Phonon-Fracton Crossover on Fractal Lattices

The spectral properties of a $d$-dimensional bond-percolating network have been examined by real-space rescaling methods. Both the triangular and face-centered-cubic lattices have been studied with the $d$-dimensional Sierpinski gasket used to model the backbone. The vibrational density of states has been calculated as a function of $p-p_c$ and we find that two excitation regimes exist which are characterized by two separate dimensionalities. There is a low-frequency phonon regime and a higher-frequency fracton regime. The specific heat exhibits a smooth crossover from phonon to fracton behavior as a function of $p-p_c$.