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Topological insulators hold promises to realize exotic quantum phenomena inelectronic, photonic, and phononic systems. Conventionally, topologicalindices, such as winding numbers, have been used to predict the number oftopologically 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 TPDWSsin a mechanical Su-Schrieffer-Heeger (SSH) model can be higher than the windingnumber depending on the strengths of beyond-nearest-neighbor interactions,revealing the breakdown of the winding number prediction. Alternatively, weresort to the Berry connection to accurately characterize the number andspatial features of TPDWSs in SSH systems, further confirmed by theJackiw-Rebbi theory proving that the multiple TPDWSs correspond to the bulkDirac cones. Our findings deepen the understanding of complex network dynamicsand offer a generalized paradigm for precise TPDWS prediction in potentialapplications involving localized vibrations, such as drug delivery and quantumcomputing.

Breakdown of Conventional Winding Number Calculation in One-Dimensional Lattices with Interactions Beyond Nearest Neighbors