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

Abstract

We study statistical properties of the ensemble of large $N\times N$ random matrices whose entries $ H_{ij}$ decrease in a power-law fashion $H_{ij}\sim|i-j|^{-α}$. Mapping the problem onto a nonlinear $σ-$model with non-local 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 one. These features are reminiscent of those typical for 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 short time scale is superdiffusive: $\langle |r|\rangle\sim t^{\frac{1}{2α-1}}$, 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 $d=(α-1/2)^{-1}$ 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.

19 pages REVTEX, 4 figures