Critical properties of the Anderson localization transition and the high dimensional limit

TL;DR

This study investigates the critical properties of Anderson localization across dimensions 3 to 6, revealing that the upper critical dimension is infinite and that high-dimensional limits better describe lower-dimensional systems.

Contribution

It provides a comprehensive analysis of the dimensional dependence of Anderson localization and establishes the infinite upper critical dimension, connecting finite-dimensional behavior with Bethe lattice results.

Findings

Critical state becomes insulating with Poisson statistics in high dimensions

Finite size corrections become logarithmic as dimension increases

The infinite-dimensional limit offers a better description of three-dimensional systems

Abstract

In this paper we present a thorough study of transport, spectral and wave-function properties at the Anderson localization critical point in spatial dimensions $d = 3$, $4$, $5$, $6$. Our aim is to analyze the dimensional dependence and to asses the role of the $d\rightarrow \infty$ limit provided by Bethe lattices and tree-like structures. Our results strongly suggest that the upper critical dimension of Anderson localization is infinite. Furthermore, we find that the $d_U=\infty$ is a much better starting point compared to $d_L=2$ to describe even three dimensional systems. We find that critical properties and finite size scaling behavior approach by increasing $d$ the ones found for Bethe lattices: the critical state becomes an insulator characterized by Poisson statistics and corrections to the thermodynamics limit become logarithmic in $N$. In the conclusion, we present physical consequences of our results, propose connections with the non-ergodic delocalised phase suggested for the Anderson model on infinite dimensional lattices and discuss perspectives for future research studies.