Spectral statistics and scattering resonances of complex primes arrays

We introduce a class of aperiodic arrays of electric dipoles generated from the distribution of prime numbers in complex quadratic fields (Eisenstein and Gaussian primes) as well as quaternion primes (Hurwitz and Lifschitz primes), and study the nature of their scattering resonances using the vectorial Green's matrix method. In these systems we demonstrate several distinctive spectral properties, such as the absence of level repulsion in the strongly scattering regime, critical statistics of level spacings, and the existence of critical modes, which are extended fractal modes with long lifetimes not supported by either random or periodic systems. Moreover, we show that one can predict important physical properties, such as the existence spectral gaps, by analyzing the eigenvalue distribution of the Green's matrix of the arrays in the complex plane. Our results unveil the importance of aperiodic correlations in prime number arrays for the engineering of gapped photonic media that support far richer mode localization and spectral properties compared to usual periodic and random media.

Figure 1

(a) $N=1050$ Eisenstein prime array, (b) $N=1068$ Gaussian prime array, (c) $N=1093$ Hurwitz prime array, and (d) $N=1081$ Lifschitz prime array. (e) to (h) are the diffraction patterns (5th root taken to enhance contrast) of the corresponding array on top of each panel.

Figure 2

The H(k) of (a) Eisenstein prime, (b) Gaussian prime, (c) Hurwitz prime, (d) Lifschitz prime.

Figure 3

Eigenvalue distributions of Eisenstein prime arrays with $\rho {\lambda }^2=$ (a) 100, (b) 10, (c) 1, (d) 0.1, (e) 0.01, (f) 0.001. The color coding shows the $lo{g}_{10}$ values of IPR for the eigenmode corresponding to each eigenvalue.

Figure 4

Eigenvalue distributions of Hurwitz prime arrays with $\rho {\lambda }^2=$ (a) 100, (b) 10, (c) 1, (d) 0.1, (e) 0.01, (f) 0.001. The color coding shows the $lo{g}_{10}$ values of IPR for the eigenmode corresponding to each eigenvalue.

Figure 5

The DOS for (a) Eisenstein prime, (b) Gaussian prime, (c) Hurwitz prime, (d) Lifschitz prime, at $\rho {\lambda }^2=100$; (e) Eisenstein prime, (f) Gaussian prime, (g) Hurwitz prime, (h) Lifschitz prime, at $\rho {\lambda }^2=10$. (i) Eisenstein prime, (j) Gaussian prime, (k) Hurwitz prime, (l) Lifschitz prime, at $\rho {\lambda }^2=0.001$. The arrows indicate the presence of spectral gaps in the DOS.

Figure 6

The first-neighbor level statistics of complex eigenvalues for (a) Eisenstein prime, (b) Gaussian prime, (c) Hurwitz prime, (d) Lifschitz prime, at $\rho {\lambda }^2=100$; (e) Eisenstein prime, (f) Gaussian prime, (g) Hurwitz prime, (h) Lifschitz prime, at $\rho {\lambda }^2=0.001$. The fittings are Poissionian for (a)–(d), as well as Ginibre (dashed curve in (e)–(h)) and critical (solid curve in (e)–(h)) distributions. The mean-square error (MSE) of fittings is less than 0.01.

Figure 7

The decay rate statistics for (a) Eisenstein prime, (b) Gaussian prime, (c) Hurwitz prime, (d) Lifschitz prime, at $\rho {\lambda }^2=100$; (e) Eisenstein prime, (f) Gaussian prime, (g) Hurwitz prime, (h) Lifschitz prime, at $\rho {\lambda }^2=10$. (i) Eisenstein prime, (j) Gaussian prime, (k) Hurwitz prime, (l) Lifschitz prime, at $\rho {\lambda }^2=0.001$.

Figure 8

A comparison of maximum IPR values among (1) Eisenstein prime, (2) Gaussian prime, (3) Lifschitz prime, and (4) Hurwitz prime arrays, at each representative optical density. (5) square, (6) triangular, and (7) uniform random arrays are included for references. The red dashed line indicate the proximity resonance of two particles with IPR = 0.5.

Figure 9

The IPR statistics for (a) Eisenstein prime, (b) Gaussian prime, (c) Hurwitz prime, (d) Lifschitz prime, at $\rho {\lambda }^2=100$; the mode with the highest IPR values are for each of the structures at this density shown respectively from (e) to (h).

Figure 10

At $\rho {\lambda }^2=10$, (a) to (d) are the most localized (highest associated IPR value) eigenmode for Eisenstein prime, Gaussian prime, Hurwitz prime, and Lifschitz prime arrays, respectively. (e) to (h) are corresponding structure's critical mode, which has the lowest IPR and small $\mathrm{Re}\mathrm{\Lambda }$ (low decay rate). (i) to (l) are spectral-gap edge modes of each corresponding structure.