Anderson transitions : multifractal or non-multifractal statistics of the transmission as a function of the scattering geometry

TL;DR

This paper investigates how different scattering geometries affect the multifractal or monofractal statistics of transmission at the Anderson transition, revealing geometry-dependent statistical behaviors through numerical analysis.

Contribution

It introduces a detailed numerical study of transmission statistics across various scattering geometries, connecting multifractal and non-multifractal behaviors at criticality.

Findings

Transmission with one incoming and many outgoing wires is multifractal.

Backward scattering channels lead to monofractal transmission statistics.

Scattering geometry significantly influences transmission statistics off criticality.

Abstract

The scaling theory of Anderson localization is based on a global conductance $g_L$ that remains a random variable of order O(1) at criticality. One realization of such a conductance is the Landauer transmission for many transverse channels. On the other hand, the statistics of the one-channel Landauer transmission between two local probes is described by a multifractal spectrum that can be related to the singularity spectrum of individual eigenstates. To better understand the relations between these two types of results, we consider various scattering geometries that interpolate between these two cases and analyse the statistics of the corresponding transmissions. We present detailed numerical results for the power-law random banded matrices (PRBM model). Our conclusions are : (i) in the presence of one isolated incoming wire and many outgoing wires, the transmission has the same multifractal statistics as the local density of states of the site where the incoming wire arrives; (ii) in the presence of backward scattering channels with respect to the case (i), the statistics of the transmission is not multifractal anymore, but becomes monofractal. Finally, we also describe how these scattering geometries influence the statistics of the transmission off criticality.