Classical observable coalescence as a discriminating diagnostic for spectral degeneracies in non-Hermitian systems

In the world of physics, most systems we encounter are closed and balanced, like a pendulum swinging in a vacuum or a planet orbiting a star. Their behavior is predictable, and their energy levels are distinct, like rungs on a ladder that never touch. However, many real-world systems are open; they exchange energy with their surroundings, gaining some and losing others. When physicists describe these open systems, they often use a mathematical framework that allows for "imaginary" numbers to represent this gain and loss. In this strange landscape, a peculiar phenomenon can occur where two different states of a system suddenly merge into one. This is called an exceptional point. Unlike ordinary crossings where two paths simply pass each other, at an exceptional point, the paths not only meet but the very nature of the states themselves fuses together, making them indistinguishable. This behavior is crucial for understanding how lasers work, how sound waves travel through special materials, and how to build sensitive sensors. Yet, finding these points in the real world has been difficult because the standard way to identify them requires looking at abstract mathematical directions that cannot be directly measured in a physical lab.

A team of researchers from universities in Cameroon and Gabon has proposed a new way to spot these elusive points using only the physical things we can actually see and measure. They focused on a specific model of a chain of interacting particles, a system that has been studied extensively in quantum mechanics but is also relevant to classical systems like electrical circuits. The researchers asked a simple but profound question: if we cannot measure the invisible mathematical directions that define an exceptional point, can we instead watch the physical positions of the particles themselves? They discovered that as the system approaches a critical threshold, the physical configurations of the particles do indeed merge. When two distinct arrangements of the system become identical, it signals the presence of an exceptional point. This "observable coalescence" provides a direct, experimental signature that does not require complex mathematical reconstruction.

To test this idea, the team analyzed a one-dimensional chain where particles interact with their neighbors and are subjected to a field that alternates between adding energy and removing it. They used a method that bridges the gap between quantum mechanics and classical physics, translating the behavior of the system into equations that describe the orientation and phase of the particles. As they adjusted the strength of the alternating field, they watched how the stable states of the system changed. They found that the system organizes itself into two distinct families of states. Within each family, there are two branches of solutions. As the field strength increased, these two branches moved closer together until they finally collided and became one. This collision happened at a precise value where the strength of the alternating field matched the strength of the interaction between the particles. At this exact moment, the physical difference between the two states vanished completely.

The researchers were careful to ensure this was not just a simple crossing of paths. In many physical systems, two energy levels can cross without the underlying states merging; this is known as a diabolic point, a common occurrence in standard physics. To distinguish between a true exceptional point and this ordinary crossing, the team looked at the behavior of the system when the field strength was zero. They found that while the energies of the states did coincide at zero field, the physical configurations of the particles remained distinct and separate. The particles did not merge. This confirmed that the zero-field crossing was an ordinary diabolic point, whereas the collision at the critical field strength was a genuine exceptional point. The key difference was that at the exceptional point, the physical states themselves became indistinguishable, whereas at the diabolic point, only their energy values happened to be the same.

To be absolutely certain of their findings, the team compared their classical observations with the exact mathematical solution of the quantum version of the same system. The quantum calculation, which is considered the gold standard for this type of problem, showed that the energy levels of the system also merged at the exact same critical field strength. Furthermore, the quantum analysis confirmed that the mathematical directions describing the states also fused together at this point, which is the defining characteristic of an exceptional point. The agreement between the classical observation of merging physical states and the quantum prediction of merging mathematical states was exact. The researchers also traced the path of the system's energy as they moved the field strength around the critical point in a circle. They found that the two energy branches swapped places after one full circle, a behavior that is unique to exceptional points and impossible for ordinary crossings.

This work establishes that the merging of physical configurations is a reliable and measurable sign of an exceptional point. It offers a new diagnostic tool for scientists working with open systems, such as those found in electronics, optics, and acoustics. Instead of needing to reconstruct invisible mathematical properties, researchers can now look for the moment when two distinct physical states of their system become identical. This approach makes the detection of these critical points much more accessible for experimental verification. The study demonstrates that the strange, topological features of non-Hermitian physics are not just abstract mathematical curiosities but are deeply embedded in the observable behavior of physical systems. By showing that classical observables can faithfully track quantum criticality, the researchers have provided a clear path for identifying and utilizing these unique points in future technologies.