Anderson localisation in infinite and finite resonator crystals with long-range interactions

Imagine a vast, perfectly ordered grid of tiny, hollow spheres, like a microscopic honeycomb made of glass. In physics, these are called resonators. When sound waves pass through such a structure, they do not simply travel in a straight line; instead, they bounce between the spheres, creating a collective vibration that can be described by a single, giant mathematical object. This object, known as a capacitance operator, captures how every single sphere talks to every other sphere, even those far away. The strength of this conversation depends on the material the spheres are made of. If the material is perfectly uniform, the sound waves can flow freely through the grid, spreading out like ripples on a calm pond. However, if the material properties of the spheres are slightly random—some slightly denser, some slightly lighter—the story changes. This randomness is known as disorder. For decades, physicists have known that in simple systems where particles only talk to their immediate neighbors, enough disorder can trap waves in place, a phenomenon called Anderson localization. But what happens when the waves can talk to neighbors across the entire grid, and those conversations fade slowly rather than stopping abruptly? This is the question that a team of mathematicians and physicists has finally answered for a specific, complex type of crystal.

The researchers focused on a crystal made of subwavelength resonators, which are objects small enough that they interact with waves much larger than themselves. In this regime, the interaction between resonators is long-range, meaning a sphere at one end of the crystal can still influence a sphere at the other end, though the effect gets weaker with distance. The team modeled a crystal where the material inside each resonator was randomly varied, simulating a real-world scenario where manufacturing imperfections create slight differences in density. They asked a fundamental question: does this randomness trap the sound waves near the top of the energy spectrum, or do they still manage to spread out? Using rigorous mathematical proofs, they demonstrated that for any fixed amount of randomness, no matter how small, the waves near the highest possible energy levels become trapped. They do not flow freely; instead, they concentrate into tight, localized spots, decaying exponentially as you move away from their center. This happens even though the waves can theoretically reach across the entire crystal. The team proved that this trapping is not just a temporary fluctuation but a permanent feature of the system's spectrum, meaning the waves are fundamentally stuck.

To understand how this works, one must look at the structure of the crystal's energy. In a perfect crystal, energy levels form a continuous band, allowing waves of different frequencies to travel. The researchers showed that when randomness is introduced, the very top of this energy band becomes a zone of total confinement. They calculated that for a specific range of energies just below the maximum, every single wave mode is localized. The size of this trapped zone depends directly on the strength of the randomness. If the disorder is weak, the trapped zone is narrow, but it exists. As the disorder increases, the zone of trapped waves expands, pushing further down into the energy band. The team also established that this behavior is not limited to infinite, theoretical crystals. They extended their proof to finite clusters of resonators, which are the kind of structures one might actually build in a laboratory. They showed that in a finite sample, almost all the wave modes in the trapped energy range concentrate within a small physical radius, while only a tiny fraction of modes near the edges of the sample behave differently. This confirms that the phenomenon is robust and observable in real, finite-sized objects.

The researchers did not stop at proving that the waves get stuck; they also quantified exactly how they behave. They found that the waves concentrate within a radius that grows very slowly as the sample size increases, specifically proportional to the size of the sample multiplied by the logarithm of its size. In practical terms, this means that even in a large crystal containing thousands of resonators, the trapped waves remain confined to a relatively small region. The team also explored what happens at lower energy levels, away from the top of the band. Here, the behavior is more complex. While they could not prove that waves are trapped everywhere in the lower band, they provided strong evidence that at higher levels of disorder, waves in the middle of the band also begin to concentrate. Their numerical simulations, which involved running hundreds of different random configurations for crystals containing up to 4,900 resonators, showed a clear trend: as disorder increased, the waves that were once spread out across the entire crystal collapsed into tight, localized clusters. The participation number, a measure of how many resonators a wave effectively uses, dropped dramatically, indicating that the wave was no longer sharing its energy with the whole system but was instead hoarding it in a small corner.

A crucial part of their work involved distinguishing between two ways of modeling a finite crystal. One method, called the truncated matrix, simply cuts off the crystal at a boundary and ignores everything outside. The other, the finite capacitance matrix, physically removes the outside resonators and recalculates the interactions based on the new, open boundary. The researchers proved that despite these different approaches, the statistical behavior of the waves is the same. Both models predict the same distribution of energy levels and the same tendency for waves to localize. This equivalence is vital because it means that the theoretical results derived for infinite crystals can be reliably applied to real, finite experiments. They also showed that the number of waves that do not follow this trapping rule is extremely small, limited to a thin layer near the physical edge of the sample. For the vast majority of the crystal, the waves are securely locked in place.

The study also addressed a long-standing question about the nature of the transition between free-flowing waves and trapped waves. In some systems, there is a sharp boundary, known as a mobility edge, that separates the two regimes. The researchers provided a precise upper bound for where this boundary could exist. They showed that for any amount of disorder, the top of the energy band is always a region of localization, and they calculated exactly how far down this region extends. While they did not prove that a mobility edge exists at lower energies, they ruled out the possibility that waves could flow freely right up to the very top of the spectrum. Their work clarifies that the "edge" of the energy band is a safe zone for localization, regardless of how weak the disorder might be. This finding is significant because it suggests that in materials with long-range interactions, the top of the energy spectrum is inherently unstable against disorder, always leading to the trapping of waves.

In the end, the paper provides a complete mathematical picture of how randomness affects waves in a specific type of crystal. It confirms that even with long-range connections, disorder wins near the top of the energy scale, forcing waves to localize. It proves that this behavior holds true for finite samples, which are the only kind that can be built in reality. And it offers a clear, quantitative description of how these waves concentrate, giving experimentalists a precise target for what to look for. The work bridges the gap between abstract mathematical theory and physical reality, showing that the complex dance of waves in a disordered crystal is not chaotic but follows strict, predictable rules. The researchers have mapped the territory where waves get lost, proving that in the presence of randomness, the highest energy states are the most likely to be found hiding in plain sight, confined to small pockets within the crystal.