Quasi-normal-mode signatures of periodicity and hyperuniformity

Imagine a material that is neither perfectly ordered like a crystal nor completely random like sand. It occupies a middle ground, a state of "hidden order" where the arrangement of its parts suppresses large-scale fluctuations in density. Scientists call this state hyperuniformity. In such materials, the particles are not arranged in a repeating grid, yet they avoid clumping together in the way random particles do. This unique balance allows them to control how waves—such as sound or light—travel through them in ways that neither perfect crystals nor random messes can. Understanding these materials is crucial for designing new types of optical devices, acoustic shields, and sensors. However, a major challenge has remained: how do we describe the behavior of these materials when they are not infinite sheets, but finite, real-world objects with edges? Traditional methods for analyzing crystals rely on the assumption of infinite repetition, which breaks down when you have a specific, bounded piece of material.

To solve this, researchers at the Universitat Politècnica de València have turned to a different way of looking at wave behavior, focusing on the "quasi-normal modes" of finite structures. Instead of asking how waves move through an endless sheet, they asked what happens when a wave hits a specific, finite patch of material and how that patch naturally vibrates and then fades away. They studied a thin, flat plate made of a uniform material, decorated with tiny point masses placed at specific locations. By changing the arrangement of these masses from a perfect grid to various forms of hyperuniform disorder, they could watch how the material's natural resonances changed. Their work reveals that the way these finite structures vibrate and lose energy contains a direct signature of the hidden order within them, offering a new tool to characterize materials that sit between order and chaos.

The researchers began by establishing a baseline using a standard grid of masses, similar to the atoms in a crystal. In a perfect, infinite crystal, waves travel in specific bands of frequency, with gaps where waves cannot exist at all. When the researchers looked at a finite version of this crystal—a square patch of 49 masses—they found that the waves did not simply disappear at the edges. Instead, the finite structure supported specific resonant states, which they identified as quasi-normal modes. These modes have two key features: a frequency at which they vibrate, and a rate at which they lose energy to the surrounding air. The team found that the resonances of the finite crystal clustered in the same frequency bands as the infinite crystal, but they also discovered a new class of vibrations. Some modes were trapped near the edges of the plate, while others were deeply localized around specific points where a mass was missing. This showed that even in a perfect grid, the finite size and any small defects create unique vibrational signatures that standard theories miss.

Next, the team replaced the perfect grid with hyperuniform patterns. They generated three different arrangements of the 49 masses, each with a different degree of "stealthiness," a measure of how strongly the particles avoid clumping. In the least stealthy arrangement, the masses were somewhat scattered, and the resulting vibrations were spread out across a wide range of frequencies, much like in a random mess. As they increased the stealthiness, the particles became more constrained, avoiding each other more strictly. This change in arrangement had a dramatic effect on the vibrations. A clear gap appeared in the spectrum of resonances, a region where very few vibrations could exist. This gap was not caused by a repeating pattern, as in a crystal, but by the collective, long-range correlations of the hyperuniform arrangement. The researchers found that as the stealthiness increased, the resonances organized themselves into distinct groups, mirroring the gaps and bands seen in the theoretical calculations of infinite, repeating supercells.

Crucially, the study showed that these resonances were not just about frequency; they also revealed how the energy was stored and lost. In the hyperuniform materials, the researchers identified three distinct types of vibrations. At low frequencies, the waves behaved as if the material were a smooth, uniform sheet. At intermediate frequencies, the waves became sensitive to the specific, correlated arrangement of the masses, forming collective patterns that spanned the whole plate. Most surprisingly, at higher frequencies, the waves began to get trapped in specific regions of the material without any defect or edge to hold them there. These "intrinsically localized" states emerged purely from the complex interference of waves bouncing off the correlated, aperiodic arrangement of masses. This finding proves that a material does not need a perfect crystal structure or a deliberate flaw to trap energy; the hidden order of a hyperuniform arrangement is enough to create these pockets of silence.

The team also tested what happened when they introduced a deliberate flaw into the hyperuniform pattern by removing one of the central masses. In a perfect crystal, removing an atom creates a trapped state deep inside a gap, which vibrates for a very long time before losing energy. In the hyperuniform material, the researchers found a similar trapped state, but it behaved differently. While the mass was indeed localized around the missing point, it lost energy much faster than a defect in a crystal would. This is because the surrounding hyperuniform material, while suppressing long-wavelength fluctuations, still allowed for other pathways for the energy to escape. The "gap" in a hyperuniform material is not a perfect wall like in a crystal; it is a porous barrier that lets some energy leak through. This distinction is vital: the quality of the resonance, or how long it lasts, depends not just on where the wave is trapped, but on the specific nature of the material surrounding it.

By mapping these complex vibrations, the researchers have provided a new way to see the invisible order in hyperuniform materials. They demonstrated that the spectrum of resonant frequencies and the way those frequencies fade away offer a complete picture of the material's internal structure. This approach bridges the gap between the theoretical description of infinite, repeating systems and the reality of finite, open objects. It shows that the hidden order of hyperuniformity leaves a clear fingerprint on how a material sings and how that song dies out. This insight allows scientists to design materials not just by how they look, but by how they vibrate, opening the door to creating structures with precisely controlled resonant properties for advanced acoustic and optical applications.