Flat Energy Bands within Antiphase and Twin Boundaries and at Open Edges in Topological Materials*

We present a model of two-dimensional electronic, photonic, and mechanical metamaterial systems, which has flat one-dimensional zero-mode energy bands and stable localized states of a topological origin confined with twin boundaries, antiphase boundaries, and at open edges. Topological origins of these flat bands are analyzed for an electronic system as a specific example, using a two-dimensional extension of the Su-Schrieffer-Heeger Hamiltonian with alternating shift of the chains. It is demonstrated that the slow group velocities of the localized flat band states are sensitively controlled by the distance between the boundaries and the propagation can be guided through designed paths of these boundaries. We also discuss how to realize this model in metamaterials.