Dilution-induced criticality is discussed using exact and scaling descriptions. A simple exact argument is used to obtain the scaling form for the density of states in dilute chains. For higher-dimensional systems it is shown that the condition for the existence of a 'fraction' edge is equivalent to the violation of a scaling relation between the fracton dimension of the infinite cluster, d and various static exponents. In addition the result dd/d f is obtained for the fracton dimensionality of the finite clusters near the percolation threshold; d f is the fractal dimension of the infinite cluster.
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.
We report the observations of the dynamical structure factors S(q,ω) for antifferomagnetic fractons in diluted three-dimensional (3d) and two-dimensional (2d) Heisenberg systems, RbMn0.4Mg0.6F3 and Rb2Mn0.598Mg0.402F4, with a magnetic concentration close to the percolation concentration, which were obtained by means of high-resolution (ΔE = 17.5 μeV) inelastic neutron scattering experiments. The peak intensity A(q) and the dispersion relation E(q) show the clear scaling laws following to A(q) ∝ q−y with y = 2.9 ± 0.1 and E(q) ∝ qz with z = 2.5 ± 0.1 for the 3d system, and y = 2.9 ± 0.2 and z = 1.8 ± 0.2 for the 2d system. The validity of the single-length-scaling postulate (SLSP) for S(q,ω) are demonstrated, for the first time. In addition, we show that the spectral dimension of antiferromagnetic fractons is unity independent of the embedding Euclidean dimension of the systems. These values are consistent with the theoretical predictions.
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.
The dynamical structure factors, S ( q ,ω), for antiferromagnetic fractons in diluted three-dimensional (3D) and two-dimensional (2D) Heisenberg systems, RbMn 0.4 Mg 0.6 F 3 and Rb 2 Mn 0.598 Mg 0.402 F 4 , respectively, with a magnetic concentration close to the percolation concentration, were observed through high-resolution (Δ E = 17.5 µeV) inelastic neutron scattering experiments well below T N . The exponents describing the peak intensity ( A ( q ) ∝ q - y ) and the dispersion relation ( E ( q ) ∝ q z ) were determined to be y = 2.9 ±0.1 and z = 2.5 ±0.1 for the 3D system and y = 2.9 ±0.2 and z = 1.8 ±0.2 for the 2D system, respectively. In addition, the validity of the single-length-scaling postulate (SLSP) was confirmed for S ( q ,ω) by using the above values of y and z . We experimentally demonstrated, for the first time, that the spectral dimension,
d~AF=1, is universal and independent of the embedding Euclidean dimensions of the systems.