In the realm of wave mechanics, the convergence of disorder and openness shapes the behavior of waves as they traverse complex media. From photonic chips and ultracold atom assemblies to intricate electrical circuits, waves invariably encounter imperfections—random irregularities known as disorder—while simultaneously engaging with non-Hermitian effects, including energy gain, loss, and directional couplings that defy reciprocal symmetry. The interplay between these factors fundamentally alters wave propagation, yielding phenomena of profound theoretical and practical interest. Historically, two distinct outcomes have been recognized: Anderson localization, where waves become trapped in the bulk due to interference in closed, disordered environments, and the non-Hermitian skin effect, an accumulation of states localized at system boundaries in open environments with nonreciprocal interactions. Until now, a unified, quantitative framework encompassing both phenomena, particularly under simultaneous disorder and non-Hermiticity, remained elusive.
Groundbreaking work by physicists Konghao Sun and Haiping Hu from the Institute of Physics at the Chinese Academy of Sciences presents a transformative approach. Their study, recently published in Science Bulletin, constructs a set of “universal Thouless relations” that magnify and generalize the classical Thouless relation formulated over five decades ago. The original framework linked the density of energy levels to wave decay rates but was restricted to simple, closed, one-dimensional chains. Sun and Hu’s generalized relations extend this foundation to encompass complex one-dimensional systems characterized by arbitrarily ranged hopping, multiple energy bands, directional couplings—capturing non-Hermitian asymmetries—and diverse disorder types. Remarkably, this new formalism bridges the divide between closed and open systems, providing a comprehensive theoretical lens for wave localization phenomena.
Central to this advancement is a conceptual pivot that reimagines energy eigenvalues as point charges distributed across the complex energy plane, forming an electrostatic landscape shaped by system parameters. The researchers leverage this analogy to reveal that the spectral structure and wave localization properties are governed by the system’s Lyapunov exponents—measures of exponential growth or decay of waves along the lattice chain. Specifically, these exponents quantify the rates at which a wave function’s amplitude evolves spatially. By linking the distribution of Lyapunov exponents to the complex eigenvalue spectrum, the team devised an approach that extracts spectral and localization information from compact, finite-dimensional “transfer matrices.” This strategy circumvents cumbersome diagonalization of large Hamiltonians, which often suffers from numerical instabilities and high computational costs, especially in non-Hermitian, open systems.
The implications of these universal relations extend beyond computational efficiency. Importantly, they illuminate the topological nature of the transition between Anderson localized bulk states and non-Hermitian skin modes. The transition is marked by the closure of a “Lyapunov gap,” a spectral signature captured by the behavior of Lyapunov exponents. This gap closure coincides with a change in an integer-valued winding number that acts as a topological invariant: it vanishes in the bulk-localized phase and assumes a nonzero value in the skin-effect regime. This topological perspective provides a robust and elegant characterization of wave localization transitions that remain robust against disorder and non-Hermitian perturbations.
Intriguingly, at the critical point where the Lyapunov gap closes, the team identified a novel class of critical wave states they term “unidirectional multifractal states.” These states exhibit asymmetrical localization properties: they appear localized when probed from one direction yet display fractal-scale spatial spreading when examined from the opposite direction. This unidirectional multifractality represents a new universality class absent in conventional Hermitian systems, underscoring the richness of non-Hermitian disordered physics and opening avenues to explore wave localization phenomena fundamentally distinct from those captured by standard paradigms.
Beyond the theoretical appeal, Sun and Hu’s framework is highly practical. Their universal Thouless relations enable rapid and precise determination of key physical quantities: spectral densities, localization lengths, mobility edges (the energies separating localized and delocalized states), and transition points. These diagnostics are computationally tractable, permitting analysis of systems with hundreds to thousands of lattice sites, surpassing traditional direct diagonalization methods both in speed and numerical stability. This advantage is especially consequential for experimental design and interpretation, where accurate spectral and localization characterizations are imperative.
Experimental platforms poised to validate these predictions abound. Synthetic structures such as photonic lattices engineered with gain and loss elements, cold-atom systems manipulated via optical potentials and dissipation, and electrical circuits configured with nonreciprocal components offer fertile terrain. These platforms support tunable disorder, energy nonconservation, and directional couplings, thereby embodying the conditions modeled theoretically. The coexistence of skin and bulk localized states predicted by this universal theory can potentially be directly visualized using modern measurement techniques like spatially resolved spectroscopy and state tomography, enabling an unprecedented view into localization phenomena.
Beyond one-dimensional chains, the authors envision their approach as a foundation for extending the analysis into higher dimensions. Though significant challenges remain, the prospect of charting the topological and localization landscapes of complex non-Hermitian, disordered systems in two and three dimensions is an enticing direction for future research. Such developments could unravel intricate localization phenomena relevant to advanced photonic materials, quantum networks, and disordered open quantum systems subjected to external fields and impurities.
The broader impact of this work lies in its conceptual unification and practical utility. By locking spectral characteristics directly to localization behavior through universal relations, it provides a coherent framework that transcends traditional limitations. It enriches the theoretical toolkit available for exploring wave dynamics in realistic, non-ideal media, thus bridging the gap between idealized models and experimentally relevant conditions. Furthermore, the revealed topological structure governing localization transitions invites a reexamination of related phenomena in condensed matter physics, optics, and beyond.
In closing, the universal Thouless relations reveal a deep intertwining of spectral topology and wave localization in disordered, non-Hermitian open systems. They capture the complex dance of waves as they succumb to disorder or cluster at boundaries, governed by underlying Lyapunov exponents and topological invariants. This breakthrough unifies previously disparate frameworks, opening fresh paths for both fundamental understanding and experimental exploration of non-Hermitian physics, promising exciting developments on multiple scientific fronts.
Subject of Research:
Wave localization in one-dimensional disordered non-Hermitian open systems.
Article Title:
Universal Thouless Relations for Wave Localization and Spectral Topology in Disordered Non-Hermitian Systems.
News Publication Date:
Not explicitly stated in the source content.
Web References:
DOI Link
References:
Sun, K., & Hu, H. (2026). Science Bulletin. DOI: 10.1016/j.scib.2026.05.055
Image Credits:
©Science Bulletin
Keywords:
Non-Hermitian systems, Anderson localization, non-Hermitian skin effect, Lyapunov exponents, universal Thouless relations, wave localization, spectral topology, disordered systems, transfer matrices, unidirectional multifractal states, topological transitions.
