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

Topological indices, such as winding numbers, have been conventionally used to predict the number of topologically protected edge states (TPESs) in topological insulators, a signature of the topological phenomenon called bulk-edge correspondence. In this work, we theoretically and experimentally demonstrate that the number of TPESs at the domain boundary of a Su-Schrieffer-Heeger (SSH) model can be higher than the winding number depending on the strengths of beyond-nearest neighbors, revealing the breakdown of the winding number prediction. Hence, we resort to the Berry connection to accurately count the number of TPESs in an SSH system with a domain boundary. Moreover, the Berry connection can elucidate wavelengths of the TPESs, which is further confirmed using the Jackiw Rebbi theory. We analytically prove that each of the multiple TPES modes at the domain boundary corresponds to a bulk Dirac cone, asserting the robustness of the Berry connection method, which offers a generalized paradigm for TPES prediction.