Critical properties in long-range hopping Hamiltonians

Some properties of d-dimensional disordered models with long-range random hopping amplitudes are investigated numerically at criticality. We concentrate on the correlation dimension d2 (for d = 2) and the nearest level spacing distribution Pc(s) (for d = 3) in both the weak (b ≫ 1) and the strong (b ≪ 1) coupling regime, where the parameter b plays the role of the coupling constant of the model. It is found that (i) the extrapolated values of d2 are of the form d2 = cdbd in the strong coupling limit and d2 = d− ad/bd in the case of weak coupling, and (ii) Pc(s) has the asymptotic form Pc(s) ∼ exp(−Adsα) for s ≫ 1, with the critical exponent α = 2 − ad/bd for b ≫ 1 and α = 1 + cdbd for b ≪ 1. In these cases the numerical coefficients Ad, ad and cd depend only on the dimensionality.