When a wave gets trapped, but not in the usual way

In a familiar picture of localization, a wave stays near a particular place because disorder prevents it from spreading. But waves in systems with directionally unequal couplings can behave differently. This directional imbalance, called non-reciprocity, can make many wave patterns accumulate together—a phenomenon known as the non-Hermitian skin effect.

Guolin Nan and colleagues ask what happens when that directional bias varies randomly from one link in a chain to the next, while the chain also has a regular-but-nonrepeating, or quasiperiodic, pattern. Their central finding is that the system can switch between two kinds of confinement, rather than between spreading and confinement. They also investigate how this picture changes when waves can hop over an additional site.

A model built to compare patterns

The researchers study a one-dimensional tight-binding model: a mathematical chain where a wave can move between neighboring sites. The model combines a quasiperiodic on-site potential with randomly varying, direction-dependent hopping amplitudes. The random directional bias is represented by what physicists call a disordered imaginary gauge field. The authors also add next-nearest-neighbor hopping to explore mobility edges—energy-dependent boundaries between different kinds of states.

They analyze the model numerically, examining wave patterns and their spatial concentration, calculating Lyapunov exponents (which indicate whether wave amplitudes grow or decay exponentially with distance), and tracking spectral winding, a measure of how the system’s complex-valued energies loop around a reference point. They also simulate the evolution of a wave packet initially placed at one site. To interpret the localization boundary, they use a mathematical transformation that maps this particular non-Hermitian model to a Hermitian quasiperiodic model. That mapping is useful here, but is not a general tool for all non-Hermitian systems.

Two ways to be confined

With only nearest-neighbor hopping, the calculations show a transition between two localized-looking regimes. At weaker quasiperiodic modulation, many states accumulate around a small number of irregular, disorder-selected positions in the chain. The authors call this the erratic non-Hermitian skin effect. At stronger modulation, states are conventionally Anderson-localized: each is concentrated around a site, but the centers of different states are distributed across the chain.

This distinction is easy to miss if one asks only whether each state is spatially confined. The paper reports that a measure called the fractal dimension tends to zero in both regimes, so it does not distinguish them. Instead, the Lyapunov exponent and the distribution of states’ centers of mass separate the two patterns in the calculations.

Adding weak next-nearest-neighbor hopping produces an intermediate regime in which the two kinds of states coexist at different energies. The resulting “anomalous mobility edge” separates Anderson-localized states from states with skin-like accumulation. It is anomalous because, in this model’s original non-Hermitian description, the boundary is not between extended and localized states—the two sides are both confined in different ways.

The authors also examine dynamics. In a particular disorder realization, a wave packet can drift in a direction associated with the sign of the spectral winding. Different realizations can favor opposite directions. When the researchers average over many realizations without sorting by that sign, the opposing drifts largely cancel; the averaged motion therefore does not show the same net directional bias. This makes the way disorder is averaged important to interpreting transport.

What remains uncertain

These are results from numerical calculations on a specific idealized model, not a report of a new experiment. The paper suggests that the signatures could be tested in engineered systems, such as photonic or electrical platforms, but does not demonstrate them there. Its phase boundaries and conclusions are tied to the model’s assumptions and parameter choices; they should not be taken as universal rules for non-Hermitian systems.

There are also finite-size caveats. In finite chains, a sample’s average directional bias may not cancel exactly, affecting whether calculated energies appear complex. The paper says this effect diminishes as system size increases, and checks some results across system sizes and numbers of disorder realizations. Such checks support the numerical analysis but do not replace experimental tests or independent replication. The authors state that their data and code are available from the corresponding author upon request.

This work is available as an arXiv preprint. A preprint is a public research manuscript, not by itself evidence that the work has passed peer review. The findings are best read as a model-based proposal: in some non-Hermitian quasicrystals, a mobility edge may divide not mobile waves from trapped ones, but two distinct forms of trapping.