Critical scaling of polarization waves on a heterogeneous chain of resonators

The intensity distribution of electromagnetic polar waves in a chain of near-resonant weakly coupled scatterers is investigated theoretically and supported by numerical analysis. Critical scaling behavior is discovered for part of the eigenvalue spectrum due to the disorder-induced Anderson transition. This localization transition (in a formally one-dimensional system) is attributed to the long-range dipole-dipole interaction, which decays inverse linearly with distance for polarization perpendicular to the chain. For polarization parallel to the chain, with inverse-squared long-range coupling, all eigenmodes are shown to be localized. A comparison with the results for Hermitian power-law banded random matrices and other intermediate models is presented. This comparison reveals the significance of non-Hermiticity of the model and the periodic modulation of the coupling.

Figure 1

Complex-valued spectrum of the TEM interaction matrix (9) with $k=1$ for (a) homogeneous ($W=0$) and disordered in regimes of (b) strong ($W=2$) and (c) weak ($W=20$) coupling. The dashed square in (c) shows the region where the corresponding TEM eigenmodes are scaling critically. The eigenmodes corresponding to the eigenvalues in the dashed regions are selected for further statistical analysis.

Figure 2

Same as Fig. 1 for TM polarization.

Figure 3

Typical TEM critical (red solid line) and TM localized (black dashed line) eigenvectors of matrices defined in Sec. 2c with $k=1$ and $W=10$. The inset shows the same plot in logarithmic scale.

Figure 4

Scaling of PDFs for TM eigenmodes for different lengths of the chain and correspondingly scaled box sizes of $b=2^{-6}\times L$. Different line types (and colors) correspond to different system sizes as indicated by the legend. $W=30$ and $k=1$. The shift of the peak to the larger values of $\mathrm{\alpha ̃}$ indicates that the eigenmodes are localized.

Figure 5

Same as Fig. 4 for TEM eigenmodes. For each figure 12% of the eigenmodes are used with the (a) most positive, (b) closest to zero, and (c) most negative real part of their eigenvalues. Critical scaling is only observed in (c). The shift in the peak of the distribution shows that the rest of the eigenmodes are localized.

Figure 6

(a) Similar to Fig. 4 for TEM modes in the regime of strong coupling with $k=1$ and $W=0.7$. The analyzed eigenmodes are selected from the spectral region indicated by the dashed triangle in Fig. 1b. (b) Typical eigenmodes used for the PDF in (a) for two system lengths $L=4096$ and $L=512$ (inset).

Figure 7

Similar to Fig. 4 for models (a) H0, (b) H1, (c) H2, and (d) C1. Critical scaling is observed for the H0 and H1 models. The eigenvectors of the H2 and C1 models are localized. These models are defined in Sec. 2d.

Figure 8

The anomalous dimensions $d_q$ (symbols) at a fixed coupling strength for (a) TEM critical modes and (b) Hermitian matrices H1 are compared with the corresponding results (solid line) (17) and (18) from perturbation analysis. For Hermitian matrices, the symmetry relation predicted in Ref. 32 is used for plotting the theoretical curve at negative $q$. The errors estimated from the least-squares fitting routine are smaller than the symbol sizes and are not shown. The largest error in $d_q$ for point $q=5.5$ on the graph is $\pm 0.02$.

Figure 9

The anomalous dimensions $d_q$ extracted from direct numerical diagonalization (symbols) for three different values of $q$ are compared with the perturbation results (lines) of Eq. (17) in the weak-coupling regime for different values of disorder and (a) $k=1$ or (b) $k=3$. The numerical results converge with theory slowly.

Figure 10

The multifractal spectrum $f(\alpha )$ for (a) TEM critical modes and (b) Hermitian matrices H0 extracted directly from the eigenvectors by using the method of Chhabra and Jensen.[31] For both graphs $W=30$ and $k=1$. The error bars indicate the standard deviation among 20 realizations of disorder and are smaller than the symbol size for most of the data points. The dashed lines are guides to the eye.

Figure 11

Summary of the scaling analysis on PDFs of eigenvectors of matrices from various models. The intermediate hypothetical models transform the Hermitian RBM to the model describing TEM coupling. The colored boxes indicate those models that show critical scaling behavior. The models indicated by white boxes have localized eigenvectors. For TEM modes, only a part of the spectrum is critical.