Anderson Localization on the Bethe Lattice: Nonergodicity of Extended States

Statistical analysis of the eigenfunctions of the Anderson tight-binding model with on-site disorder on regular random graphs strongly suggests that the extended states are multifractal at any finite disorder. The spectrum of fractal dimensions $f(\alpha )$ defined in Eq. (3) remains positive for $\alpha$ noticeably far from 1 even when the disorder is several times weaker than the one which leads to the Anderson localization; i.e., the ergodicity can be reached only in the absence of disorder. The one-particle multifractality on the Bethe lattice signals on a possible inapplicability of the equipartition law to a generic many-body quantum system as long as it remains isolated.

Figure 1

Numerical results for $f(\alpha )$ on the RRG with the connectivity $K+1=3$ after linear extrapolation $f(\alpha ,N)=f^{\text{ext}}(\alpha )+c/\ln N$ to $1/\ln N\to 0$ for different values of disorder $W$. The dashed straight lines show the slope $k<1/2$ for the localized ($W=22.5$) and $k=1/2$ for the critical ($W=17.5$) states.

Figure 2

$N$ dependence of $f(\alpha ,N)$ on the RRG with $K=2$ for $N=2,4,8,16\times {10}^3$ (from green to blue in ascending order) in the extended phase $W=10$. The $f^{\text{ext}}(\alpha )$ obtained by linear extrapolation of $f(\alpha ,N)$ to $1/\ln N\to 0$ is shown by a thin red line. The maximal value $f_{\text{max}}\approx 1.03$ of $f^{\text{ext}}(\alpha )$ is very close to the theoretical expectation $f_{\text{max}}=1$. We also show the fractal dimensions $D_2$ and $D_1=\underset{q\to 1+}{\lim }{D}_q$ corresponding to $f^{\text{ext}}(\alpha )$. The plots for different $N$ show apparent fixed points at $\alpha \approx 0.5$ and $\alpha \approx 1.6$ indicated by arrows. Similar fixed points with $W$-dependent positions are seen at any strength of disorder studied. This rules out that $f(\alpha ,N)$ approaches at $\ln N\to \infty$ the ergodic limit $f(\alpha )=1$ at $\alpha =1$, $f(\alpha )=-\infty$ otherwise. In the insert: the linear extrapolation of $f(\alpha ,N)$ for $\alpha =1.0$; the red points show $f(\alpha =1.0,N)$ at $N=2,4,8,16,32\times {10}^3$.

Figure 3

$f^{\text{ext}}(\alpha )$ obtained by linear extrapolation (see Fig. 2) of $f(\alpha ,N)$ to $1/\ln N\to 0$ (red) and $f(\alpha ,N)$ for $N=2,4,8,16\times {10}^3$ at disorder strength $W=5$. The fixed points are shown by arrows. In the insert: verification of the symmetry Eq. (5) for the extrapolated $f^{\text{ext}}(\alpha )$ (coinciding blue and red thick curves), and for $f(\alpha ,N)$ at $N=16\times {10}^3$ (distinctly different thin blue and red curves).