Spectral statistics and scattering resonances of complex primes arrays

Abstract

We introduce a novel 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 novel gapped photonic media that support far richer mode localization and spectral properties compared to usual periodic and random media.

Aperiodic Photonics of Elliptic Curves

“…at the metal-insulator threshold where it is known that all the wave functions exhibit multifractal scaling properties [46]. We remark that the presence of a critical statistics in the spectral behavior of EC structures occurs over a broad range of optical densities compared to the case of random media in which criticality is achieved only at the threshold density ρ c [25,31,43].…”

“…For two-dimensional (2D) arrays, this function characterizes the distribution of the diffracted intensity peaks contained within a square region, centered at the origin, with a maximum size of 2k × 2k in the reciprocal space [43]. Interestingly, Eq.…”

“…On the contrary, for structures with absolutely continuous Fourier spectra the integrated intensity function is continuous and differentiable. Finally, in the case of structures with singular-continuous spectra like the ones generated by the distribution of the prime number on complex quadratic fields and quaternion rings [43], the Bragg peaks are no longer well-separated but clustered into a hierarchy of self-similar contributions giving rise to a weak continuous component in the spectrum that smoothly increases the value of Z(q) in between consecutive plateaus. In the multiple scattering orders.…”

“…However, the level statistics of deterministic aperiodic systems displays different features with respect to the uniform random scenario. In our previous works, we have investigated the transition from the presence to the absence of level repulsion by increasing ρλ 2 in different open, deterministic, and aperiodic planar systems [25,31,43]. We have found that the distribution obeys at ρλ 2 ¡1 the critical cumulative probability density:…”

“…These more extended, long-lived resonances are similar to the critical scattering resonances that are typical of fractal and multifractal systems. These optical modes characterized by a power-law envelope localization and multifractal field intensity oscillations[25,26,33,39,43,71].To gain additional insights on the nature of the scattering resonances of EC-based arrays compared to U R systems, we have evaluated the proportion of modes that extend over a number of particles specified by the range of the MSE values considered. In particular, we computed the probability of the number of scattering resonances in different MSE ranges and at different optical densities.Fig.5shows the results of this study.…”