Scattering at the Anderson transition: Power--law banded random matrix model

TL;DR

This paper investigates the scattering properties at the critical point of a one-dimensional power-law banded random matrix model, revealing how delay times and resonance widths scale with system size at the metal-insulator transition.

Contribution

It provides a novel analysis of the scaling behavior of scattering delay times and resonance widths at criticality in a power-law banded random matrix model.

Findings

Typical delay times scale as L^{D_1}

Resonance widths scale as L^{-(2-D_2)}

Scaling relates to eigenfunction dimensions D_1 and D_2

Abstract

We analyze the scattering properties of a periodic one-dimensional system at criticality represented by the so-called power-law banded random matrix model at the metal insulator transition. We focus on the scaling of Wigner delay times $\tau$ and resonance widths $\Gamma$. We found that the typical values of $\tau$ and $\Gamma$ (calculated as the geometric mean) scale with the system size $L$ as $\tau^{\tiny typ}\propto L^{D_1}$ and $\Gamma^{\tiny typ} \propto L^{-(2-D_2)}$, where $D_1$ is the information dimension and $D_2$ is the correlation dimension of eigenfunctions of the corresponding closed system.