Return probability for the Anderson model on the random regular graph

We study the return probability for the Anderson model on the random regular graph and give evidence of the existence of two distinct phases: a fully ergodic and nonergodic one. In the ergodic phase, the return probability decays polynomially with time with oscillations, being the attribute of the Wigner-Dyson-like behavior, while in the nonergodic phase the decay follows a stretched exponential decay. We give a phenomenological interpretation of the stretched exponential decay in terms of a classical random walker. Furthermore, comparing typical and mean values of the return probability, we show how to differentiate an ergodic phase from a nonergodic one. We benchmark this method first in two random matrix models, the power-law random banded matrices, and the Rosenzweig-Porter matrices, which host both phases. Second, we apply this method to the Anderson model on the random regular graph to give further evidence of the existence of the two phases.

Figure 1

Mean $\overline{\mathcal{R}}\left(t\right)$ and typical $e^{\overline{log\mathcal{R}}\left(t\right)}$ return probability as a function of time $t$ (a), (b) or versus each other (c), (d) for PLRBM and RPRM models. (a) $\overline{\mathcal{R}}\left(t\right)$ and $e^{\overline{log\mathcal{R}}\left(t\right)}$ versus $t$ shown in log-log scale in the multifractal phase of the PLRBM ($a=1,b=0.5$) for a fixed system size $L=2^{14}$. The dashed lines are guides for the eyes emphasizing the power-law decay of $\overline{\mathcal{R}}\left(t\right)$ and $e^{\overline{log\mathcal{R}}\left(t\right)}$ in time with different powers. (b) $\overline{\mathcal{R}}\left(t\right)$ and $e^{\overline{log\mathcal{R}}\left(t\right)}$ versus $t$ shown in log-linear scale in the fractal phase of RPRM ($\gamma =1.25$) for $L=2^{14}$. $\overline{\mathcal{R}}\left(t\right)\sim e^{\overline{log\mathcal{R}}\left(t\right)}\sim e^{-E_{\text{Th}}t}$ show exponential decay with the same exponent $E_{\text{Th}}$ being the Thouless energy of the model ($E_{\text{Th}}\sim L^{1-\gamma }$). (c) ${\overline{{\mathcal{R}}_{\infty }}}^{-1}$ versus $e^{-\overline{log{\mathcal{R}}_{\infty }}}$ in two different phases of the PLRBM: In the ergodic phase ($a=0.5,b=1$), the mutual dependence is linear ${\overline{{\mathcal{R}}_{\infty }}}^{-1}\sim e^{-\overline{log{\mathcal{R}}_{\infty }}}$, while in the multifractal phase ($a=1,b=0.5$) ${\overline{{\mathcal{R}}_{\infty }}}^{-1}\sim e^{-\alpha \overline{log{\mathcal{R}}_{\infty }}}$ with $\alpha <1$. The inset shows the scaling of the typical ${\text{IPR}}_x$ versus the typical $\mathcal{R}\left(\infty \right)$ in the linear scale and confirms the same $L$-scaling of both typical quantities. (d) ${\overline{{\mathcal{R}}_{\infty }}}^{-1}$ versus $e^{-\overline{log{\mathcal{R}}_{\infty }}}$ for the RPRM in two phases. Both in ergodic ($\gamma =0.5$) and fractal ($\gamma =1.25$) phases, the mutual dependence is linear ${\overline{{\mathcal{R}}_{\infty }}}^{-1}\sim e^{-\overline{log{\mathcal{R}}_{\infty }}}$.

Figure 2

The return probability $\overline{\mathcal{R}}\left(t\right)$ versus $t$ for the RRG with the energy shell $\left|\mathrm{\Delta }E\right|=2\delta E=E_{BW}/32$ being a fraction of the whole energy bandwidth $E_{\text{BW}}$ for several disorder strengths $W$ and system sizes $L$. (a) $\overline{\mathcal{R}}\left(t\right)$ in the ergodic phase $W=3$ for several system sizes $L=2^{16},2^{18},2^{20}$; The inset shows that $\overline{\mathcal{R}}\left(t\right)$ for $W=3$ and $L=2^{20}$ almost coincides with the ergodic solution Eq. (6) ${\mathcal{R}}_{\text{erg}}\left(t\right)={\left[sin\left(2\delta Et\right)/\left(2\delta Et\right)\right]}^2$ (shown by a black dashed line) [52, 53, 54, 55]; (b) $\overline{\mathcal{R}}\left(t\right)$ in the nonergodic phase $W=12$ at several system sizes $L=2^{16},2^{18},2^{20}$. The inset shows a nontrivial plateau in the ratio $log\overline{\mathcal{R}}\left(t\right)/\overline{log\mathcal{R}}\left(t\right)<1$. (c) $\overline{\mathcal{R}}\left(t\right)$ for a fixed $L=2^{20}$ and several disorder strengths $W=3$, 6, 8, 9, 10, 12.

Figure 3

The return probability $\overline{\mathcal{R}}\left(t\right)$ versus $t^{1-W/W_{\text{AT}}}$ in the nonergodic phase for several disorder strengths $W=8$, 9, 10, 12, and fixed $L=2^{20}$ [rescaled data from Fig. 2], confirming the stretch-exponential behavior $\overline{\mathcal{R}}\left(t\right)\sim e^{-\mathrm{\Gamma }{t}^{\beta }}$, with $\beta =1-W/W_{\text{AT}}$ and $W_{\text{AT}}=18.2$.

Figure 4

Scaling of mean and typical IPR in RRG versus the system size $L$ and versus each other. (a) $\overline{{\text{IPR}}_x}$ and $e^{\overline{log{\text{IPR}}_x}}$ as a function of $L$ at the disorder strength $W=11$. Both curves show significant deviations from the power-law scaling ${\text{IPR}}_x\sim L^{-D}$. (b) ${\overline{{\text{IPR}}_x}}^{-1}$ versus $e^{-\overline{log{\text{IPR}}_x}}$ ($\overline{{\text{IPR}}_x}\sim e^{\alpha \overline{log{\text{IPR}}_x}}$) for several $W$. Different points of the curve correspond to the different system sizes $L=2^9-2^{17}$. Panel (c) shows $L$-dependent power ${\alpha }_L$ of the mutual IPR scaling as a function of $W$. ${\alpha }_L$ has been extracted from the linear fitting of $log\overline{{\text{IPR}}_x}$ versus $\overline{log{\text{IPR}}_x}$ with an enlarging $L$. The inset of panel (b) shows ${\alpha }_L$ and $D_L$ as a function of $L$. $D_L$ has been extracted from the linear fitting of $\left(-log\overline{{\text{IPR}}_x}\right)$ versus $logL$. The subscript $L$ in ${\alpha }_L$ and $D_L$ indicates the largest system size considered in the fit starting with $L=2^9$.