Topological edge states with BNN

This study explores the limitations of topological invariants, such as the winding number, traditional predicting topologically protected domain-wall states (TPDWSs) within one-dimensional Su-Schrieffer-Heeger (SSH) lattices that incorporate interactions beyond nearest neighbors (BNNs). Through both theoretical analysis and experimental validation using mechanical lattices, it is demonstrated that BNN interactions can lead to a higher number of TPDWSs than predicted by conventional winding number calculations. The research introduces the Berry connection as an alternative tool to accurately characterize these states, with further confirmation provided by Jackiw-Rebbi theory. These findings offer a deeper understanding of complex network dynamics and present a generalized approach for precise TPDWS prediction.