$\mathcal{PT}$ and anti-$\mathcal{PT}$ phase transitions in a trimerized Su--Schrieffer--Heeger chain with nonreciprocal Rashba spin-orbit coupling

In the quiet corners of quantum physics, scientists are exploring a world where the usual rules of energy and stability no longer apply. In the standard view of the universe, energy levels in a system are like solid, unchanging numbers; they are real and predictable. However, a growing field of study looks at "open" systems, where energy can leak in or out, much like a room with an open window. In these environments, the mathematical descriptions of the system become "non-Hermitian," a technical term meaning the energy values can become complex numbers, possessing both a real part and an imaginary part. For decades, physicists believed that if such a system possessed a specific kind of balance called parity-time symmetry, its energy would remain real and stable, behaving as if it were a closed, perfect system. But recent discoveries have shown that this balance can break in surprising ways, leading to a phenomenon where waves of particles pile up at the edges of a material rather than spreading evenly through it. This behavior, known as the non-Hermitian skin effect, challenges our understanding of how materials conduct electricity or light, and it raises a fundamental question: can a single material host different types of stability in its interior versus its edges?

A team of researchers has now mapped out a new territory where these questions are answered with striking clarity. By designing a theoretical model of a one-dimensional chain of atoms, they discovered a way to split the system into two independent parts based on the spin of the electrons, a property that can be thought of as a tiny internal compass. In this specific setup, the chain is arranged in repeating groups of three atoms, and the electrons are subjected to a special kind of interaction that pushes them differently depending on which way they are moving and which way their internal compass points. The researchers found that this arrangement creates a unique separation of duties within the material. While the vast majority of the electrons inside the chain might settle into a state of perfect stability with real energy values, the electrons trapped at the very ends of the chain can simultaneously exist in a state of pure imaginary energy, a condition that is usually considered unstable in other contexts.

To understand how this happens, imagine a long line of people passing a ball. In a normal line, the ball moves back and forth with equal ease. In the system the researchers studied, the rules of the game are rigged so that the ball is much more likely to be passed in one direction than the other, and this bias changes depending on whether the person holding the ball is facing left or right. The scientists built a mathematical model of a chain where the atoms are grouped in threes, and they introduced this directional bias, known as nonreciprocal coupling, specifically for electrons with different spins. They found that the spin-flip symmetry of the system allowed them to treat the "spin-up" and "spin-down" electrons as two separate, non-interacting worlds. In each of these worlds, the researchers could calculate exactly where the energy levels would be real, where they would be imaginary, and where they would become a confusing mix of both.

The results revealed a rich landscape of possibilities that had not been seen before. The researchers identified four distinct phases for the bulk, or the main body, of the material. In some regions, the entire bulk was stable with real energy. In others, the bulk was entirely imaginary. There were also mixed phases where one band of energy remained real while the others became complex. Most remarkably, they found that the edges of the material could behave differently from the bulk. In a specific range of conditions, the bulk of the material could be perfectly stable with real energy, while the edge states were locked into a purely imaginary energy state. This "bulk-edge symmetry class separation" means that the interior and the boundary of the same material obey different fundamental rules of stability at the same time.

The team did not just guess at these outcomes; they derived precise mathematical boundaries that define exactly when these transitions occur. They used a method that extends the concept of a crystal's repeating pattern to account for the directional bias, allowing them to predict the behavior of the system under open conditions where the ends are exposed. To verify their findings, they ran detailed computer simulations that tracked the movement of electrons and the shape of their energy levels. These simulations confirmed that the edge states remained pinned to the imaginary axis while the bulk stayed real, and vice versa, depending on the specific settings of the atomic chain. They also calculated a geometric quantity known as the Berry phase, which acts as a topological fingerprint, confirming that these edge states are protected by the underlying symmetry of the system and cannot be easily destroyed by small disturbances.

What makes this discovery particularly significant is the level of control it offers. By simply adjusting the strength of the directional bias or the spacing between the atoms, the researchers showed that they could switch the system between these different phases. They found that the behavior of the spin-up electrons was the mirror image of the spin-down electrons, shifting the conditions for stability in opposite directions. This spin-dependent control suggests that such materials could be engineered to host specific types of edge states while keeping the rest of the material in a different state. The study provides a minimal and clear platform for realizing these effects, showing that the interplay between spin, directionality, and symmetry can create a material where the inside and the outside live in different physical realities. This work opens a door to designing new types of electronic or photonic devices where the flow of information or energy can be selectively guided or isolated based on the spin of the particles, all while maintaining a stable connection between the bulk and the edge.