Nonergodic Phases in Strongly Disordered Random Regular Graphs

We combine numerical diagonalization with semianalytical calculations to prove the existence of the intermediate nonergodic but delocalized phase in the Anderson model on disordered hierarchical lattices. We suggest a new generalized population dynamics that is able to detect the violation of ergodicity of the delocalized states within the Abou-Chakra, Anderson, and Thouless recursive scheme. This result is supplemented by statistics of random wave functions extracted from exact diagonalization of the Anderson model on ensemble of disordered random regular graphs (RRG) of $N$ sites with the connectivity $K=2$. By extrapolation of the results of both approaches to $N\to \infty$ we obtain the fractal dimensions $D_1(W)$ and $D_2(W)$ as well as the population dynamics exponent $D(W)$ with the accuracy sufficient to claim that they are nontrivial in the broad interval of disorder strength $W_E<W<W_c$. The thorough analysis of the exact diagonalization results for RRG with $N>10^5$ reveals a singularity in $D_{1,2}(W)$ dependencies which provides clear evidence for the first order transition between the two delocalized phases on RRG at $W_E\approx 10.0$. We discuss the implications of these results for quantum and classical nonintegrable and many-body systems.

Figure 1

Fractal dimensions $D_2$ and $D_1$ for $K=2$ RRG and the population dynamics exponent $D$ as functions of disorder strength $W$. The $W$ dependence of $D$ extrapolated to $N\to \infty$ is presented by the “brush-painted” blue line where the width corresponds to the uncertainty of extrapolation. In spite of this uncertainty, $D$ is distinctly different from 0 and 1 in a broad interval of $W$ manifesting the nonergodic (multifractal) nature of extended wave functions. The red solid line with black data points is a “running” fractal dimension $D_2(N,W)=-d⟨\ln I_2⟩/d\ln N$ obtained by exact diagonalization at the maximal size $N=128000$ of a disordered RRG. The fat red line is a sketch of the fractal dimension $D_2(N\to \infty ,W)\equiv D_2(W)$ extrapolated to infinite $N$. Inset: the jump singularity in the running fractal dimensions $D_1(N=60000,W)$ and $D_2(N=128000,W)$ manifesting the ergodic transition at $W=W_E\approx 10.0$.

Figure 2

The population dynamic exponent $D(W)$ (blue points with gray error bars) extrapolated to $N=\infty$ and $\delta =0$ for $K=2$. The condition $D(W_c)=0$ yields $W_c=18.4_{-0.2}^{+0.4}$. In a broad interval of $W<W_c$ we obtained $D(W)$ distinctly different from the ergodic limit $D(W)=1$. Lower inset: The collapse of data for a fixed $W$ and different $N_p$, $\delta$ to a function $D(W,X)$ of $X=-1/\ln (N_p^{-1}+a{\delta }^b)$. Extrapolation to $X=0$ gives the population dynamic exponent $D(W)$. The delocalized phase corresponds to $1\ge D(W)>0$, whereas in the localized phase $D(W)<0$. Upper inset: the finite-size critical disorder $W_c(X)$ defined as $D[W_c(X),X]=0$ and its extrapolation to $X=0$ by the power-law fit $W_c(X)=W_c-a{X}^{1/\nu }$ with $W_c=18.4$, $\nu =0.56$ (blue), and $W_c=19.0$, $\nu =0.7$ (red). Without extrapolation the value of $W_c$ at maximal population size $N_p^{*}\sim 10^8$ is $W_c(N_p^{*})\approx 17.5$.

Figure 3

Left panel: $D_2(N,W)$ deep in the delocalized phase. The curves tend to converge to two different values of $D_2$ for $W=W_E+0$ and $W_E-0$, where $W_E\approx 10.0$. Right panel: Formation of a jump in $D_2(W)$.

Figure 4

$D_2(N,W)$ close to the localization transition at $W=W_c$. The $N$ dependence shows minima (red spot) for $W<W_c$ at $N_{\min }\to \infty$ as $W\to W_c$ [19]. Inset: $D_2(N_{\min },W)$ as a function of $W$. Extrapolation by a second-order polynomial gives $W_c=18.1\pm 0.5$.