MOBILITY EDGE IN QUANTUM PERCOLATION - FRACTAL CHARACTER OF EIGENFUNCTIONS AND A RELATION FOR THE CORRELATION DIMENSION

A numerical study on large cubic lattices (edge length L up to 30 atomic sites) of eigenfunctions at the mobility edge in quantum percolation establishes a well-defined correlation dimension d of 1.26, which is significantly less than earlier estimates. This result is established by means of an innovative scaling argument that does not require the identification of an extended region of linear behavior on a log-log plot in order to measure a fractal dimension. The method has the advantage of showing, in a simple graphical way, the range and intensity of finite-size effects. In the case at hand, it indicates that lattice sizes on the order of at least L = 30 are necessary for obtaining reliable values for the correlation dimension. In addition, a simple quantitative relation between the correlation dimension and the relative localization length is proposed. This relation is shown to hold quite accurately (within 5%) in the case at hand.