Interacting Electronic Topology of Nonlocal Crystals

In the world of materials science, scientists have long been fascinated by insulators—substances that block the flow of electricity. For decades, the standard view was that these materials worked simply because their electrons were stuck in place, localized to specific atoms like cars parked in a garage. However, a revolution in physics revealed that some insulators are far more mysterious. These "topological" insulators possess hidden, non-local properties that cannot be explained by looking at individual atoms alone; instead, they arise from the collective behavior of electrons across the entire material. This discovery led to a new classification of matter based on symmetry and entanglement, where the rules of the game are set by how the material's structure repeats itself in space. Yet, a fundamental assumption has always held these theories together: that the interactions between electrons must be local, meaning an electron can only feel the presence of its immediate neighbors. If an electron could instantly influence another electron on the far side of the crystal, the very concept of a one-dimensional chain would dissolve, and the established rules of topology would no longer apply.

A team of researchers has now challenged this assumption by exploring what happens when they deliberately break the rule of locality. They asked a simple but profound question: if we allow electrons to interact across infinite distances within a crystal that still maintains its repeating pattern, can we create a state of matter that is impossible under normal conditions? Their answer is a definitive yes. By constructing a theoretical model of a one-dimensional crystal where electrons interact with every other electron regardless of distance, they discovered a unique, stable state of matter that possesses a specific symmetry property forbidden in all known local materials. This state is not just a mathematical curiosity; it is a robust, gapped ground state that exists only because the interactions are non-local, proving that relaxing the constraint of locality opens up entirely new sectors of physical reality that were previously thought to be inaccessible.

The researchers focused on a specific type of crystal arrangement: a one-dimensional ring of atoms with a repeating pattern, where electrons carry a property called spin and are subject to time-reversal symmetry. In such a system, physicists use a specific number, known as an inversion eigenvalue, to describe how the entire collection of electrons behaves when the crystal is flipped inside out. In standard, non-interacting materials, this number is always positive. In more complex, interacting materials where electrons only talk to their neighbors, this number can be negative, but only if it changes depending on the total size of the crystal. Specifically, for a local crystal, the sign of this number flips every time you add a single unit of length to the chain. This size-dependence is a strict law for local materials; a negative sign that remains constant regardless of how large the crystal grows is strictly forbidden.

To test the limits of this law, the team built an exact mathematical model using a type of interaction originally proposed by Hatsugai and Kohmoto, which is diagonal in momentum space. In physical terms, this means the interaction is not limited to nearby atoms but extends across the entire system, effectively making the crystal behave like a zero-dimensional object where every part is connected to every other part. They populated this model with electrons and solved for the lowest energy state, or ground state, of the system. The result was striking. The ground state they found was unique and stable, possessing a gap to excited states, meaning it is a true insulator. Most importantly, this state carried an inversion eigenvalue of negative one that did not change, no matter how many unit cells were added to the chain. This size-independent negative sign is a value that symmetry theory allows for a fixed-size system, but which local physics strictly prohibits. The model successfully realized a state that sits in a "forbidden zone" for all local crystals, demonstrating that non-locality is a concrete route to new phases of matter.

The researchers also examined what would happen if they tried to force this interaction to be local, limiting the range of influence to a few neighboring atoms. When they did this, the energy gap that protected the unique state vanished as the system grew larger, and the material became gapless, losing its insulating character. This confirmed that the exotic state they found is not a subtle variation of a local phase but a direct consequence of the infinite-range interaction. Furthermore, they calculated a geometric quantity related to the center of mass of the electrons in the ring. In standard local insulators, this value is always an integer. In their non-local model, however, this value was exactly one-half, a fractional number that signals the electrons are pinned to a specific, non-trivial position within the repeating unit. This fractional value, combined with the size-independent negative sign, provides a clear fingerprint of this new non-local crystalline phase.

While the study was conducted in one dimension, the authors showed that their framework can be extended to higher dimensions. In two dimensions, they found that while some complex patterns of symmetry are possible, the specific size-independent negative sign they found in one dimension cannot be realized, even with non-local interactions, due to constraints imposed by another topological quantity known as the Chern number. This highlights that the rules of non-local crystals are not just a simple extension of local ones but involve their own distinct set of constraints and possibilities. The work suggests that while we cannot yet build a perfect non-local crystal in a lab, the theoretical landscape has expanded. Systems realized with trapped ions, ultracold atoms, or Rydberg arrays, where long-range interactions are naturally present, could potentially host these exotic states. By proving that symmetry-allowed but locality-forbidden states exist, the study opens a new chapter in the classification of matter, showing that the universe of possible insulators is larger than we thought, provided we are willing to let electrons reach across the void to touch one another.