Disordered stealthy hyperuniform materials exhibit the statistical isotropy of a liquid combined with some of the structural properties of a crystal—including homogeneity, suppressed density fluctuations at large length scales, bounded holes, and a structure factor $S\left(k\right)$ that vanishes for a finite range of wave vectors, $0<k\le K$. Numerous studies have shown that this combination leads to highly unusual physical properties for a disordered medium, including optical transparency, effective delocalization, ultrafast spreadability, optimal diffusion coefficients and conductivities, and complete isotropic photonic band gaps. Gyromorphs, point patterns whose structure factor includes rings of intense Bragg-like peaks arranged with discrete $G$-fold rotational symmetry, were recently introduced as counterexamples: disordered media that can somehow achieve the same physical properties, in some cases with higher performance, without stealthiness or hyperuniformity. In this paper, we resolve the puzzle of how gyromorphs fit consistently with the stealthy hyperuniform studies. We first show that gyromorphs are actually hyperuniform and, in the large-$G$ limit where they become nearly isotropic, they belong to the weakest form of hyperuniformity, known as class III. Thus, gyromorphs should have comparatively degraded physical properties compared to stealthy hyperuniform media, which belong to the strongest form of hyperuniformity, known as class I. We verify this expectation using the rigorous spectral Green's-matrix method for the calculation of the density of states and Purcell factors in large arrays of electric dipoles. We find that gyromorphs display size-dependent pseudogaps richly populated by localized states rather than smooth band gaps like those found for highly stealthy hyperuniform (class I) materials or in deterministic structures such as Vogel spirals and triangular lattices. Furthermore, we predict similar disorder-induced degradation relative to stealthy hyperuniformity with regard to transparency, spreadability, and diffusion properties.

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• Photonics
• Amorphous materials
• Random & disordered media
• Electromagnetic wave theory
• Green's function methods