Generalized Wigner-Smith theory for perturbations at exceptional and diabolic point degeneracies

Abstract

Spectral degeneracies, including diabolic (DP) and exceptional (EP) points, exhibit unique sensitivity to external perturbations, enabling powerful control and engineering of wave phenomena. We present a residue-based perturbation theory that quantifies complex resonance splitting of DP and EP type spectral degeneracies using generalized Wigner-Smith operators. We validate our theory using both analytic Hamiltonian models and numerical electromagnetic simulations, demonstrating excellent agreement across a range of cases. Our approach accurately predicts degenerate resonance splitting using only scattering data, offering a powerful framework for precision tuning, inverse design, and practical exploitation of non-Hermitian phenomena.

Figures (2)

• Figure 1: (a) Trajectories of poles in the polarizability of an initially spherical nanoparticle subjected to an axial elongation (inset) along its first principal axis ($a_1 \rightarrow a_1 + \Delta a$) as computed by root searching (RS, $\circ$ markers), our GWS residue formula ($\times$ markers), and full FEM simulations ($+$ markers). Colors indicate perturbation amplitude $\Delta a$ on a logarithmic scale. (b) Magnitude of the relative difference in pole shifts for both branches as obtained from the RS and GWS methods as a function of $\Delta a$.
• Figure 2: Splitting in the complex plane of (a) EP$_2$, (b) EP$_3$, and (c) EP$_4$ degenerate resonances supported in TO designed plasmonic nanowire structures (insets) upon a perturbation, $\Delta\varepsilon$, of the host permittivity. Complex frequency shifts were calculated from direct solution of the TO dispersion relation ($\circ$ markers) and our GWS residue formula ($\times$ markers). Marker colour encodes $\Delta\varepsilon$ on a logarithmic scale. (d) Log–log plot of the magnitude of the frequency shift $|\Delta\omega|$ versus $\Delta\varepsilon$ for each EP. Gradients of $1/2$, $1/3$, and $1/4$ are found for EP$_2$, EP$_3$ and EP$_4$ respectively (black dashed fits).