Remarks on invertible phases with non-onsite symmetry

In the quantum world, matter can organize itself into states that are far more exotic than the familiar solids, liquids, and gases. Among these are "invertible phases," a special class of quantum states that are stable and gapped, meaning they have a clear energy gap that prevents them from easily changing. These states are unique because they can be paired with another state to cancel each other out, returning the system to a simple, empty baseline. When these phases are protected by symmetry—a rule that says the system looks the same after a specific transformation—they form a structured landscape of possibilities. For decades, physicists have understood how these phases behave when the symmetry acts locally, meaning the rule applies to each particle individually. However, a new frontier has opened up: what happens when the symmetry is "non-onsite," acting in a way that links particles across distances in a complex, non-local pattern? This question is crucial because such symmetries often carry "anomalies," which are subtle inconsistencies that usually prevent the system from settling into a simple, short-range entangled state.

A team of researchers has now explored this uncharted territory, focusing on a specific type of quantum material known as a p-plus-ip superconductor in two spatial dimensions. In this system, electrons pair up in a way that creates a swirling, chiral flow, characterized by a specific number called the chiral central charge. For a long time, it was believed that if a system possessed a certain four-fold symmetry, it could only support invertible phases where this chiral central charge was a whole number. The researchers set out to test this rule when the symmetry was not local but rather a complex, non-onsite version. They constructed an exact mathematical model of a p-plus-ip superconductor and demonstrated that it possesses a precise, non-local symmetry that acts on the entire system. This symmetry is not a simple rotation of individual particles but a sophisticated operation that respects the quantum rules of the whole material.

The team discovered that this non-onsite symmetry behaves in a way that defies previous expectations. While the symmetry carries a specific "lattice anomaly" that forbids the system from existing in a simple, short-range entangled state, it does not destroy all order. Instead, it shifts the rules of the game. The researchers found that this symmetry strictly forbids the usual whole-number states. If the system tries to settle into a state with a whole-number chiral central charge, the symmetry breaks down. However, the system is not left empty-handed. It finds a new home in a set of states where the chiral central charge is a half-integer, specifically numbers like one-half, three-halves, and so on. The p-plus-ip superconductor, which naturally has a chiral central charge of one-half, turns out to be the perfect candidate for this new regime. The researchers proved that the symmetry operator for this material is mathematically exact and exponentially localized, meaning its influence fades quickly with distance, yet it remains a non-local operation that cannot be simplified into a local one.

To understand the deeper implications, the team looked at what happens when the "fermion parity"—a fundamental property of the electrons—is turned into a gauge symmetry, effectively making the invisible visible. This process transforms the superconductor into an "Ising topological order," a state of matter that hosts exotic particles called anyons. In this new state, the original four-fold symmetry re-emerges, but its structure is enriched by the lattice anomaly. The researchers described this using a sophisticated mathematical framework called a fusion 2-category, which tracks how different symmetry operations combine and interact. They found that the symmetry in this topological state is not a simple group but a more complex structure where the symmetry operations fuse in unexpected ways, creating non-invertible lines that braid with the anyons. This structure confirms that the anomaly prevents the system from behaving like a standard topological order with a whole-number central charge.

Furthermore, the study revealed that this p-plus-ip superconductor possesses an even larger, continuous symmetry that is also non-local. This enlarged symmetry acts like a fractional quantum Hall effect, a phenomenon where electrical conductivity is quantized in fractions rather than whole numbers. In a standard system with local symmetry, such fractional responses are impossible for invertible phases. Here, however, the non-local nature of the symmetry allows the system to exhibit a response that is half-quantized. The researchers showed that if one were to introduce a vortex, a point where the symmetry twists, it would bind a special zero-energy mode and carry a topological spin that leads to this half-quantized Hall response. This suggests that the non-onsite symmetry fundamentally alters the landscape of possible quantum phases, shifting the allowed states away from the familiar whole numbers and into a new realm of half-integer possibilities.

The work resolves a puzzle that had confused physicists: how a system with a non-trivial lattice anomaly can still support an invertible phase. The answer is that the anomaly does not merely block the system; it redirects it. By forbidding the conventional whole-number states, the anomaly forces the system to adopt a shifted set of states with half-integer characteristics. This finding challenges the traditional view that anomalies are simply obstructions to be avoided. Instead, they act as selectors that define a new, shifted classification of matter. The researchers have provided a concrete example of this phenomenon, showing that the p-plus-ip superconductor is not just a candidate for a specific phase but is the very embodiment of a new class of invertible phases protected by non-onsite symmetries. This opens the door to a broader understanding of how microscopic anomalies can shape the macroscopic properties of quantum materials, suggesting that there may be many other classes of spin liquids and topological states waiting to be discovered in the presence of these complex, non-local symmetries.