational density of states of general two-component random mixtures near percolation thresholds

The authors study scalar vibrational properties of random networks where two types of harmonic springs, fA and fB=hfA (h<<1), are present with probability p and 1-p, respectively. They develop a scaling theory for the density of states N( omega ) near the critical concentration pc, and test it by numerical simulations on the square lattice. For p>pc and h sufficiently small. They recover the standard fracton-phonon crossover at a characteristic frequency, determined by the correlation length xi , omega xi>> omega B, where omega B approximately fB1/2 is the maximum frequency of the B sublattice. For finite h and p sufficiently close to pc, they obtain a new type of fracton-phonon crossover determined by two characteristic frequencies: omega h (which depends on h) and omega B, with omega h<< omega B. In this case, a hump mediates the crossover from fracton-like modes (occurring for omega >> omega B), to Debye phonons (occurring for omega <> omega xi.