  We theoretically and experimentally demonstrate singular topological edge
states in a locally resonant metamaterial with a configuration based on
inversion-symmetric extended Su-Schrieffer-Heeger chains. Such states arise
from a topological gap transition from a conventional Bragg-type gap (BG) to a
local resonance-induced gap (LRG), accompanied by a topological phase
transition. Remarkably, nontrivial topological states can emerge in the
vicinity of the singularity in the imaginary parts of the wavenumber within the
bandgap, leading to highly localized modes on a scale comparable to a single
subwavelength unit cell. We experimentally demonstrate distinct differences in
topologically protected modes-highlighted by wave localization-between the BG
and the LRG in locally resonant granular-based metamaterials. Our findings
suggest the scope of topological metamaterials may be extended via their
bandgap nature.
