Multifractality in high-dimensional graphs induced by correlated radial disorder

TL;DR

This work demonstrates that radial disorder, a correlated disorder pattern on rooted high-dimensional graphs, induces robust multifractality of single-particle eigenstates. The authors map the problem to an emergent Krylov 1D chain where conventional Anderson localization competes with the exponential growth of row sites, producing broad IPR distributions and a clear distinction between mean and typical multifractal measures. They derive exact or near-exact expressions for the multifractal exponents $ au_q$ as functions of the single control parameter $\xi ext{---the localization length along the Krylov chain}---$, and validate these results with comprehensive numerical diagonalization on Cayley trees and hypercubes. The findings highlight how strong disorder correlations can qualitatively alter localization properties in high-dimensional graphs, offering a transparent mechanism for multifractality with potential implications for Fock-space and many-body localization contexts.

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.

Figures (15)

• Figure 1: Left panel: Rooted Cayley tree, illustrated for connectivity $K=2$. The graph has $L+1$ rows, with $N_{r}=K^{r}$ sites/vertices on row $r$ (and $r=0$ the root site). Hoppings (bonds/edges) are as indicated, each of strength $\Gamma$. With radial disorder, all sites on a given row have the same site energies (indicated by a common colour); while sites on different rows have distinct site energies. Right panel: Corresponding tree under the symmetrised basis transformation specified in Sec. \ref{['section:CTs']}. Hoppings are again as indicated, all being of strength $\sqrt{K}\Gamma$. The tree has fragmented into a disconnected set of 1d chains. Full discussion in text (see Sec. \ref{['section:Puretree']}).
• Figure 2: $L$-dimensional hypercube graph (illustrated for $L=6$). The graph has $L+1$ rows, with $N_{r}=\binom{L}{r}$ sites/vertices on row $r$ ($r=0$ is the apex site). Under $\hat{H}_{\Gamma}$, hoppings (of strength $\Gamma$) connect only sites on adjacent rows. Each site on row $r$ is connected under $\hat{H}_{\Gamma}$ to $L$ others: to $r$ distinct sites on row $(r-1)$, and to $(L-r)$ sites on row $(r+1)$. With radial disorder, all sites on a given row have the same site energies (indicated by a common colour); while sites on different rows have distinct site energies. For full discussion see Sec. \ref{['section:hcgraph']}.
• Figure 3: Mean $\mathcal{L}_{2}^{\mathrm{mean}}$ (left) and typical $\mathcal{L}_{2}^{\mathrm{typ}}$ (right) IPR for $q=2$ (with $K=2$), computed from the full analytical expressions in Eqs. \ref{['eq:Calc4']},\ref{['eq:Calc7']} over the range of $\xi$ indicated. $\mathcal{L}_{2}^{\mathrm{mean}}$ is plotted vs the total number of tree generations, $L$, while $\mathcal{L}_{2}^{\mathrm{typ}}$ is plotted against the total number of sites on the tree, $N\propto K^{L}$. Red dashed line in the $\ln \mathcal{L}_{2}^{\mathrm{mean}}$ plot indicates a slope of $-1$.
• Figure 4: Left panel: multifractal exponent $\tau_{q}$ as a function of $q$, shown for $\xi =0.1, 3$ and $10$ (with $K=2$). Solid lines show results computed numerically via Eqs. \ref{['eq:Calc9']},\ref{['eq:Calc7']} using the full sum expression in Eq. \ref{['eq:Calc4']}, while dashed lines show the exact results given in Sec. \ref{['section:tauq']} (Eq. \ref{['eq:mf1']} for $\xi =3,10$, and Eq. \ref{['eq:mf3']} for $\xi =0.1$); the two are indistinguishable. Right panel: $\tau_{q}$vs$\xi$, shown for $q=4,2$ and $0.5$,and given exactly by Eq. \ref{['eq:mf4']} for $q=2,4$ and Eq. \ref{['eq:mf5']} for $q=1/2$.
• Figure 5: Distribution $P_{\mathcal{L}_2}(\mathcal{L}_2)$ of $\mathcal{L}_2$ obtained from ED (shown for $W=1$ with $K=2$), for the number of generations $L$ as indicated. For clarity, $L\times P_{\mathcal{L}_2}(\mathcal{L}_2)$vs$\mathcal{L}_2$ is plotted (as explained in text). Left panel shows data with linear bins, right panel with logarithmic bins. In right panel, the black dashed line shows the $P_{\mathcal{L}_2}(\mathcal{L}_2) \propto 1/{\cal L}_2$ behaviour, Eq. \ref{['eq:Calc15']}.