Criticality without self-similarity: a 2D system with random long-range hopping

TL;DR

This paper studies a 2D quantum system with long-range hopping, revealing a non-fractal critical point where eigenfunctions do not exhibit traditional multifractal scaling, differing from known Anderson transitions.

Contribution

It demonstrates that the 2D long-range hopping model exhibits a unique criticality without self-similarity, contrasting with conventional Anderson transitions.

Findings

Eigenfunctions are not fractal at the transition.

Density moments scale with the logarithm of system size.

The transition differs from 1D and higher-dimensional Anderson transitions.

Abstract

We consider a simple model of quantum disorder in two dimensions, characterized by a long-range site-to-site hopping. The system undergoes a metal-insulator transition -- its eigenfunctions change from being extended to being localized. We demonstrate that at the point of the transition the eigenfunctions do not become fractal. Their density moments do not scale as a power of the system size. Instead, in one of the considered limits our result suggests a power of the logarithm of the system size. In this regard, the transition differs from a similar one in the one-dimensional version of the same system, as well as from the conventional Anderson transition in more than two dimensions.