Observation of a Topological Phase Transition in Random Coaxial Cable Structures with Chiral Symmetry

In the quiet world of condensed matter physics, researchers often look for patterns that hold true even when things are messy. One such pattern involves the way waves travel through a chain of connected points, like beads on a string. If the connections between these points alternate in strength—some tight, some loose—the system can enter a special state where waves get stuck at the very ends, refusing to move into the middle. This is known as a topological state, a condition that is remarkably stable against disorder. Imagine a long line of people passing a ball; if the rules of passing are slightly random, the ball usually gets lost in the middle. But in these special systems, the ball is guaranteed to stay at the end, no matter how chaotic the passing rules become in the center. This stability comes from a hidden symmetry in the way the chain is built, a property that scientists call chiral symmetry. Understanding how these states behave when the chain is not just a straight line but a closed loop, and when the disorder is truly random, has been a difficult theoretical challenge. It requires knowing if the system can switch between different topological states and what happens to the waves right at the moment of that switch.

A team of physicists at the University of Sheffield and the Max Planck Institute for the Science of Light has now observed these phenomena directly, using a system of coaxial cables to mimic the behavior of complex quantum materials. Instead of working with atoms or electrons, which are tiny and difficult to control, the researchers built a macroscopic model using radio frequency signals traveling through cables. They arranged these cables into chains where the electrical properties of each segment were randomly chosen from two specific values. This setup created a perfect, physical version of a mathematical model known as the Su-Schrieffer-Heeger model, but with the added complexity of randomness. By measuring how signals bounced off and traveled through these cable chains, the team demonstrated that a specific state remained locked at a precise frequency, changing by less than 0.2 percent across dozens of different random arrangements. This tiny variation proved that the system maintained its hidden symmetry with near-perfect accuracy, protecting the state from the chaos of the random connections.

The researchers then took the experiment a step further by connecting the ends of these chains to form closed loops. In a straight chain, the special state sits at the end, but in a loop, the ends meet, and the system must decide whether to keep the state or let it disappear. The team found that by adjusting the ratio of the two types of cables used in the loop, they could drive the system through a topological phase transition. This transition was marked by the closing of an energy gap, a region where no signals could normally exist. At the exact point of this transition, the signals that were previously trapped at the ends became delocalized, spreading out evenly across the entire loop despite the strong disorder. This behavior confirmed a long-standing prediction that at the boundary between two topological phases, waves can become free to roam the whole structure, a phenomenon that is usually hidden in theoretical calculations but is now clearly visible in this physical setup.

To understand how this works, one must look at the specific design of the experiment. The researchers used a vector network analyzer, a standard piece of equipment for measuring radio waves, to probe the cables. They measured the impedance, which is essentially how much the cable resists the flow of electricity, at different frequencies. When they measured a straight chain of sixteen cables, they found a sharp peak in the signal at a frequency of approximately 114 megahertz. This peak corresponded to the topologically protected state. The remarkable part was that when they repeated this measurement on forty-one different chains, each built with a different random sequence of cable types, the frequency of this peak barely moved. The standard deviation of the frequency was only 0.22 megahertz. This level of consistency showed that the protection of the state was not an accident of a specific arrangement but a fundamental property of the system's symmetry. The slight variations observed were attributed to tiny, unavoidable errors in the physical lengths of the cables, not to a failure of the symmetry itself.

When the chains were connected to form loops, the behavior changed in a way that revealed the nature of the phase transition. The researchers defined a parameter based on the ratio of the two cable types used in the loop. When this parameter was zero, the system was in a marginal state, sitting right on the edge between two different topological phases. In this specific condition, the energy gap closed, and two distinct states appeared at the zero-energy point. By measuring the signal at adjacent points in the loop, the team showed that these two states were localized on different halves of the loop's structure, one on the "black" sites and one on the "white" sites, a separation that is a hallmark of the underlying symmetry. As they moved away from this marginal point, the gap reopened, and the states became localized again, but this time they were confined to one end of the chain if it were cut open.

The transmission measurements provided the final piece of the puzzle, showing how the waves moved through the system. When the researchers sent a signal through the center of a straight chain that corresponded to the marginal loop condition, the signal passed through with perfect efficiency, as if the disorder did not exist. This perfect transmission is a signature of the topological phase boundary. In chains that were not at this boundary, the signal was blocked or absorbed, depending on which side of the transition the chain fell. The data showed that the transmission strength depended only on the ratio of the cable types, following a smooth curve that matched theoretical predictions. This confirmed that the transition was not just a change in the energy levels but a fundamental shift in how the waves could travel through the material.

The study also addressed the question of what happens in very long chains. Theoretical models predict that in an infinite system, the density of states at the transition point should show a sharp, singular peak. However, in the finite chains used in this experiment, which were sixteen cables long, this sharp peak was replaced by a broader hump. The researchers noted that while their current setup was too short to see the sharp singularity, the broadening they observed was consistent with simulations. They suggested that if they were to build chains with fifty to one hundred cables, the sharp feature might become visible. This limitation highlights the difference between theoretical infinities and practical experiments, but it does not diminish the clarity of the phase transition they observed.

The work confirms that coaxial cables are an excellent platform for studying topological physics. The ability to create structures with high symmetry and introduce disorder in a controlled way allowed the team to observe effects that are difficult to see in other systems. Unlike photonic or acoustic lattices, where controlling every connection is difficult, the cable system allowed for a precise mapping of the mathematical model to a physical reality. The researchers demonstrated that the topological protection is robust, surviving the randomness of the cable selection, and that the transition between phases is marked by a clear change in how waves are distributed. This experimental validation of the random SSH model provides a concrete example of how topology can protect states against disorder, a principle that could eventually inform the design of more robust electronic or optical devices.

In the end, the experiment showed that even in a system built from random parts, order can emerge in the form of protected states and clear phase transitions. The researchers did not just simulate these effects; they measured them directly, observing the frequency of the protected state and the flow of signals through the loops. The results were consistent with the predictions of topological physics, showing that the gap closes and the states delocalize at the transition point. The study stands as a clear demonstration that the abstract concepts of topological phases can be realized and explored in simple, tangible systems, opening the door to further investigations into more complex networks and the behavior of waves in disordered environments.