Scaling Theory of the Anderson Transition in Random Graphs: Ergodicity and Universality

We study the Anderson transition on a generic model of random graphs with a tunable branching parameter $1<K<2$, through large scale numerical simulations and finite-size scaling analysis. We find that a single transition separates a localized phase from an unusual delocalized phase that is ergodic at large scales but strongly nonergodic at smaller scales. In the critical regime, multifractal wave functions are located on a few branches of the graph. Different scaling laws apply on both sides of the transition: a scaling with the linear size of the system on the localized side, and an unusual volumic scaling on the delocalized side. The critical scalings and exponents are independent of the branching parameter, which strongly supports the universality of our results.

Figure 1

Single parameter scaling of $⟨\ln \mathrm{Im}G⟩$ versus number of generations $M_g$, for $M_{\text{pool}}=400$, $M_g$ from $2M_{\text{pool}}$ to $10^5$, $p=0.06$, and $W\in [1.3,1.65]$ (see color code). $⟨\ln \mathrm{Im}G⟩\sim -{M_g}^{\rho }$ with $\rho \approx 0.28$ at $W=1.46\approx W_c(M_{\text{pool}})$. Finite-size scaling of $-⟨\ln \mathrm{Im}G⟩/{M_g}^{\rho }$ following Eq. (1). The scaling parameter $\xi$ diverges at the threshold as $\xi \sim |W-W_c(M_{\text{pool}}){|}^{-\nu }$ with $\nu \approx 1.4$ and $W_c(M_{\text{pool}})\approx 1.46$. Inset: critical disorder $W_c(M_{\text{pool}})$ versus $M_{\text{pool}}$. The line is a fit by $W_c(M_{\text{pool}})=W_c^{\infty }+A_0{M_{\text{pool}}}^{-\beta }$ with $A_0$ a constant, $W_c^{\infty }=1.77$, and $\beta =0.33$. The error bars are of the same order as the fluctuations of the data points.

Figure 2

Scaling of moments $⟨P_2⟩$ with $N$ for $p=0.06$ and $W\in [0.7,3]$. For $W>W_c$ (localized phase), linear scaling of $⟨P_2⟩/({\log }_{2}N{)}^{-{\tau }_2}$ following Eq. (2) with ${\tau }_2=0.42$. For $W<W_c$ (delocalized phase), volumic scaling following Eq. (3). Data for $W=W_c$ have been shifted horizontally for visibility. Dotted line shows asymptotic ergodic behavior $N^{-1}({\log }_{2}N{)}^{{\tau }_2}$ corresponding to $⟨P_2⟩\sim N^{-1}$. Upper inset (note the log scale): the critical behavior of $⟨P_2⟩$ at $W=1.6\approx W_c$ is very well fitted (black line) by $⟨P_2⟩=A_0({\log }_{2}N{)}^{-{\tau }_2}$ with $A_0$ a constant and ${\tau }_2\approx 0.42$. Lower inset: correlation volume $\mathrm{\Lambda }$ (circles) and localization length $\xi$ (squares) on both sides of the transition. Solid lines are the fits: $\ln \mathrm{\Lambda }=A_1+A_2(W_c-W{)}^{-\kappa }$ (for $W\in [1,1.5]$) and $\xi =A_3(W-W_c{)}^{-{\nu }_l}$ with $W_c=1.6$ and $\kappa =0.5$ (assigned values), yielding ${\nu }_l\approx 1$. Error bars on $⟨P_2⟩$ are below the symbol size, and not taken into account in the scaling analysis.

Figure 3

$⟨P_2⟩$ versus box volume $\ell$ for $N=2^{21}$ with $p=0.06$ and $W=1.0$ (square), $1.3$ (circle), $1.6$ (triangle), $2.4$ (diamond). The gray shaded area on the left delimits the small $\ell <1/2p$ multifractal behavior (the dotted lines have a slope ranging from 0.44 to 0.66). The light-blue shaded area on the right delimits the large scale $\ell \gg \lambda$ ergodic behavior (the dashed line has slope 1).

Figure 4

Scaling behavior of the NEV $\lambda$ with $N=2^{10}$ to $2^{21}$, $p=0.06$, and $W\in [0.8,2.4]$. At the threshold $W=W_c\approx 1.6$, $\lambda \sim N/{\log }_{2}N$. In the localized phase, the data collapse is excellent when plotted as a function of $({\log }_{2}N)/\xi$ (linear scaling, see text). In the delocalized phase, a volumic scaling as function of $N/\mathrm{\Lambda }$ makes the curves collapse. The dotted line shows asymptotic ergodic behavior $N^{-1}({\log }_{2}N)$ corresponding to $\lambda =\mathrm{\Lambda }$ [42]. Inset: determination of the NEV $\lambda$ through rescaling of ${\tilde{\pi }}_2(\ell )$ data (see text) versus $\ell /\lambda$, in the large scale $\ell \gg \lambda$ ergodic regime, for $W=1.6$ and $N=2^{10}$ (square), $2^{14}$ (circle), $2^{18}$ (triangle), and $2^{21}$ (diamond).