  Topological insulators hold promises to realize exotic quantum phenomena in
electronic, photonic, and phononic systems. Conventionally, topological
indices, such as winding numbers, have been used to predict the number of
topologically protected domain-wall states (TPDWSs) in topological insulators,
a signature of the topological phenomenon called bulk-edge correspondence.
Here, we demonstrate theoretically and experimentally that the number of TPDWSs
in a mechanical Su-Schrieffer-Heeger (SSH) model can be higher than the winding
number depending on the strengths of beyond-nearest-neighbor interactions,
revealing the breakdown of the winding number prediction. Alternatively, we
resort to the Berry connection to accurately characterize the number and
spatial features of TPDWSs in SSH systems, further confirmed by the
Jackiw-Rebbi theory proving that the multiple TPDWSs correspond to the bulk
Dirac cones. Our findings deepen the understanding of complex network dynamics
and offer a generalized paradigm for precise TPDWS prediction in potential
applications involving localized vibrations, such as drug delivery and quantum
computing.
