Anderson localisation in spatially structured random graphs

TL;DR

This work introduces spatially structured high-dimensional random graphs by embedding random regular graphs into complete graphs and introducing distance-dependent long-range hopping via ExpRRG and ExpRRG-RH models. By tuning the hopping range ξ, the authors map a phase diagram where localisation and delocalisation compete against onsite disorder, finding a direct, KT-like transition without an intervening multifractal phase. Self-consistent locator theory and numerical ED broadly agree on the critical lines, with distinct scaling for average versus typical correlations and IPRs, and a finite critical correlation length at the transition. The results have implications for understanding localisation on complex graphs and offer avenues for exploring many-body localisation physics on Fock-space analogues and related dynamical phenomena.

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.

Figures (12)

• Figure 1: Schematic representation of the ExpRRG model. The underlying graph is the skeleton RRG with connectivity $K=3$. To represent the spatially structured long-ranged hopping, we consider a reference site (marked as red) and the rest of the sites are coloured according to their distance from the reference with lighter colours representing farther distances. The long-ranged hoppings indicated by the additional edges are also coloured according to their strength with darker colours representing larger hopping amplitudes.
• Figure 2: Phase diagram of the ExpRRG model obtained numerically from ED in the $W$-$\xi$ plane. The heatmap shows the IPR exponent, $\tau_2$ with darker colours denoting $\tau_2\approx 0$ in the localised phase and lighter colours corresponding to $\tau_2\approx 1$ in the delocalised phase. The red circles denote the critical points obtained from the analysis of the level statistics and the green line shows the locus of critical points obtained analytically from a self-consistent mean-field theory in Sec. \ref{['sec:selfconsisten']}.
• Figure 3: Level statistics for the ExpRRG model. The top panels shows the mean level spacing ratio, $\braket{r}$, as a function of $W$ for a fixed $\xi$ (left) and as a function of $\xi$ for a fixed $W$ (right), for different system sizes. These correspond to the horizontal and vertical cuts shown by the dashed lines in Fig. \ref{['fig:pd-exprrg']}. The crossing in the data for different $N$ indicates the Anderson transition, and crossings such as the ones seen here were used to estimate the phase boundary in Fig. \ref{['fig:pd-exprrg']}. The lower panels show the drift of the data with $N$ at parameter values representative of the delocalised and localised phases. For the former, the data approaches the GOE value of $\approx 0.53$ with increasing $N$ whereas for the latter it approaches the Poisson value of $0.386$.
• Figure 4: The generalised IPRs, $\braket{I_q}$ for the ExpRRG model, for $q=1/4$ (top) and $q=2$ (bottom). The left panel shows the data as a function of $N$ for different $W$ (different colours) for a fixed $\xi$ whereas in the right panel, different colours denote different $\xi$ values at a fixed $W$.
• Figure 5: Flowing fractal exponent $\tilde{\tau}_q(N)$ as a function of $q$ (defined in Eq. \ref{['eq:flowing-fractal-exponent']}) at three different points in the parameter space for different $N$. The upper panels correspond to the delocalised phase at weak disorder (left) and strong disorder but large $\xi$ (right). The dashed lines denote the asymptotic $q-1$ behaviour. The lower panel corresponds to the localised phase at strong disorder and small $\xi$. The inset shows the same data as the main panel but zoomed in to focus on the low $q$ regime. The red dashed line shows the fit to form in Eq. \ref{['eq:qstar']} which is used to extract $q_\ast$.