Boundary effect on in-gap edge states in nonlocal Su-Schrieffer-Heeger model

The field of topological states of matter has garnered considerable attention across electronic, photonic, and phononic systems due to its remarkable abilities in waveguiding and localization, which remain robust against disorders and defects. A fundamental challenge in material physics lies in understanding the interplay between intrinsic properties and those induced by boundaries. In infinite periodic materials, resonant modes are notably absent within band gaps. However, when the material is truncated to form a finite periodic structure, these modes may appear as localized edge modes within the band gap. In this study, we introduce a generalized system by incorporating nonlocal interactions into the well-established Su-Schrieffer-Heeger (SSH) model. This generalized system exhibits a broader range of topological properties, including non-trivial topological phases and associated localized edge states. We conduct a detailed investigation of the zero-energy edge states, exploring their characteristics and behavior. Additionally, we discuss the influence of boundaries on the existence of edge states and consider the impact of Fifth Nearest Neighbors (FNNs) within the system.