Disorder and openness are everywhere. In photonic chips, ultracold atoms, and electrical circuits, waves rarely travel through a perfect, sealed medium. Instead they encounter imperfections (disorder) while also gaining or losing energy and feeling one-way couplings (non-Hermiticity). Two very different things can happen to a wave under these conditions. In a closed, disordered medium, interference can freeze a wave in place—a phenomenon known as Anderson localization. In an open system, a large number of states can instead pile up against the edges, the so-called non-Hermitian skin effect. Until now, physicists had no single, quantitative framework that could describe both behaviors at once, especially once disorder is added to the mix, because the usual band-theory tools assume a perfectly repeating lattice that disorder destroys.
In a study published in Science Bulletin, Konghao Sun and Haiping Hu of the Institute of Physics, Chinese Academy of Sciences, introduce a set of "universal Thouless relations" that close this gap. The classical Thouless relation, written down more than fifty years ago, connected the density of energy levels to the rate at which waves decay—but only for the simplest closed chains. The new relations extend that idea dramatically: they hold for one-dimensional systems with any finite range of hopping, any number of bands, one-way (nonreciprocal) couplings, and many kinds of disorder, covering both closed and open systems on the same footing.
The key is a change of viewpoint. The team treats each energy level as a tiny electric charge in the complex plane and the spectrum as an electrostatic landscape, then shows that this landscape is governed by the system's Lyapunov exponents—numbers that measure how fast a wave grows or shrinks as it moves along the chain. This connection lets the researchers read off the spectrum and the localization behavior directly from small "transfer matrices," sidestepping the need to diagonalize enormous matrices, a process that is notoriously unstable and slow for open systems.
Using the relations, the authors show that the switch between Anderson-style bulk trapping and skin-effect edge pile-up is genuinely topological: it is triggered when a single "Lyapunov gap" closes, and it can be tracked by an integer winding number that is zero for bulk-trapped states and nonzero for edge-piled states. Exactly at the transition, they find a new kind of critical state—christened a unidirectional multifractal state—that is localized when viewed from one direction but spreads out in a fractal, scale-dependent way when viewed from the other. Such a lopsided critical state has no analog in ordinary closed (Hermitian) systems.
Beyond the conceptual unification, the framework is practical. It pinpoints spectral densities, localization lengths, mobility edges, and transition points efficiently and accurately, and the authors report that in standard numerical precision it can outperform direct diagonalization for systems of a thousand sites. The results should be testable in today's synthetic platforms—photonic lattices, cold-atom setups, and electrical circuits—where gain, loss, nonreciprocity, and disorder can all be tuned, and where the predicted coexistence of edge and bulk states could be mapped directly.
"Our relations lock the spectrum and the localization together, so the shape of the spectrum tells you where the waves go," the authors note. They add that the approach extends naturally to systems with impurities, applied fields, certain open quantum systems, and random networks, and they highlight the move to higher dimensions as an open and inviting direction for future work.
