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 random matrices whose entries H₈₉ decrease in a power-law fashion H₈₉|i-j|^-. Mapping the problem onto a nonlinear model with nonlocal interaction, we find a transition from localized to extended states at =1. At this critical value of 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 =1, parametrized by the value of the coupling constant of the model. At >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/ (2-1), 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= (-1/2) ^-1 dimensions. Finally, the region 1/2 is equivalent to the corresponding Gaussian ensemble of random matrices (=0). The theoretical predictions are compared with results of numerical simulations. 1996 The American Physical Society.