We show that the amplitude fluctuations of fractons on a percolating cluster have well-defined multifractal properties. The multifractal spectrum turns out to be explicitly dependent on the frequency of the modes and rather stable for modes with similar frequency. The physical implications of this result are that self-similar fluctuations are an intrinsic characteristic of fractons. Therefore, smoothing or averaging procedures can be questionable in this respect. This property could also help clarify the ambiguities that are present in the definition of the correlation length. It is interesting to note that the localization of fractons is of a different nature with respect to Anderson localized wave functions because these are multifractal only at the mobility edge.
We discuss the localization of electronic eigenfunctions and vibrational excitations (``fractons'') on self-similar percolation clusters at criticality on the Cayley tree, in
d=2, and in
d=3, both in topological
(l) and in Euclidian
(r) space. We find that the localization behavior of electrons and fractons is very similar, but does depend on the type of average performed and on the number of configurations
N taken into account. It can be characterized by three different localization regimes. In the first regime, at small distances
l and
r from the localization center, electrons and fractons are superlocalized in
l space, while the localization is not exponential in
r space. In the intermediate regime, which is the most important one, we find stretched exponential localization (``sublocalization''), $\mathrm{ln}〈\ensuremath{\psi}〉\ensuremath{\sim}\ensuremath{-}{\mathcal{l}}^{{d}_{\ensuremath{\Psi},\mathcal{l}}}$ or $\ensuremath{\sim}\ensuremath{-}{r}^{{d}_{\ensuremath{\Psi},r}}$ with effective localization exponents ${d}_{\ensuremath{\Psi},\mathcal{l}}\ensuremath{\approx}{d}_{\ensuremath{\Psi},r}\ensuremath{\approx}0.6$. In the third regime, for large
l and
r, the averages strongly depend on the number of configurations, even in
l space, and superlocalization ${(d}_{\ensuremath{\Psi},\mathcal{l}}{,d}_{\ensuremath{\Psi},r}>1)$ is observed, converging to simple exponential behavior asymptotically. To understand this complicated localization behavior, we present an analytical calculation of $\mathrm{ln}〈\ensuremath{\psi}〉$ for the third regime, which is based on the fact that the local amplitudes at large distances from the localization center obey a log-normal distribution.
We present a study of the one-magnon density of states in the three-dimensional dilute Heisenberg ferromagnet. The ensemble-averaged density of states is evaluted by calculating a small number of spectral moments and employing a maximum-entropy criterion. Finite-size-lattice data, via common numerical simulation techniques, are also presented. The results demonstrate the fracton excitations at the percolation concentration
pc as expected from the Alexander-Orbach conjecture and show the presence of the magnon-to-fracton crossover. At the percolation threshold
pc, we estimate a value for the spectral dimension of
ds\ensuremath{\approxeq}1.52. The corresponding problem of phonons in random materials is also discussed in connection with the available experimental data.
The fracton dynamics of percolating Heisenberg antiferromagnets are studied in terms of a dynamic scaling argument and large-scale numerical analysis. It is shown that the spectral (or fracton) dimension for antiferromagnetic fractons,
d~AF, is bounded by the relation
d~AF≤1. The densities of states (DOS) are calculated for very large percolating antiferromagnets at the critical concentration p c formed on d =2, 3, and 4 simple cubic lattices. It is concluded, from the calculated DOS's, that the spectral dimensions
d~AF are very close to unity for any Euclidean dimension d (≥2). We have also calculated the DOS for percolating antiferromagnets above p c in order to clarify the crossover behavior from extended magnons to fractons. In addition, the dynamical structure factor S ( q ,ω) is investigated both analytically and numerically. It is found, postulating a single-length scaling, that the asymptotic behaviors can be characterized by two exponents, the dynamical exponent z AF and a new exponent y . The numerical results for S ( q ,ω) support the validity of the single-length-scale postulate.
A numerical study of the spin-wave density of states on the infinite percolating cluster of the site diluted Heisenberg ferromagnet is presented. The Gaussian elimination algorithm is employed for the accurate determination of the integrated density of states as a function of concentration in large two-dimensional strips. At the percolation threshold the energy dependence of the density of states is \ensuremath{\rho}(E)\ensuremath{\propto}${E}^{\mathrm{\ensuremath{-}}\ensuremath{\varphi}}$. We obtained the value of \ensuremath{\varphi}=0.32\ifmmode\pm\else\textpm\fi{}0.02 consistent with the presence of fracton excitations at all length scales, in agreement with previous estimates. For concentrations above
pc the density of states is constant in the magnon or long-wavelength regime while a crossover to the fracton regime occurs at shorter wavelengths. The magnon-fracton crossover in energy is smooth and no indication of a fracton-edge peak is found for this system.