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 x N random matrices whose entries H-ij decrease in a power-law fashion H-ij similar to \i-j\(-alpha). Mapping the problem onto a nonlinear a model with nonlocal interaction, we find a transition from localized to extended states at alpha=1. At this critical value of alpha 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 alpha=1, parametrized by the value of the coupling constant of the sigma model. At alpha>1 all states are expected to be localized with integrable power-law tails. At the same time, for 1<alpha<3/2 the wave packet spreading at a short time scale is superdiffusive: [\r\] similar to t(1/(2 alpha-1)), which leads to a modification of the Altshuler-Shklovskii behavior of the spectral correlation function. At 1/2<alpha<1 the statistical properties of eigenstates are similar to those in a metallic sample in d=(alpha-1/2)(-1) dimensions. Finally, the region alpha<1/2 is equivalent to the corresponding Gaussian ensemble of random matrices (alpha=0). The theoretical predictions ate compared with results of numerical simulations.