Time-dependent reflection at the localization transition

A short quasimonochromatic wave packet incident on a semi-infinite disordered medium gives rise to a reflected wave. The intensity of the latter decays as a power law, $1/t^{\alpha }$, in the long-time limit. Using the one-dimensional Aubry-André model, we show that in the vicinity of the critical point of Anderson localization transition, the decay slows down, and the power-law exponent $\alpha$ becomes smaller than both $\alpha =2$ found in the Anderson localization regime and $\alpha =3/2$ expected for a one-dimensional random walk of classical particles.

Figure 1

Grayscale plot of the average IPR of eigenstates of the (a) Anderson and (b) Aubry-André models. Averaging was performed over 100 realizations of random potential for a system of $L={10}^3$ sites with periodic boundary conditions. The dashed vertical line in (b) shows the critical disorder strength $W_c=4$ of the Aubry-André model. White space corresponds to regions of parameters where no states exist.

Figure 2

Average time-dependent reflection coefficient for the Anderson model computed for a system of length $L={10}^3$, disorder strength $W=4$, and the central energy of the wave packet (a) $E_0=-1.2$ or (b) 0. Solid and dashed lines correspond to the Gaussian [Eq. (8)] and parabolic [Eq. (9)] spectra, respectively, and to two different spectral widths, $\sigma =0.1$ (black and green lines) and 0.05 (red and blue lines). The results are averaged over ${10}^4$ independent realizations of random potential. Dashed straight lines illustrate the power-law decay at long times.

Figure 3

Same as Fig. 2, but for the Aubry-André model. The letters G and P mark lines corresponding to the Gaussian and parabolic spectra of the incident wave packet, respectively.

Figure 4

Same as Fig. 3, but for (a) and (b) $W=3$ and (c) and (d) 5.

Figure 5

The power-law exponent $\langle \alpha \rangle$ of the average time-dependent reflection coefficient for the Anderson and Aubry-André models obtained from the fits to the results presented in Figs. 2, 3, 4, and 4 and similar results for other values of $W$. Long times, $100<t<3000$, were used for the fits for all $W$ in the case of the Anderson model and for $W>3.5$ in the case of the Aubry-André model. For the Aubry-André model with $W\le 3.5$, only $100<t<500$ were used. The long-time behavior of $\langle R\left(t\right)\rangle$ for the Aubry-André model starts to deviate from a pure power law for $W<3.5$ and $W>6$ (shown by dashed lines). The error bars show the uncertainties due to both statistical fluctuations in the numerical data and the differences between $\alpha$ obtained for the different shapes and widths of the incident-wave-packet spectrum. The vertical dashed line shows the critical point $W_c=4$ of the Aubry-André model; the two horizontal dashed lines show $\alpha =1.5$ and 2, expected for diffuse and localized waves, respectively. Inset: The slowest decay of $\langle R\left(t\right)\rangle$ observed for $W=3.75$.

Figure 6

Illustration of finite-size effects in the average reflection coefficient $\langle R\left(t\right)\rangle$ of the Aubry-André model in (a) the extended and (b) critical regimes. Different lines correspond to different system lengths $L=100$, 200, ${10}^3$ in (a) and $L=10$, 20, 40, 100, ${10}^3$ in (b). The incident wave packet is assumed to have a parabolic shape (11).