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Topological Degeneracy Induced Flat Bands in two-Dimensional Holed Systems

Systems hosting flat bands offer a powerful platform for exploring strong correlation physics. Theoretically topological degeneracy rising in systems with non-trivial topological orders on periodic manifolds of non-zero genus can generate ideal flat bands. However, experimental realization of such geometrically engineered systems is very difficult. In this work, we demonstrate that flat planes with strategically patterned hole defects can engineer ideal flat bands. We constructing two families of models, singular flat band systems where degeneracy is stabilized by non-contractible loop excitations tied to hole defects and perfectly nested van Hove systems where degeneracy arises from line excitations in momentum space. These models circumvent the need for exotic manifolds while retaining the essential features of topological flat bands. By directly linking defect engineering to degeneracy mechanisms, our results establish a scalable framework for experimentally accessible flat band design.