Anderson localization and ergodicity on random regular graphs

A numerical study of Anderson transition on random regular graphs (RRGs) with diagonal disorder is performed. The problem can be described as a tight-binding model on a lattice with $N$ sites that is locally a tree with constant connectivity. In a certain sense, the RRG ensemble can be seen as an infinite-dimensional $\left(d\to \infty \right)$ cousin of the Anderson model in $d$ dimensions. We focus on the delocalized side of the transition and stress the importance of finite-size effects. We show that the data can be interpreted in terms of the finite-size crossover from a small $\left(N\ll N_c\right)$ to a large $\left(N\gg N_c\right)$ system, where $N_c$ is the correlation volume diverging exponentially at the transition. A distinct feature of this crossover is a nonmonotonicity of the spectral and wave-function statistics, which is related to properties of the critical phase in the studied model and renders the finite-size analysis highly nontrivial. Our results support an analytical prediction that states in the delocalized phase (and at $N\gg N_c)$ are ergodic in the sense that their inverse participation ratio scales as $1/N$.

Figure 1

Mean adjacent gap ratio $r$ as a function of disorder $W$ at various $N$ (see legend). Inset: apparent crossing point $W_{*}$ as a function of the system size, $lnN$.

Figure 2

Mean adjacent gap ratio $r$ as a function of system size, $lnN$, at various disorder levels $W$.

Figure 3

$lnN{P}_2$ as a function of the system size at various disorder levels. Dots: simulation; lines: smooth interpolation.

Figure 4

(a) Fractal exponent $\mu$ as a function of the system size at various disorder levels. (b) Zoom-in for selected disorders $W=13–17)$. Inset: position of the minimum of $\mu \left(N\right)$ as a function of disorder. (c) Critical behavior of the correlation length $\xi \left(W\right)$ at $W<W_c$ as extracted from the position of the crossing point $N_{*}$ of $r\left(W\right)$ curves (red) and minima $N_{min}$ in $\mu \left(N\right)$ (blue). The dashed line corresponds to $\xi \left(W\right)\propto {(W_c-W)}^{-1/2}$.