Single-particle limit of a topological edge state in a locally resonant band gap

Topological metamaterials promise unprecedented wave control. 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 the frequency where 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 while preserving the band topology. Crucially, when the resulting topological edge state intersects this singularity, it 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.