Transition from localized to extended eigenstates in the ensemble of power-law random banded matrices

We study statistical properties of the ensemble of large N×{}N random matrices whose entries Hᵢⱼ decrease in a power-law fashion Hᵢⱼ{~}|i-j|^-α. Mapping the problem onto a nonlinear {σ} model with nonlocal interaction, we find a transition from localized to extended states at {α}=1. At this critical value of {α} the system exhibits multifractality and spectral statistics intermediate between the Wigner-Dyson and Poisson statistics. These features are reminiscent of those typical of the mobility edge of disordered conductors. We find a continuous set of critical theories at {α}=1, parametrized by the value of the coupling constant of the {σ} model. At {α}>1 all states are expected to be localized with integrable power-law tails. At the same time, for 13/2 the wave packet spreading at a short time scale is superdiffusive: 〈|r|〉{~}t^1/(2α-1), which leads to a modification of the Altshuler-Shklovskii behavior of the spectral correlation function. At 1/21 the statistical properties of eigenstates are similar to those in a metallic sample in d=({α}-1/2)^-1 dimensions. Finally, the region {α}1/2 is equivalent to the corresponding Gaussian ensemble of random matrices ({α}=0). The theoretical predictions are compared with results of numerical simulations. {} 1996 The American Physical Society.