Diffusion in the Anderson model in higher dimensions

We present an extended microcanonical Lanczos method for a direct evaluation of the diffusion constant and its frequency dependence within the disordered Anderson model of noninteracting particles. The method allows to study systems beyond ${10}^6$ sites of hypercubic lattices in $d=3–7$ dimensions. Below the transition to localization where we confirm dynamical scaling behavior of interest is a wide region of incoherent diffusion, similar to percolating phenomena and to interacting many-body localized systems.

Figure 1

Dynamical diffusion $D\left(\omega \right)$ in the Anderson model at the intermediate disorder $W/t=20$, (a) for hypercubic lattices $d=2–7$ and (b) for $d=3,4$ showing the influence of the damping $\eta$.

Figure 2

Diffusion coefficient $D_0$ vs disorder strength $W$ within the Anderson model in hypercubic lattices with $d=3–7$ (in the inset in the normal scale for $D_0/t<1$) for $D_0$, in the vicinity of the critical regime also fitted with the scaling form $D_0\propto {(W_c-W)}^s$ (see the text for details).

Figure 3

Dynamical diffusion response $D\left(\omega \right)$ in the vicinity of the Anderson transition $W\lesssim W_c$ compared to the scaling form $D\left(\omega \right)=w^{s}F\left(w^{-\nu }{L}_{\omega }\right)$ for (a) $d=3$ and (b) $d=4$ Anderson model (see the text for details).

Figure 4

Imbalance stiffness $C_0$ vs $W$ for the Anderson model in $d=2,3$ dimensions. Results for $d=3$ are fitted to critical behavior $C_0\propto {(W-W_c)}^{\nu }$.