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

TL;DR

This paper investigates the transition from localized to extended eigenstates in power-law random banded matrices, revealing a critical point at alpha=1 with multifractality and intermediate spectral statistics, supported by numerical simulations.

Contribution

It introduces a nonlinear sigma-model approach to analyze the localization transition in power-law random matrices, identifying a critical point at alpha=1 with multifractal eigenstates.

Findings

Transition at alpha=1 from localized to extended states

Multifractality and intermediate spectral statistics at criticality

Superdiffusive wave packet spreading for 1<alpha<1.5

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 $H_{ij}\sim|i-j|^{-\alpha}$. 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 $\alpha$ 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 $\alpha=1$, 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 short time scale is superdiffusive: $\langle |r|\rangle\sim t^{\frac{1}{2\alpha-1}}$, 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 are compared with results of numerical simulations.