Topological interface modes in aperiodic subwavelength resonator chains

TL;DR

The paper addresses topologically protected interface modes in one-dimensional block disordered chains of subwavelength resonators by modeling the spectrum with a capacitance operator. It develops a $\\mathbb{Z}_2$-valued Zak-like bulk index from the chiral-symmetric off-diagonal part and proves a bulk-edge correspondence: joining two semi-infinite chains with different topological character yields a localized interface mode at frequency $\\lambda= \\frac{2}{1-\\delta^2}$. The results are supported by a dynamical propagation-matrix viewpoint and extensive numerics, and extended to quasi-periodic arrangements such as Fibonacci tilings. These findings provide a rigorous foundation for robust interface states in disordered subwavelength metamaterials and set the stage for higher-dimensional generalizations and practical designs.

Abstract

We consider interface modes in block disordered subwavelength resonator chains in one dimension. Based on the capacitance operator formulation, which provides a first-order approximation of the spectral properties of dimer-type block resonator systems in the subwavelength regime, we show that a two-fold topological characterization of a block disordered resonator chain is available if it is of dominated type. The topological index used for the characterization is a generalization of the Zak phase associated with one-dimensional chiral-symmetric Hamiltonians. As a manifestation of the bulk-edge correspondence principle, we prove that a localized interface mode occurs whenever the system consists of two semi-infinite chains with different topological characters. We also illustrate our results from a dynamic perspective, which provides an explicit geometric picture of the interface modes, and finally present a variety of numerical results to complement the theoretical results.

Figures (6)

• Figure 1: Spectrum of $\mathcal{D}^{\omega}$ for the two domination regimes ($K=2000, \delta=10^{-1}$ in both cases). The highlighted regions contain the interface modes together with the critical lines $\frac{8}{3+\sqrt{1+8\delta^2}}$ (red dots) and $\frac{2}{1-\delta^2}$ (red dashes). We can see that, depending on the domination regime, the interface mode lies on the lower or upper critical line.
• Figure 2: Spectrum and interface mode of the off-diagonal operator $\widetilde{\mathcal{C}}^{\omega}$ for a left-$II$, right-$I$ dominated sequence $\omega$ ($K=1000, \delta=10^{-1}$). Left: Spectrum $\text{Spec}(\widetilde{\mathcal{C}}^{\omega})$ with the interface eigenvalue at $\lambda=0$ highlighted in red. Right: The interface mode $u$ corresponding to $\lambda=0$. We can see that it is indeed exponentially localised at the interface $j=1000$.
• Figure 3: Source (stable) $s(\mathcal{P}^\lambda_d)$ and sink (unstable) $u(\mathcal{P}^\lambda_d)$ phase for both block types $d=I,II$ on the projective space $\mathbb{RP}^1\simeq S^1$ as $\lambda$ passes through the critical value $c_1$ (for $\delta=10^{-1}$). We observe that the SSC is violated for $\lambda<c_1$, that $u(\mathcal{P}_I^{\lambda})=s(\mathcal{P}_{II}^{\lambda})$ for $\lambda=c_1$ and that the SSC is fulfilled afterwards.
• Figure 4: Spectrum $\text{Spec}(\mathcal{C}^\omega)$ for a left$-I$, right-$II$ dominated block disordered system ($K=2000, \delta=10^{-1})$, together with the distinct spectral regions determined by the block properties. Here, shared pass band denotes $\lambda$ in the band of both blocks, SSC violation denotes $\lambda$ in the shared gap but with violated SSC, and bandgap denotes the bandgap of the composite system corresponding to $\lambda$ in the shared gap together with a fulfilled SSC.
• Figure 5: Interface mode for two block disordered left-$I$, right-$II$ dominated systems with varying domination strengths ($K=400, \delta=10^{-1}$). The expected left and right decay, as predicted by $\xi^{\varepsilon_-}$ and $\xi^{\varepsilon_+}$, is plotted as a blue dashed and red dashed line respectively. Left: Equal strength left and right domination. Right: Weak left and strong right domination.

Theorems & Definitions (27)

• Definition 2.1
• Proposition 2.2
• Proposition 2.3: graf2018bec_disorder_chiraltauber2022chiral_finite_chain
• Remark 2.5
• Proposition 2.6
• Remark 2.7
• Remark 2.8
• Remark 2.9
• Definition 3.1: Subordinate solution
• Theorem 3.2