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

The gist The long-range radiative coupling through a passive reciprocal chiral medium produces an unusual non-Hermitian skin effect in a chain of electric dipoles, where chirality generates polarization-dependent phase accumulation and attenuation, leading to degenerate pairs of skin modes localized at opposite boundaries.

Non-Hermitian Skin Effect Mechanism

The system exhibits an unusual non-Hermitian skin effect characterized by the boundary accumulation of an extensive number of bulk eigenmodes under open boundary conditions The chirality generates polarization-dependent phase accumulation and attenuation, leading to degenerate pairs of skin modes localized at opposite boundaries. Reciprocity reverses this bias for the opposite circular polarization and produces equal-frequency mode pairs localized at opposite boundaries.

Spectral Structure and Evolution

The spectral signature is sensitive to boundary conditions, as the large-N spectrum under open boundary conditions (OBCs) generally differs from the Bloch spectrum under periodic boundary conditions (PBCs) and is instead determined by the GBZ. The retarded interaction makes this comparison more subtle, as the radiative Bloch symbol is an infinite Laurent series rather than a finite-order characteristic polynomial. The conventional finite-range expectation of a size-independent OBC spectrum separated from the Bloch spectrum therefore cannot be assumed.

Spatial Profile and Localization

The skin mode profile consists of an exponential contribution from an isolated complex-β pole and a longer-range tail proportional to 1/[n(ln n)2], where n is the distance from the occupied boundary measured in lattice sites. The boundary response contains an exponential decay due to complex-β pole and a universal branch-cut tail, directly relating the finite-size skin profile to the analytic structure of the Green’s tensor. Finite-chain calculations show that part of the OBC spectrum contracts toward the Bloch spectrum as the chain length increases, while the number of boundary-localized modes remains extensive over the calculated range.

Boundary Response and Decay Rates

The boundary solution is expressed as a generating function of the complex variable β, where isolated zeros of the analytically continued denominator produce exponential terms, and logarithmic light-line branch points produce branch cuts. The amplitude at a distance n from the occupied boundary has an asymptotic form [4] pn = Ape−ξ−1βn+iqpn + ALLe−δ−n+iqLLn(ln n)2[1 + o(1)]. The isolated complex-β pole produces the exponential profile observed over an intermediate spatial range, while the light-line branch cut produces the second term, whose exponential factor is fixed by the attenuation constant δ− of the nearest branch point and whose algebraic factor follows from the logarithmic singularity.

Influence of Chirality and Dissipation

Chirality gives the two propagation directions polarization-dependent phase accumulation and attenuation. The leading radiative coupling is equivalent to a polarization-dependent complex shift of the Bloch wave number, with its imaginary part being the familiar imaginary gauge field. Reversing σ reverses this shift and exchanges the two boundaries, as required by reciprocity. Varying Im κ shifts δ± = η0[Im nh ± Im κ] oppositely while leaving their mean unchanged at leading far-field order, which is a diagonal similarity transformation of the open-chain matrix.

Conclusion on Localization

The resulting profile contains an exponential contribution from an isolated complex-β pole and a longer-range branch-cut tail proportional to 1/[n(ln n)2], explaining the crossover of the spatial decay observed in finite chains. This analytic structure explains the crossover of the spatial decay observed in finite chains. Finite-chain calculations further show that part of the OBC spectrum contracts toward the Bloch spectrum as the chain length increases, whereas the number of boundary-localized modes remains extensive. Thus, this partial spectral contraction coexists with reciprocal skin localization rather than eliminating it. The more visible open-chain response occurs in the directional decay of the eigenvectors. This work is supported by RGC Hong Kong (AoE/P-502/20, CRS HKUST601/23, and JLFS/P-603/24).

--- Page 1 ---

Non-Hermitian Skin Effect from Radiative Coupling in a Reciprocal Chiral Medium Kin Hung Fung,1, ∗ Changhao Meng,1 Yixin Xiao,1 and C. T. Chan1, 2 Department of Physics, The Hong Kong University of Science and Technology, Hong Kong, China Institute for Advanced Study, The Hong Kong University of Science and Technology, Hong Kong We find that long-range radiative coupling through a passive reciprocal chiral medium produces an unusual non-Hermitian skin effect in a chain of electric dipoles Although the system is reciprocal, the chirality generates polarization-dependent phase accumulation and attenuation, leading to degenerate pairs of skin modes localized at opposite boundaries We find that the skin mode profile consists of an exponential contribution from an isolated complex-β pole and a longer-range tail proportional to 1/[n(ln n)2], where n is the distance from the occupied boundary measured in lattice sites Finite-chain calculations show that part of the OBC spectrum contracts toward the Bloch spectrum, whereas the number of boundary-localized modes remains extensive

--- Page 2 ---

Coupled dipoles in a reciprocal chiral medium.—With time dependence e−iωt for given frequency ω, the reciprocal chiral constitutive relations are D = ϵ0ϵrE + iκc H, B = µ0µrH − iκc E, (1) where D is the electric displacement field, E is the electric field, B is the magnetic flux density, H is the magnetic field, ϵ0 and µ0 are the permittivity and permeability of free space, ϵr and µr are the relative permittivity and permeability of free space The dipoles satisfy α−1 radpi − Xj!=iGEE[(i − j)azˆ]pj = 0, (2) where the electric Green tensor obeys∇×∇×−2ωκc∇×−ω2c2/2(n2h−κ2)GEE(R)=ω2µ0µr δ(R)I with nh =√ϵrµr. On the chain axis, axial rotational symmetry makes the transverse response diagonal in the circular basis pσ = (px + iσpy)/√2 To link our equations to the standard NHSE models, we consider the case of weak dispersive κ at a reference frequency ω0 (the singleparticle resonance frequency) The off-diagonal matrix elements in Eq. (2) are then [24] G Rσ(m) = g−σeix−σm x2 −σm + ix−σm2 − 1/2m3 − gσeixσm2 m3 G L σ(m) = g σeixω0a/c, η0 = ω0a/c, x0 = (x+ + x−)/2, and gσ is the helicity-dependent coupling coefficient incorporating the resonator-strength normalization and the host response The leading 1/m terms give G R,Lσ(m) ∼ C R,Lσ m eix0me∓iσ(x+−x−)m/2 The real part of (x+ − x−) produces opposite phase shifts for the two propagation directions, whereas the imaginary part produces opposite attenuation shifts Equation (5) therefore shows that the leading radiative coupling is equivalent to a polarization-dependent complex shift of the Bloch wave number; its imaginary part is the familiar imaginary gauge field Reversing σ reverses the shift and exchanges the two boundaries, as required by reciprocity The full Maxwell interaction contains additional polarization-dependent prefactors and the m−2 and m−3 near-field terms in Eq. (4). These terms are retained in all spectra and eigenvectors, while Eq. (5) isolates the physical origin of the directional bias Evaluating the Green tensor and the radiative selfenergy at the reference frequency ω0 gives a linearized coupled-dipole eigenvalue problem. We denote its complex eigenvalue by omega = ω/ω0 The two circular-polarization subspaces have identical eigenvalues, and their right eigenvectors are related by reversal of the site index The Supplemental Material extends the calculation to the nonlinear outgoing-wave eigenvalue problem, with the constitutive parameters and retarded Green tensor evaluated self-consistently at complex frequency. The resulting modes preserve the same exact reciprocal pairing and display the strongest dispersive shifts near the light lines [24]

--- Page 3 ---

To connect the reciprocal boundary pairing in Fig. 1 with its spectral origin, we first examine the Bloch and complex-β structure in Fig. 2, before comparing it with the open-chain spectra and profiles in Fig.

1. Bold spectral analysis for system stability: The improved AI can perform part of the OBC spectrum contracts toward the Bloch spectrum as the chain length increases by analyzing how boundary-localized modes evolve relative to bulk modes as a function of chain length, allowing for predictive modeling of finite-chain spectral contraction.

2. Predictive localization mapping: The system can predict the spatial profile crossover by analyzing the boundary response contains an exponential decay due to complex-β pole and a universal branch-cut tail, directly relating the finite-size skin profile to the analytic structure of the Green’s tensor. This allows for accurate determination of whether a mode is β-pole-controlled or light-line limited.

3. Nonlinear response simulation: The AI can simulate nonlinear dynamics by utilizing results from the supplemental material, specifically evaluating the nonlinear outgoing-wave eigenvalue problem, enabling it to model how modes exhibit the strongest dispersive shifts near the light lines.

4. Chirality tuning for localization control: The system can dynamically tune boundary localization by exploiting the parameter sensitivity described in Figure 5, noting that Im κ can tune localization without appreciably shifting the open-chain resonances when compared to PBCs.

We show that long-range radiative coupling in a passive chiral medium produces an unusual non-Hermitian skin effect in a chain of dipoles. The skin-mode profile consists of a long-range light-line branch-cut contribution, a non-exponential factor 1/[n(n) 2] with n being the site number. Finite-chain calculations show that part of the open-boundary spectrum contracts toward the Bloch spectrum, whereas the number of boundary-localized modes remains extensive. The retarded radiative coupling makes both the real-space profile and the finite-size spectral signatures of boundary localization depart from standard finite-range and pure exponential coupling theory.