Effects of deterministic and random discrete scale invariance on spin wave spectra

The properties of magnon spectra in low dimensional random Sierpinski fractals, deterministic fractals, and percolation clusters are calculated and compared. While deterministic scale invariance leads to singular continuous spectra with gaps and degenerated levels, random scale invariance leads to continuous density of states which enables to define low frequency exponents; the exponents associated to random percolation clusters and to Sierpinski fractals are significantly different, even if their fractal dimensions are very close. The study of spacing levels shows quantitatively that the degeneracy is linked to geometrical symmetry in deterministic fractals while it is strongly reduced by random discrete scale invariance.