Single-particle edge state in a local-resonance-induced topological band gap

Abstract

Topological metamaterials promise unprecedented wave control. Here, we theoretically and numerically investigate a one-dimensional Su-Schrieffer-Heeger (SSH) inspired stiffness dimer modified with a local resonator, which imparts a frequency-dependent effective stiffness to the unit cell. We demonstrate a two-step mechanism to create a topological local-resonance-induced band gap (LRG): first, a conventional Bragg-type band gap (BrG) is made topologically non-trivial via band inversion at a Dirac point; second, by tuning a dimerization parameter, the character of this non-trivial BrG is switched to that of an LRG via an intermediate flat band state. This process preserves the non-trivial topology without requiring gap closure within the LRG. Crucially, we find that when the resulting topological edge state intersects a characteristic frequency of the LRG -- specifically, an attenuation singularity where the effective stiffness vanishes -- it achieves extreme localization of vibrational energy. This state is confined to a single particle at the boundary, resulting in an inverse participation ratio of exactly unity, the theoretical limit for localization in a discrete system. Further, we demonstrate that while random disorder scatters the frequency of this mode, introducing tuned boundaries stabilizes the single-particle mode over a broad parameter range. Our findings provide a clear pathway to designing ultra-localized, topologically protected states in low-frequency regimes.

Figures (10)

• Figure 1: Comparison of topological edge modes in a Bragg-type band gap (BrG) and an effective stiffness local-resonance-induced band gap (LRG). (a) Band closure and reopening driving topological transition in a BrG. (b) Pathway to topologically non-trivial LRG via gap-type switching from a non-trivial BrG. (c) Unit cell of conventional stiffness dimer chain. (d) Unit cell of effective stiffness dimer chain with local resonators ($m_3, k_3$). (e) Band structure of conventional dimer; blue region denotes BrG, red line indicates edge mode. (f) Left: Band structure of effective stiffness dimer with BrG (blue, II) and LRG (yellow, I). Right: $K_{1,\text{eff}}$ versus a normalized $\Omega^2=\omega^2M/(K_1+K_2)$; orange dashed line marks $K_{1,\text{eff}}=0$. (g) Edge mode profile in conventional dimer showing exponential decay. (h) Single-particle mode (SPM), localized on the edge with IPR$=1$, in the effective stiffness dimer. Parameters: ${K_1=1}$, ${K_2=1.4286}$, ${M=1}$, ${k_3=0.5}$, ${m_3=0.2}$, ${\lambda=1}$, ${N=60}$ unit cells with fixed boundaries.
• Figure 2: Band gap type switching via a flat band. Orange dashed line marks $K_{1,\text{eff}}=0$; its location defines LRG (yellow) versus BrG (blue). (a) $\delta < \delta_f$: singularity in upper gap (II = LRG). (b) $\delta = \delta_f$: singularity traversal through the flat band. (c) $\delta > \delta_f$: singularity in lower gap (I = LRG). Offset $\epsilon = 0.15\delta_f$ used in (a) and (c). Parameters: $K=1$, $M=1$, $k_3=0.5$, $m_3=0.2$, $\lambda=1$.
• Figure 3: Topological phase transitions and band inversion. (a) Band structure evolution with $\delta$. Color map: phase difference $|\arg(U_2)-\arg(U_1)|$ (in-phase = even = blue = 0, out-of-phase = odd = red = $\pi$). DP1/DP2 mark Dirac points; dashed yellow line and the vertical gray plane indicate the flatband and $\delta_f$, respectively. (b) Phase along the Brillouin zone versus $\delta$ for each band; phase change along $q=\pi$ shows inversion at $\delta_{\pm}$. (c) Zak phase versus $\delta$, confirming transitions only at $\delta_{\pm}$. Parameters: $K=1$, $M=1$, $k_3=0.43$, $m_3=0.2$, $\lambda=1$.
• Figure 4: Finite-chain eigenspectra and edge states. (a) Schematic of finite chain with fixed boundaries. (b) Standard dimer spectrum; edge modes (red markers) emerge for $\delta > 0$. (c) Effective stiffness dimer spectrum; orange curve marks $K_{1,\text{eff}}=0$. Vertical lines: $\delta_-$, $\delta_f$, $\delta_s$, $\delta_+$. (d) Edge mode shape in standard dimer at $\delta=0.3$. (e) Edge mode in upper BrG at $\delta=0.37$. (f) Edge modes in lower LRG at $\delta_s-0.3$, $\delta_s$, and $\delta_s+0.3$; center shows SPM (IPR$=1$). Parameters: $K=1$, $M=1$, $k_3=0.43$, $m_3=0.2$, $\lambda=1$, $N=60$.
• Figure 5: Localization and disorder with tuned boundaries. IPR shown by color (yellow = 1, blue = minimum). Lower panels: IPR for modes 60 and 61 (LRG edge modes). (a) No disorder: IPR $\to 1$ at $\delta_s$. (b) 10% disorder: frequencies scatter but high IPR maintained near $\delta_s$. (c) Tuned boundaries, no disorder: edge modes track $K_{1,\text{eff}}=0$, achieving IPR $= 1$ broadly. (d) Tuned boundaries, 10% disorder: robustness in "No mixing" region; hybridization in "Mixing" region.