Boundaries Powerfully Confine Energy in One-Dimensional Materials

Shu-Xuan Wang and colleagues at Sun Yat-sen University observed eigenstates localising at the boundary of a system exponentially under open boundary conditions. The localisation length for these states scales with the system’s length under generalised boundary conditions, a phenomenon termed scale-free skin effect. This model-independent approach offers a new perspective on finite size effects within non-Hermitian systems and provides valuable insight for future research in this area. Universal scaling of eigenstate localization lengths reveals a model-independent skin effect Eigenstate localization lengths now scale directly with the length of one-dimensional systems under generalised boundary conditions, marking a significant improvement over previous limitations. The scale-free skin effect (SFSE) reveals a universal principle governing energy state concentration at material edges, a phenomenon previously understood only through case-by-case analysis of specific models. The non-Hermitian skin effect (NHSE), a well-established phenomenon in non-Hermitian systems, typically manifests as an exponential localization of eigenstates at the boundary when subjected to open boundary conditions (OBC). However, conventional NHSE exhibits a localization length that remains independent of the system size, presenting a limitation in understanding finite-size effects. This new research demonstrates a departure from this behaviour, revealing that for certain conditions, the localization length is, in fact, proportional to the system’s length. Further analysis establishes that SFSE can be induced even in Hermitian chains with purely imaginary impurities, opening new avenues for investigating finite size effects in non-Hermitian physics and providing a fresh perspective on these complex systems. This is particularly significant as it bridges the gap between Hermitian and non-Hermitian physics, suggesting that non-Hermitian behaviour can emerge from seemingly Hermitian systems through carefully engineered perturbations. Simulations consistently demonstrated this behaviour across system lengths of 75, 100, and 125, with the mean position of eigenstates remaining consistently at either 0 or 1, confirming scale-free modes. This consistent positioning indicates a robust localization at the boundaries, irrespective of the specific system size within the tested range. When the impurity strength was less than the difference between left and right hopping parameters, theoretical predictions and numerical results for the Hatano-Nelson model with onsite boundary impurities showed accurate agreement. The Hatano-Nelson model, a paradigmatic example in non-Hermitian physics, served as a crucial testbed for validating the theoretical framework and demonstrating its applicability to established models. Analysing Generalised Boundary Conditions via Perturbation Theory for Scale-Free Skin Effect Perturbation theory proved key to unlocking this understanding of SFSE, allowing the team to reframe the problem as a slight deviation from a simpler, well-understood scenario instead of directly solving complex models. The core of their approach lies in the application of perturbation theory to a system initially governed by periodic boundary conditions. In a system with periodic boundary conditions, energy states are delocalized and can propagate freely around the loop. By introducing generalised boundary conditions (GBC), the researchers effectively modified the edges of the system, transitioning them from rigid, fixed boundaries to partially permeable membranes. This subtle alteration allows for a controlled introduction of non-Hermiticity and the emergence of the scale-free skin effect. The analysis considers a Hamiltonian with hopping matrices, representing the interactions between adjacent sites, and a system length denoted by ‘L’, alongside wave functions described by creation and annihilation operators, standard tools in quantum mechanical descriptions of many-body systems. The GBC are mathematically expressed as modifications to the boundary terms in the Hamiltonian, allowing for a systematic investigation of their impact on the eigenstate localization. This approach differs from the typical skin effect with size-independent localization, as localized states do not decay proportionally to system length. In conventional NHSE, the localization length is a constant, regardless of the system size. However, in SFSE, the localization length increases linearly with the system length, leading to a more pronounced boundary localization in larger systems. This scaling behaviour is a direct consequence of the GBC and the underlying physics governing the SFSE. Edge state localisation linked to system size in one-dimensional materials The scale-free skin effect promises advances in areas like sensing and wave control, yet current explanations remain incomplete. Clarifying how energy concentrates at edges in one-dimensional systems, independent of specific material properties, offers a foundational understanding for designing new sensors and potentially guiding research into real-world materials containing imperfections. The ability to predict and control edge state localization has significant implications for the development of novel sensing technologies. By engineering materials with specific boundary conditions, it may be possible to create sensors that are highly sensitive to external stimuli, such as changes in temperature, pressure, or strain. Researchers demonstrated a mechanism for the scale-free skin effect in one-dimensional systems, revealing that the extent of localisation of energy at the boundary increases proportionally with the system’s length. This differs from the typical skin effect where localisation remains constant regardless of size. The work provides a new perspective on understanding how finite system size impacts non-Hermitian systems and clarifies the relationship between boundary conditions and edge state localisation. The authors suggest this understanding is foundational for designing new sensors and guiding research into materials with imperfections. 👉 More information 🗞 Mechanism for scale-free skin effect in one-dimensional systems 🧠 ArXiv: https://arxiv.org/abs/2604.01638 NON-HERMITIAN SKIN EFFECT ONE-DIMENSIONAL SYSTEMS