Ioffe-Regel criterion for Anderson localization in the model of resonant point scatterers

We establish a phase diagram of a model in which scalar waves are scattered by resonant point scatterers pinned at random positions in the free three-dimensional space. A transition to Anderson localization takes place in a narrow frequency band near the resonance frequency provided that the number density of scatterers $\rho$ exceeds a critical value ${\rho }_c\simeq 0.08k_0^3$, where $k_0$ is the wave number in the free space. The localization condition $\rho >{\rho }_c$ can be rewritten as $k_0{\ell }_0<1$, where ${\ell }_0$ is the on-resonance mean free path in the independent-scattering approximation. At mobility edges, the decay of the average amplitude of a monochromatic plane wave is not purely exponential and the growth of its phase is nonlinear with the propagation distance. This makes it impossible to define the mean free path $\ell$ and the effective wave number $k$ in a usual way. If these last are defined as an effective decay length of the intensity and an effective growth rate of the phase of the average wave field, the Ioffe-Regel parameter ${\left(k\ell \right)}_c$ at the mobility edges can be calculated and takes values from 0.3 to 1.2 depending on $\rho$. Thus, the Ioffe-Regel criterion of localization $k\ell <{\left(k\ell \right)}_c=\mathrm{const}.\sim 1$ is valid only qualitatively and cannot be used as a quantitative condition of Anderson localization in three dimensions.

Figure 1

Probability densities $p_{\omega ,N}$ of the logarithm of Thouless conductance $g$ at a fixed density of scatterers $\rho /k_0^3=0.15$, for a fixed frequency $\omega ={\omega }_0+0.255{\mathrm{\Gamma }}_0$ close to the critical frequency ${\omega }_c^{\mathrm{I}}$ of the localization transition, and for different numbers of scatterers $N$. A $(q\times 100)\mathrm{th}$ percentile is defined as a limit of integration $ln{g}_q$ (shown by a vertical arrow) up to which $p_{\omega ,N}(lng)$ should be integrated to obtain $q$ (equal to the grey area in the figure) as a result. The illustration in the figure is for the fifth percentile ($q=0.05$).

Figure 2

Illustration of the procedure applied to determine the mobility edges ${\omega }_c$. Percentiles $ln{g}_q$ of the probability distribution of the logarithm of the Thouless conductance $g$ are calculated numerically as functions of the real part $\mathrm{Re}\mathrm{\Lambda }$ of the eigenvalues $\mathrm{\Lambda }$ of the Green's matrix (1), for two different numbers $N$ of point scatterers in a sphere (see the inset), at a fixed density of scatterers $\rho$ (red and blue data points with error bars). The values $\mathrm{Re}{\mathrm{\Lambda }}_c$ of $\mathrm{Re}\mathrm{\Lambda }$ at which lines $ln{g}_q\left(\mathrm{Re}\mathrm{\Lambda }\right)$ corresponding to different $N$ cross determine the mobility edges ${\omega }_c={\omega }_0-({\mathrm{\Gamma }}_0/2)\mathrm{Re}{\mathrm{\Lambda }}_c$ (dashed vertical lines). Our results are obtained from at least $5.5\times {10}^6$ eigenvalues for each $\rho$ and $N$. Final estimations of ${\omega }_c$ are averages over results obtained for ten equispaced values of $q$ in the interval $q=0.01\text{–}0.1$.

Figure 3

Phase diagram of the model of resonant point scatterers: grey and white areas correspond to the regions of localized and extended modes, respectively. Data of Table 1 are shown by full circles and squares for the mobility edges I and II, respectively. Straight solid lines are linear fits showing the estimated mobility edges as functions of scatterer density for $\rho /k_0^3\ge 0.121$. Dashed lines are quadratic fits for $\rho /k_0^3\le 0.121$. Their crossing point identifies ${\rho }_c/k_0^3\simeq 0.08$ as an estimation of the lowest density at which localized modes may appear.

Figure 4

(a) The exponential decay of the amplitude and (b) the linear growth of the phase of the average wave field as a function of depth $z$ into the medium at a low scatterer number density $\rho /k_0^3=0.05$, for several frequencies $\omega$. Averaging is performed over ${10}^6$ scatterer configurations for each curve. The inset of panel (b) shows the geometry of a cylindrical layer of radius $k_{0}R=20$ and thickness $k_{0}L=12$, illuminated by a monochromatic plane wave, used for the calculations.

Figure 5

Decay of the amplitude [first row, panels (a) and (b)] and growth of the phase [second row, panels (c) and (d)] of the average wave field as a function of depth $z$ into the medium at the mobility edges I (first column) and II (second column) for 7 scatterer densities from $\rho /k_0^3=0.1$ to 0.4 listed in Table 1. Averaging is performed over ${10}^5\text{–}{10}^6$ scatterer configurations for each curve. These results are obtained in the same geometry as those of Fig. 4 [see the inset of Fig. 4] but using $k_{0}L=10$.

Figure 6

Values of the Ioffe-Regel parameter $k\ell$ at the mobility edges $\omega ={\omega }_c^{\mathrm{I},\mathrm{II}}$ as functions of scatterer number density $\rho /k_0^3$ (low-frequency mobility edge I shown by red circles, high-frequency mobility edge II are the blue squares). (a) $\ell$ and $k$ were estimated by the integral formulas (9) and (10) with $k_0{z}_{\max }=4$. (b) $\ell$ and $k$ were estimated from linear fits of Eqs. (7) and (8) to the numerical data in the interval $k_{0}z\in (1,4)$. The horizontal dashed lines show ${\left(k\ell \right)}_c=1$ expected at a mobility edge.