Abstract
We study statistical properties of the ensemble of large N×N random matrices whose entries decrease in a power-law fashion ∼|i-j. 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 1<α<3/2 the wave packet spreading at a short time scale is superdiffusive: 〈|r|〉∼, which leads to a modification of the Altshuler-Shklovskii behavior of the spectral correlation function. At 1/2<α<1 the statistical properties of eigenstates are similar to those in a metallic sample in d=(α-1/2 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.
- Received 26 April 1996
DOI:https://doi.org/10.1103/PhysRevE.54.3221
©1996 American Physical Society