Critical properties in long-range hopping Hamiltonians

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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 (bᵈ ≫ 1) and the strong (bᵈ ≪ 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 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) P (s) has the asymptotic form Pc(s) (-Adsα) for s≫, with the critical exponent α=2-ad/bᵈ for bᵈ ≫ 1 and α=1+cd bᵈ for bᵈ ≪ 1. In these cases the numerical coefficients Ad, ad and cd depend only on the dimensionality.