Single-particle limit of a topological edge state in a locally resonant 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. The resonator introduces an attenuation singularity: at a frequency at which the effective stiffness vanishes, the spatial attenuation of waves diverges. By tuning a dimerization parameter, we migrate this singularity from one band gap to the other via an intermediate flat-band state, transferring the dominant local-resonance character between the gaps without closing either gap and while preserving the underlying band topology. Crucially, when the resulting topological edge state intersects the attenuation singularity, the edge state collapses onto a single boundary particle, forming a single-particle mode (SPM). This yields an inverse participation ratio of exactly unity, the theoretical limit for localization in a discrete system. Moreover, this extreme localization can be realized at low frequencies, below the first Bragg-type band gap. Further, we demonstrate that while random disorder detunes this mode, merely tuning the boundaries stabilizes the single-particle mode over a broad parameter range. Our findings provide a clear pathway to designing ultra-localized edge states in low-frequency regimes, where band topology guarantees the edge mode and local resonance drives its single-particle confinement.