Transition from localized to extended eigenstates in the ensemble of power-law random banded matrices
Transition from localized to extended eigenstates in the ensemble of power-law random banded matrices
We study statistical properties of the ensemble of large $N\times N$ random matrices whose entries $ H_{ij}$ decrease in a power-law fashion $H_{ij}\sim|i-j|^{-\alpha}$. Mapping the problem onto a nonlinear $\sigma-$model with non-local interaction, we find a transition from localized to extended states at $\alpha=1$. At this critical value of $\alpha$ …