🔬 mesoscale physics

Non-Hermitian Skin Effect from Radiative Coupling in a Reciprocal Chiral Medium

This paper demonstrates that long-range radiative coupling within a passive reciprocal chiral medium induces a non-Hermitian skin effect in an electric dipole chain, characterized by degenerate boundary-localized modes with a unique profile combining an exponential decay and a long-range algebraic tail.

Original authors: Kin Hung Fung, Changhao Meng, Yixin Xiao, C. T. Chan

Published 2026-08-28

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CC BY 4.0

Original authors: Kin Hung Fung, Changhao Meng, Yixin Xiao, C. T. Chan

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of physics, materials are often categorized by how they handle energy. Some materials, known as passive systems, simply absorb energy or let it pass through without adding any. Others are non-reciprocal, meaning they treat waves traveling in one direction differently than waves traveling in the opposite way, much like a one-way street for light or sound. For decades, scientists have been fascinated by a phenomenon called the non-Hermitian skin effect. In simple terms, this effect causes a massive number of waves, which usually spread out evenly through a material, to suddenly pile up and crowd against the edges or boundaries of that material. This happens when the connections between the tiny building blocks of a system are lopsided, pushing everything to one side. However, a new study challenges the assumption that this crowding effect requires a one-way street. The researchers investigated what happens when a material is perfectly reciprocal—meaning it treats forward and backward travel equally—but possesses a property called chirality, which gives it a specific handedness or twist.

A team of physicists at the Hong Kong University of Science and Technology set out to explore this specific scenario using a chain of electric dipoles, which are tiny sources of electric fields, placed inside a special type of material. This material is passive, meaning it does not generate energy, and it is chiral, meaning its internal structure twists electromagnetic waves in a way that depends on their polarization, or the direction in which they spin. The researchers modeled a long line of these dipoles and calculated how they would interact with each other over long distances through the medium. They found that even though the system is reciprocal and obeys the standard rules of symmetry, the chiral nature of the medium creates a surprising result. The chirality causes waves traveling in one direction to accumulate a different amount of phase and energy loss compared to waves traveling in the opposite direction. This difference acts like a subtle, invisible push that depends on the spin of the wave.

The most striking discovery was that this subtle push creates pairs of waves that crowd against the boundaries, but in a perfectly balanced way. Because the system is reciprocal, the rules for one type of spinning wave are the exact mirror image of the rules for the opposite spin. Consequently, waves with one type of spin pile up against the left end of the chain, while waves with the opposite spin pile up against the right end. These two groups of waves are degenerate, meaning they share the exact same frequency, yet they are locked to opposite sides of the material. This creates a unique state where the boundary accumulation is extensive, involving a large number of the system's modes, but it is split evenly between the two ends rather than all going to one side.

To understand exactly how these waves behave, the researchers looked closely at the mathematical structure of the interactions. They found that the profile of these crowded waves is not a simple, smooth curve. Instead, it is made of two distinct parts. The first part is a rapid, exponential drop-off that dominates the wave's shape for a significant distance from the edge. This part is driven by a specific, isolated point in the complex mathematical description of the system. However, as the wave gets further away from the edge, a second, longer-range tail takes over. This tail decays much more slowly, following a specific pattern related to the distance and the natural limits of how light travels through the material. This slow decay is caused by the way the material handles the transition between different types of wave propagation, creating a "branch cut" in the mathematical landscape that forces the wave to linger further out than a simple exponential curve would suggest.

The team also examined how the size of the chain affects the system. In many similar systems, making the chain longer would cause the behavior to settle into a predictable pattern that looks very different from the behavior of an infinite chain. Here, the researchers found a more complex evolution. As the chain grows longer, a portion of the system's energy levels moves closer to the pattern seen in an infinite, repeating chain. However, the number of waves that remain crowded against the boundaries does not shrink or disappear; it stays large and extensive. This means that even as the system gets bigger, the skin effect persists, with a significant fraction of the waves remaining trapped at the edges. The study confirms that this effect is robust and is directly tied to the fundamental analytic structure of the material's response, specifically the presence of logarithmic points that define the limits of light propagation.

One of the most counterintuitive findings concerns the role of energy loss in the material. The researchers discovered that changing the amount of energy dissipation in the chiral medium dramatically alters the behavior of waves in a repeating, infinite loop. Yet, for the finite chain with open ends, the overall energy levels of the system remain almost completely unchanged. While the specific shape of the waves and how tightly they are packed against the edges can be tuned by adjusting this dissipation, the fundamental frequencies at which the system resonates stay put. This suggests that the boundary localization is a very stable feature that can be manipulated without shifting the system's core identity. The work demonstrates that long-range radiative coupling in a chiral medium creates a new form of skin effect that is reciprocal, balanced, and governed by a complex interplay between exponential decay and long-range tails, offering a fresh perspective on how waves behave in structured materials.

Technical Summary: Non-Hermitian Skin Effect from Radiative Coupling in a Reciprocal Chiral Medium

Problem Statement
The Non-Hermitian Skin Effect (NHSE) is typically characterized by the accumulation of extensive bulk eigenmodes at boundaries under open boundary conditions (OBCs), often driven by asymmetric couplings or non-reciprocal gauge fields. While recent studies have shown that reciprocal systems with internal polarization degrees of freedom can support paired skin effects, the behavior of NHSE in systems with long-range radiative coupling remains subtle. Specifically, in radiative systems, the Bloch symbol is an infinite Laurent series rather than a finite-order polynomial, possessing logarithmic light-line branch points and a limited convergence annulus in the complex- $\beta$ plane. It is unclear how the finite-chain OBC spectrum evolves relative to the infinite-chain Bloch spectrum in such long-range systems and whether boundary localization persists as spectral differences change. This paper addresses the nature of NHSE in a chain of electric dipoles coupled via long-range radiative interactions within a passive, reciprocal chiral (Pasteur) medium.

Methodology
The authors model a straight chain of electric dipoles embedded in a reciprocal chiral medium with constitutive relations incorporating a chirality parameter $\kappa$ . The system is governed by a coupled-dipole equation where the electric Green tensor couples every pair of sites. The analysis proceeds in two stages:

Linearized Analysis: At a reference frequency $\omega_0$ , the system is treated as a linearized eigenvalue problem. The transverse eigenchannels are identified as exactly circularly polarized modes. The authors derive the radiative coupling coefficients ( $G^R_\sigma$ and $G^L_\sigma$ ) which include $1/m$ , $1/m^2$ , and $1/m^3$ terms, where $m$ is the distance in lattice sites.
Spectral and Spatial Analysis: The authors examine the Bloch spectrum and the complex- $\beta$ structure. They utilize the analytically continued equal-modulus condition to define the Generalized Brillouin Zone (GBZ) for this long-range system, distinguishing between isolated complex- $\beta$ poles and logarithmic light-line branch cuts. Finite-chain calculations ( $N=400$ to $1200$) are performed to compute OBC spectra and eigenvector profiles. The spatial decay of boundary modes is analyzed by decomposing the boundary response into contributions from isolated poles and branch cuts.

Key Contributions and Results

Reciprocal Paired Skin Modes: The study demonstrates that chirality induces polarization-dependent phase accumulation and attenuation. While the system is reciprocal, the two transverse circular-polarization subspaces acquire opposite directional biases. Consequently, the system supports degenerate pairs of skin modes localized at opposite boundaries. Reversing the helicity ( $\sigma \to -\sigma$ ) reverses the bias, localizing the partner mode at the opposite edge.
Dual-Component Skin Mode Profile: The spatial profile of the skin modes is found to consist of two distinct components:
1. An exponential contribution arising from an isolated complex- $\beta$ pole, which dominates the intermediate spatial range.
2. A longer-range algebraic tail proportional to $1/[n(\ln n)^2]$ , originating from the logarithmic light-line branch points of the retarded interaction.
  This tail eventually controls the envelope at sufficiently large distances, limiting the observable localization rate.
Spectral Evolution and Partial Contraction: Finite-chain calculations reveal a nuanced evolution of the OBC spectrum as chain length $N$ increases. A portion of the OBC spectrum contracts toward the Bloch spectrum (PBC spectrum), while the most displaced eigenvalues remain near the analytically continued equal-modulus locus. Crucially, despite this partial spectral contraction, the number of boundary-localized modes remains extensive (e.g., rising from 159/200 to 1080/1200 modes in the studied range).
Role of Dissipative Chirality ( $\text{Im } \kappa$ ): The authors show that varying the dissipative part of the chirality parameter strongly deforms the Bloch spectrum by shifting light-line branch points. However, the OBC spectrum remains nearly invariant. This is because the change in $\text{Im } \kappa$ acts primarily as a diagonal similarity transformation on the open-chain matrix, altering eigenvector envelopes (localization strength) without significantly shifting eigenvalues.
Analytic Structure of Decay: The paper derives an expression for the finite-size crossover rate $\xi^{-1}_\times(N)$ , which connects the effective decay rate measured in finite chains to the light-line attenuation constant $\delta_-$ . This explains why fitted decay rates for strongly localized modes deviate from the simple $|\ln|\beta||$ prediction and approach a plateau determined by the branch-cut contribution.

Significance and Claims
The paper claims to identify an unusual NHSE mechanism driven by long-range radiative coupling in a reciprocal medium, distinct from short-range non-Hermitian models. The primary significance lies in demonstrating that:

Reciprocity and NHSE Coexist: Skin effects can occur in reciprocal systems through polarization-dependent phase and attenuation biases, producing symmetric pairs of boundary-localized modes.
Long-Range Effects are Fundamental: The analytic structure of the Green's tensor (specifically light-line branch cuts) fundamentally alters the skin mode profile, introducing a universal algebraic tail that limits localization rates and modifies the spectral convergence behavior.
Partial Spectral Contraction: The study establishes that in long-range radiative systems, the OBC spectrum does not simply converge to the Bloch spectrum; rather, a partial contraction occurs while the extensive nature of boundary localization persists. This challenges the conventional expectation of a size-independent OBC spectrum separated from the Bloch spectrum in finite-range lattices.

The work provides a direct link between the finite-size skin profile and the analytic structure of the Green's tensor, offering a refined understanding of NHSE in radiative and photonic systems.

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