Multifractality in high-dimensional graphs induced by correlated radial disorder

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.