Tunable Localisation in Parity-Time-Symmetric Resonator Arrays with Imaginary Gauge Potentials

TL;DR

This work addresses how reciprocal gain/loss and non-reciprocal imaginary gauge potentials interact in a PT-symmetric, one‑dimensional array of deep subwavelength resonators. It develops an asymptotic framework based on a gauge capacitance matrix C^{θ,γ} and leverages Chebyshev polynomials and extended Toeplitz analysis to characterize eigenvalues and eigenvectors. The authors show that crossing an exceptional point decouples almost all eigenmodes and triggers a PT-phase transition with edge-condensation governed by topological indices, with the density of exceptional points increasing as the system size N grows. The results provide a rigorous bridge between non-Hermitian wave physics and quantum tight-binding models, offering insights for designing PT-symmetric metamaterials and understanding topological non-Hermitian phenomena in larger dimensional settings.

Abstract

The aim of this paper is to illustrate both analytically and numerically the interplay of two fundamentally distinct non-Hermitian mechanisms in a deep subwavelength regime. Considering a parity-time symmetric system of one-dimensional subwavelength resonators equipped with two kinds of non-Hermiticity - an imaginary gauge potential and on-site gain and loss - we prove that all but two eigenmodes of the system decouple when going through an exceptional point. By tuning the gain-to-loss ratio, the system changes from a phase with unbroken parity-time symmetry to a phase with broken parity-time symmetry. At the macroscopic level, this is observed as a transition from symmetrical eigenmodes to condensated eigenmodes at one edge of the structure. Mathematically, it arises from a topological state change. The results of this paper open the door to the justification of a variety of phenomena arising from the interplay between non-Hermitian reciprocal and non-reciprocal mechanisms not only in subwavelength wave physics but also in quantum mechanics where the tight binding model coupled with the nearest neighbour approximation can be analysed with the same tools as those developed here.

Figures (8)

• Figure 1: A chain of $2N$ one-dimensional identical and equally spaced resonators. Material parameters and sign of the imaginary gauge potentials depend on the resonator's position.
• Figure 2: Distribution of the exceptional points for varying $N$. For any $N$, the system exhibits a trivial exceptional point at $\theta=\frac{\pi}{2}$. All other exceptional points concentrate in the interval $[0,e/N]$ and become increasingly dense as $N$ grows.
• Figure 3: Geometrical interpretation of the eigenvalues of $\mathcal{C}^{\theta,\gamma}$ as given by \ref{['prop: real eig means on level set of ratio of poly']}. In this view, we can also clearly see the exceptional points, where two real eigenvalues meet and become complex. Namely, this happens exactly when $\mu^\theta(\mathop{\mathrm{\mathbb{R}}}\nolimits)$ goes from passing through one of the inner regions in (B) to moving past them and two red crosses meet.
• Figure 4: Eigenvalue locations close to the two line segments $(\mu^\theta)^{-1}([-1,1])\cup (\mu^{-\theta})^{-1}([-1,1])$ for $\theta = 0.2$, $\gamma = 1$ and $N=60$.
• Figure 5: Decoupling of the eigenvectors of the gauge capacitance matrix. The macroscopic behaviour of the eigenvectors (exponential decay/growth) is predicted by the location of the eigenvalues in the complex plane with respect to the region of topological convergence defined in \ref{['eq:def_E1_E2']} displayed here as trace of \ref{['eq: symbol def']}. Looking at the two highlighted eigenvalues (red and blue), Figure (A-C) correspond to point (2) in \ref{['lemma:exponential decay and decoupling']} while (D) corresponds to point (1).

Theorems & Definitions (67)

• Theorem 2.1
• Definition 2.2
• Proposition 2.3
• proof
• Proposition 2.4
• Proposition 2.5
• Lemma 2.6
• proof
• Theorem 3.1
• proof