Anderson localisation in spatially structured random graphs

TL;DR

This paper investigates Anderson localisation on high-dimensional, spatially structured graphs with long-range hopping, revealing how hopping range influences the localisation transition and phase diagram.

Contribution

It introduces a model interpolating between short-range and fully connected graphs, combining numerical and analytical methods to analyze localisation phenomena.

Findings

Increasing hopping range shifts the localisation transition to stronger disorder.

Beyond a critical hopping range, the localised phase disappears regardless of disorder strength.

The transition appears to be Kosterlitz-Thouless-like with distinct critical behaviours.

Abstract

We study Anderson localisation on high-dimensional graphs with spatial structure induced by long-ranged but distance-dependent hopping. To this end, we introduce a class of models that interpolate between the short-range Anderson model on a random regular graph and fully connected models with statistically uniform hopping, by embedding a random regular graph into a complete graph and allowing hopping amplitudes to decay exponentially with graph distance. The competition between the exponentially growing number of neighbours with graph distance and the exponentially decaying hopping amplitude positions our models effectively as power-law hopping generalisation of the Anderson model on random regular graphs. Using a combination of numerical exact diagonalisation and analytical renormalised perturbation theory, we establish the resulting localisation phase diagram emerging from the interplay of the lengthscale associated to the hopping range and the onsite disorder strength. We find that increasing the hopping range shifts the localisation transition to stronger disorder, and that beyond a critical range the localised phase ceases to exist even at arbitrarily strong disorder. Our results indicate a direct Anderson transition between delocalised and localised phases, with no evidence for an intervening multifractal phase, for both deterministic and random hopping models. A scaling analysis based on inverse participation ratios reveals behaviour consistent with a Kosterlitz-Thouless-like transition with two-parameter scaling, in line with Anderson transitions on high-dimensional graphs. We also observe distinct critical behaviour in average and typical correlation functions, reflecting the different scaling properties of generalised inverse participation ratios.