Power-law eigenstates of a regular Vicsek fractal with hierarchical interactions

The nature of transverse vibrational eigenmodes of a regular Vicsek fractal with hierarchical nearest-neighbor interactions has been analyzed using a real-space renormalization-group approach. Different types of spatial scaling behavior for the nondegenerate transverse vibrational modes are obtained depending on a positive hierarchical parameter R: for R>${\mathit{R}}_{\mathit{c}}$, the nondegenerate modes are all extended in nature, while for R<${\mathit{R}}_{\mathit{c}}$, they exhibit a type of power-law behavior, with the exponent of power depending explicitly on R.