Steering of topological criticality in the Su-Schrieffer-Heeger chain via non-Hermitian substitutional impurities

In the microscopic world of materials science, there exists a class of substances known as topological insulators. These are materials that act as insulators in their interior but conduct electricity perfectly along their edges, a property that is remarkably robust against imperfections. Imagine a highway where traffic flows smoothly on the shoulders regardless of potholes or debris in the middle lanes; this is the essence of topological protection. For decades, scientists have studied a specific theoretical model, often called the Su-Schrieffer-Heeger chain, to understand how these edge currents emerge and how they might be controlled. This model describes a one-dimensional line of atoms where electrons hop from one to the next, with the strength of the hop alternating between strong and weak. When the pattern of these hops is right, the system enters a special topological state that hosts these protected edge currents. However, real-world materials are never perfect; they contain impurities and defects. A major question in the field has been how these imperfections affect the delicate topological order. While some disorder can sometimes create new topological states, it often destroys them. Recently, physicists have also begun exploring "non-Hermitian" systems, where energy can be gained or lost, or where the rules of movement are not the same in both directions, adding a layer of complexity that mimics open systems interacting with their environment.

A team of researchers has now investigated what happens when a pristine, perfect line of atoms is disrupted not by random noise, but by specific, localized defects that break the symmetry of movement. They focused on a chain of atoms where they replaced a few of the native atoms with foreign ones, creating substitutional impurities. Crucially, these impurities were designed to be non-reciprocal, meaning an electron hopping toward the defect experiences a different force than one hopping away from it. The researchers wanted to know if these isolated, asymmetric defects could change the fundamental rules of the material. They found that a single defect, no matter how strange, cannot destroy the topological protection of the entire chain. The system is resilient; the defect simply traps a few electrons locally without disturbing the global flow. However, the story changes dramatically when many of these defects are introduced. When a finite concentration of these asymmetric impurities is scattered throughout the chain, they begin to interact with one another. Instead of acting as isolated obstacles, they collectively reshape the landscape of the material.

The most striking discovery is that the researchers found a way to deterministically move the boundary between the topological state and the normal state. By carefully adjusting the internal structure of the impurities—specifically, by making the hopping strength different for the two types of connections an impurity makes—they could push the critical point where the phase transition occurs. If they configured the impurities to favor connections between different cells of the chain, the topological phase expanded, surviving in regions where it normally would have vanished. Conversely, if they configured the impurities to favor connections within a single cell, the topological phase was compressed, disappearing much earlier than expected. This means that by simply changing the local geometry of the defects, one can actively reconfigure the macroscopic properties of the entire material. The researchers validated this by simulating chains of hundreds of atoms and averaging over thousands of random arrangements to ensure the effect was real and not just a fluke of a specific setup. Their calculations showed that the transition from one phase to another became "smeared" or broadened, a sign that the disorder was fundamentally altering how the material behaves.

The study also revealed that this reconfiguration is linked to a change in the nature of the energy levels within the material. In a perfect system, the energy levels are real numbers. In the presence of these non-reciprocal defects, the energy levels can become complex numbers, a mathematical sign that the system is losing energy or gaining it in a way that breaks time-reversal symmetry. The researchers observed that as the concentration of defects increased, the system underwent a transition where these complex energy levels began to appear in large numbers. This transition was accompanied by a phenomenon known as the non-Hermitian skin effect, where the electron waves, instead of spreading out evenly, were pushed exponentially toward the edges of the chain. This accumulation of waves at the boundary is a direct consequence of the non-reciprocal hopping introduced by the impurities. The researchers noted that while the topological phase boundary could be shifted, the system remained surprisingly stable against the emergence of these complex energy states, provided the asymmetry of the defects was not pushed to an extreme.

To understand the depth of this change, the team performed a detailed analysis of how the system behaved as they varied its size. In physics, the way a system changes as it grows larger often reveals its fundamental nature. They found that the critical exponent, a number that describes how the transition width changes with system size, was different from that of a standard, clean chain. For the clean chain, this number is exactly one. For their disordered, non-Hermitian chains, the number increased to approximately 1.455 for symmetric defects and 1.397 for asymmetric ones. This shift confirms that the presence of these defects has moved the material into a new universality class, a distinct category of physical behavior that cannot be described by the old rules of the clean system. The researchers also noted that at very high concentrations of defects, the transition becomes so broad that the concept of a sharp phase transition breaks down entirely, suggesting a complete macroscopic smearing of the topological order.

The implications of this work are significant for the design of future quantum materials. It demonstrates that topological protection, often thought of as a rigid property, can be actively tuned and manipulated through the strategic placement and design of local defects. By breaking the internal symmetry of these defects, scientists can expand or compress the topological phase without needing to alter the entire material globally. This offers a new tool for engineering robust quantum states that can withstand disorder or, conversely, for creating states that are sensitive to specific structural changes. The findings suggest that the interplay between disorder and non-reciprocity is not just a source of noise to be eliminated, but a resource that can be harnessed to control the macroscopic behavior of topological systems. The researchers conclude that their work establishes a clear link between the microscopic asymmetry of isolated scatterers and the macroscopic shift of topological boundaries, opening a path for future research into how these effects might be observed in real-world platforms like photonic crystals or electrical circuits.