Anderson transition for elastic waves in three dimensions

We use two different fully vectorial microscopic models featuring nonresonant and resonant scattering, respectively, to demonstrate the Anderson localization transition for elastic waves in three-dimensional (3D) disordered solids. Critical parameters of the transition determined by finite-time and finite-size scaling analyses suggest that the transition belongs to the 3D orthogonal universality class. Similarities and differences between the elastic-wave and light scattering in strongly disordered media are discussed.

Figure 1

Density of states of the model with nonresonant scattering. The red arrow indicates the position of the mobility edge determined in Sec. 2b.

Figure 2

Results for the model with nonresonant scattering. (a) An example of a fit with $m_1=2$, $n_1=1$, and $m_2=n_2=0$. Symbols with error bars show numerical data for different times $t=100$ (black), 200 (red), 400 (green), 800 (blue), 1600 (orange), 3200 (purple), 6400 (cyan), 12800 (magenta). Solid lines are polynomial fits; the dashed vertical line shows the mobility edge ${\omega }_c\simeq 12.09$. (b) The critical exponent $\nu$ extracted from fits similar to the one in (a) using only the data corresponding to $t\ge t_{\min }$ as a function of $t_{\min }$. Error bars are equal to one standard deviation of $\nu$. The dashed horizontal line shows the average value of $\nu$, the gray area shows the error of the average.

Figure 3

Analysis of the model with resonant scattering. (a) Second percentile of the distribution $p(lng)$ at a fixed density $\rho$ of scatterers in a sphere for eight different total numbers of scatterer $N=2000$ (black), 4000 (red), 6000 (green), 8000 (blue), 10000 (orange), 12 000 (purple), 14 000 (cyan), 16 000 (magenta). Vertical dashed lines show approximate positions of mobility edges where lines corresponding to different $N$ all cross. (b) A polynomial fit to the data in (a) in the vicinity of one of the mobility edges ($m_1=n_1=3$, $m_2=n_2=1$). The best-fit position of the mobility edge $\mathrm{Re}{\mathrm{\Lambda }}_c\simeq -0.99$ is shown by the dashed vertical line; the critical exponent $\nu$ following from the fit is indicated on the graph.

Figure 4

Results for the model with resonant scattering. (a) The mobility edge and (b) the critical exponent determined from the fits to the $q\mathrm{th}$ percentiles as a function of $q$. The horizontal dashed line shows the values of $\mathrm{Re}{\mathrm{\Lambda }}_c$ and $\nu$ averaged over all $q=0.001$–0.05. The gray areas correspond to the uncertainties of the averages.

Figure 5

Eigenvalues of the Green's matrix (12) are shown by dots for a single random configuration of $N$ scatterers in a sphere at a fixed number density $\rho /k_0^3=0.15$. The gray scale of each dot corresponds to the IPR (20) of the corresponding eigenvector. Fully extended eigenvectors have $\mathrm{IPR}\sim 1/N\ll 1$ and are shown in light gray. $N=2000$ in (a) and 10 000 in (b). Dashed vertical lines show the mobility edges.