On compact topological edge modes in photonic ribbon lattices

Topological insulation and the associated unidirectional propagation of edge modes have opened new avenues for light control, enabling the design of novel photonic devices with enhanced performance and stability [1]. The number of topologically protected edge modes, determined via the bulk-edge correspondence, is linked to changes in the bulk k- space Hamiltonian and is quantified by topological invariants such as the winding number (Zak phase) in 1D and Chern numbers in 2D systems. This framework inherently implies a nonzero band curvature. In contrast, flatband (dispersionless) systems offer a distinct route for light control by supporting highly compact localized modes (CLMs). While topological protection and flatband localization are typically considered opposing effects, their interplay can significantly enhance the robustness of edge modes. Recent studies have demonstrated the existence of compact topologically protected modes [2,3]. Here, we systematically analyze this phenomenon in quasi-1D ribbon photonic lattices, considering four graphene-like ribbon configurations where the band curvature can be tuned by introducing an artificial magnetic flux in specific plaquettes. We investigate the emergence of compact topological edge modes in the presence of ribbon symmetries and geometric constraints. Additionally, we examine the robustness of CLMs against disorder and nonlinear effects and identify optimal lattice configurations for device design. Initial experimental efforts in realizing SP states [2], provide promising confirmation of our approach.