DENSITY OF STATES, LEVEL-STATISTICS AND LOCALIZATION OF FRACTONS IN 2-DIMENSIONAL AND 3-DIMENSIONAL DISORDERED-SYSTEMS

We calculate by the Lanczos method the density of spin wave states, and its fluctuation properties on the infinite percolating cluster of a randomly site-dilute Heisenberg ferromagnet. Our results demonstrate that the averaged density follows the fracton laws with spectral dimension values d(s) = 1.32 and d(s) = 1.30 in two and three dimensions, respectively, and is smooth at the magnon-fracton crossover. Similar laws are also shown in the case of continuous disorder on the bonds of the clusters. The density fluctuations are studied via the nearest energy-level-spacing distribution function P(S), which is shown to obey the Wigner surmise with level-repulsion far from the percolation threshold p(c) and an almost Poisson law with uncorrelated spectrum at p(c). The localization properties of excitations are investigated by considering the density of states fluctuations and also via the participation ratio of the eigenvector amplitudes. It is seen that the fracton states are sharply localized. Our results are further discussed in connection to previous theories and numerical data.