A model for two-dimensional electronic, photonic, and mechanical metamaterial
systems is presented, which has flat one-dimensional zero-mode energy bands and
stable localized states of a topological origin confined within 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.Comment: 5 page main text, 2 page supplemental materia
