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 d(2) (for d = 2) and the nearest level spacing distribution P-c(s) (for d = 3) in both the weak (b(d) much greater than 1) and the strong (b(d) much less than 1) coupling regime, where the parameter b(-d) plays the role of the coupling constant of the model. It is found that (i) the extrapolated values of d(2) are of the form d(2) = C(d)b(d) in the strong coupting limit and d(2) = d - a(d)/b(d) in the case of weak coupling, and (ii) P-c(s) has the asymptotic form P-c(s) similar to exp (-A(d)S(a)) for s much greater than 1, with the critical exponent a = 2 - a(d)/b(d) for b(d) much greater than 1 and a = 1 + C(d)b(d) for b(d) much less than 1. In these cases the numerical coefficients A(d), a(d) and c(d) depend only on the dimensionality. (C) 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.