Why finite-size non-Hermitian eigenvalues can miss asymptotic eigenstates in topological physics

In the quantum world, the behavior of particles is often described by mathematical objects called Hamiltonians, which act like instruction manuals for how energy moves and changes within a system. For decades, physicists have relied on a specific type of these instructions, known as Hermitian, which guarantee that energy levels are real numbers and that the system's states are stable and predictable. However, the real world is rarely so tidy. Many modern systems, from lasers to biological networks, are open to their environment, exchanging energy and matter. To describe these, scientists use non-Hermitian models, which allow for gain and loss. These models reveal strange new behaviors, such as the "skin effect," where particles that should spread out instead pile up against the edges of a material. A central challenge in this field has been figuring out how to predict the behavior of these systems when they become infinitely large, a state known as the thermodynamic limit. Traditionally, researchers have tried to guess this infinite behavior by studying smaller, finite versions of the system and watching how their energy levels change as they grow. This paper argues that for a specific class of non-Hermitian systems, this common method is fundamentally broken, leading to a complete misreading of what the system actually does.

The researchers, working with a team from Sweden and France, set out to understand why this extrapolation fails. They focused on a simple model called the Hatano-Nelson chain, which describes particles hopping along a line with a bias toward one direction. In a standard finite system, if you calculate the energy levels, you find they all collapse into a single point. If you were to follow this trend to an infinitely large system, you would conclude that the system has only one possible energy state. However, the authors show that this conclusion is an illusion created by the size of the system. When they looked at the actual states of the particles rather than just their energy numbers, they discovered that the infinite system supports a vast, continuous range of energy states. In the finite versions, these states exist as "almost" eigenstates—states that behave like perfect energy levels for a very long time before eventually drifting away. As the system grows larger, these states become more stable and last longer, eventually becoming perfect, permanent states in the infinite limit. The finite-size calculation misses them entirely because it only looks at the exact numbers, ignoring the fact that the system is teetering on the edge of a much richer reality.

To explain why this happens, the authors introduce the idea of a "quasi-Jordan structure." In simpler terms, this is a hidden architectural feature of the system where states are linked together in a chain. In a normal system, every state stands alone. In these non-Hermitian systems, the states are stacked so that one state can turn into another, creating a hierarchy. In a finite system, this structure is cut off at the edges, hiding its true nature. But in an infinite system, this chain becomes macroscopic, stretching across the entire material and creating a whole new spectrum of behaviors that the simple energy numbers cannot see. The researchers demonstrated that this isn't just a quirk of the simplest model. They applied the same logic to a more complex system known as the non-Hermitian SSH chain, which is often used to study topological phases—states of matter defined by their global shape rather than local details. In this model, the standard method of looking at finite energy levels suggested that certain edge states, which are crucial for the material's topology, did not exist or appeared at the wrong conditions.

By switching their focus to the "eigenstate spectrum"—a way of looking at the system based on how stable the states are rather than just their exact energy values—the researchers found a different picture. They showed that the edge states were indeed present, behaving exactly as the theory of infinite systems predicted, even though the finite energy calculations said otherwise. The finite-size eigenvalues were failing to detect these states because the system was so sensitive to its boundaries that the numbers shifted wildly with size, while the physical states remained robust. The authors proved that this discrepancy is not a mistake in calculation but a fundamental property of these systems. They showed that the "skin effect," where particles accumulate at the boundaries, is directly linked to these hidden structures. When the system has a non-trivial winding number—a mathematical measure of how the system's properties twist around a loop—it inevitably develops these macroscopic chains of states. These chains act like a scaffold for the system's behavior, allowing it to support a wide variety of stable states that are invisible to the standard tools of finite-size analysis.

The implications of this work are significant for how we understand and design materials with non-Hermitian properties. The paper clarifies that relying solely on the energy levels of small, computer-simulated systems can lead to incorrect conclusions about the topology and stability of large, real-world materials. It suggests that the apparent failure of the "bulk-boundary correspondence"—the rule that usually connects the properties of the inside of a material to its edges—is not a failure of the theory, but a failure of the method used to test it. The researchers conclude that to truly understand these systems, scientists must look beyond the simple list of energy numbers and consider the stability and structure of the states themselves. This shift in perspective reveals that the thermodynamic limit of these systems is far more complex and interesting than previously thought, filled with hidden modes that only reveal themselves when the system is large enough to let them breathe. The work serves as a cautionary tale for the field, urging researchers to be careful when drawing conclusions about infinite systems from finite data, especially in the presence of the skin effect.