Multifractality in high-dimensional graphs induced by correlated radial disorder

TL;DR

This paper studies robust multifractality in high-dimensional disordered graphs caused by correlated radial disorder, revealing broad multifractal spectra and emergent one-dimensional chain fragmentation.

Contribution

It introduces a novel model of radial disorder in high-dimensional graphs, demonstrating analytically and numerically the presence of robust multifractal eigenstates with broad IPR distributions.

Findings

Multifractality is induced by radial disorder correlations.

Mean and typical IPRs scale differently, highlighting multifractality.

Graph fragmentation into effective chains underpins the multifractal states.

Abstract

We introduce a class of models containing robust and analytically demonstrable multifractality induced by disorder correlations. Specifically, we investigate the statistics of eigenstates of disordered tight-binding models on two classes of rooted, high-dimensional graphs -- trees and hypercubes -- with a form of strong disorder correlations we term `radial disorder'. In this model, site energies on all sites equidistant from a chosen root are identical, while those at different distances are independent random variables (or their analogue for a deterministic but incommensurate potential, a case of which is also considered). Analytical arguments, supplemented by numerical results, are used to establish that this setting hosts robust and unusual multifractal states. The distribution of multifractality, as encoded in the inverse participation ratios (IPRs), is shown to be exceptionally broad. This leads to a qualitative difference in scaling with system size between the mean and typical IPRs, with the latter the appropriate quantity to characterise the multifractality. The existence of this multifractality is shown to be underpinned by an emergent fragmentation of the graphs into effective one-dimensional chains, which themselves exhibit conventional Anderson localisation. The interplay between the exponential localisation of states on these chains, and the exponential growth of the number of sites with distance from the root, is the origin of the observed multifractality.