Finite-size scaling analysis of localization transition for scalar waves in a three-dimensional ensemble of resonant point scatterers

We use the random Green's matrix model to study the scaling properties of the localization transition for scalar waves in a three-dimensional (3D) ensemble of resonant point scatterers. We show that the probability density $p\left(g\right)$ of normalized decay rates of quasimodes $g$ is very broad at the transition and in the localized regime and that it does not obey a single-parameter scaling law for finite system sizes that we can access. The single-parameter scaling law holds, however, for the small-$g$ part of $p\left(g\right)$ which we exploit to estimate the critical exponent $\nu$ of the localization transition. Finite-size scaling analysis of small-$q$ percentiles $g_q$ of $p\left(g\right)$ yields an estimate $\nu \simeq 1.55\pm 0.07$. This value is consistent with previous results for the Anderson transition in the 3D orthogonal universality class and suggests that the localization transition under study belongs to the same class.

Figure 1

(a) Eigenvalues of a single random realization of the Green's matrix (1) are shown by points on the complex plane for $N=8000$ scatterers randomly distributed in a sphere with a density $\rho /k_0^3=0.15$. (b) Same as (a) but with the vertical axis in logarithmic scale. Arrows indicate eigenvalues corresponding to spatially localized eigenvectors and two-scatterer proximity resonances, respectively. The latter are concentrated along the dashed line given by Eq. (3). The inset illustrates the geometry, with points inside an imaginary sphere corresponding to scatterers at random positions $\left\{{\mathbf{r}}_j\right\}$.

Figure 2

(a) Average logarithm of the Thouless conductance as a function of frequency $\mathrm{Re}\mathrm{\Lambda }$ at a given number density $\rho /k_0^3=0.15$ and for several total numbers $N$ of scatterers from $N=2000$ to $N=16000$ (the order of curves corresponding to increasing $N$ is indicated by an arrow). Vertical dashed lines show the estimated locations of the two mobility edges (denoted as I and II) where curves corresponding to different $N$ all cross. (b) Zoom on the region around the mobility edge I, shown by a gray rectangle in panel (a). These results are obtained by averaging over 6385, 3140, 1798, 1250, 1120, 880, 744, and 665 numerically generated and diagonalized [31] realizations of random Green's matrices for $N$ from 2000 to 16 000, respectively.

Figure 3

The variance (a) and the third central moment (b) of $lng$. The order of curves corresponding to increasing $N$ is indicated by arrows. Vertical dashed lines are the same as in Fig. 2.

Figure 4

Evolution of the probability density of $lng$ from the extended regime (a) through the critical point (b) to a regime in which localized states appear (c), in the vicinity of the localization transition I. The order of curves corresponding to increasing $N$ is indicated by arrows. Dashed line in (b) shows the percentile $g_q$ for $q=0.05$. Distributions at the critical point can be considered independent of $N$ on the left from the dashed line.

Figure 5

Same as Fig. 4 but for the localization transition II.

Figure 6

Probability density of $lng$ at large negative $\mathrm{Re}\mathrm{\Lambda }$ where subradiant states localized on pairs of scatterers (proximity resonances) come into play. To improve statistics, averaging is performed over a range ${\delta }_{\mathrm{Re}\mathrm{\Lambda }}=1$ around the values of $\mathrm{Re}\mathrm{\Lambda }$ given on the plots instead of ${\delta }_{\mathrm{Re}\mathrm{\Lambda }}=0.01$ in Figs. 4 and 5. This is made possible by the slow dependence of the properties of proximity resonances on $\mathrm{Re}\mathrm{\Lambda }$.

Figure 7

Examples of second-order polynomial fits (solid lines) to the numerical data (symbols) for the $\beta$ function $\beta (ln{g}_q)$. Fits for four different values of $q$ are shown. Only data points corresponding to $\beta >-5$ were used for fitting and are shown in the figures. Different symbols correspond to estimations of the $\beta$ function for $N=2$ and $4\times {10}^3$ (full circles), $N=4$ and $6\times {10}^3$ (squares), $N=6$ and $8\times {10}^3$ (triangles), $N=8$ and $10\times {10}^3$ (upside-down triangles), $N=10$ and $12\times {10}^3$ (diamonds), $N=12$ and $14\times {10}^3$ (open circles), $N=14$ and $16\times {10}^3$ (stars). Values of the critical exponent $\nu$ following from the fits are shown on the figures.

Figure 8

Examples of fits (solid lines) to the numerical data (symbols with error bars) for four different values of $q$ and $m_1=2,n_1=1,m_2=n_2=0$. Different curves correspond to different $N=4$, 6, 8, 10, 12, and $16\times {10}^3$. The order of curves corresponding to increasing $N$ is indicated by an arrow in the panel (a). The best-fit parameters $\mathrm{Re}{\mathrm{\Lambda }}_c$ and $\nu$ are given on the plots.

Figure 9

Best-fit values of the mobility edge $\mathrm{Re}{\mathrm{\Lambda }}_c$ (a) and of the critical exponent $\nu$ (b) obtained from the fits to the data for $q=0.001$–0.05 using $m_1=2,n_1=1$, and $m_2=n_2=0$ (symbols with error bars). The horizontal dashed lines show mean values of $\mathrm{Re}{\mathrm{\Lambda }}_c$ and $\nu$, respectively. The shaded regions correspond to the uncertainties of the means.

Figure 10

Demonstration of single-parameter scaling of percentiles $g_q$ for four different values of $q$. Points with error bars are numerical data of Fig. 8. Lines show the scaling function (8) as determined from the fits of Fig. 8.