  Motivated by the recent theoretical studies on a two-dimensional (2D) chiral
Hamiltonian based on the Su-Schrieffer-Heeger chains, we experimentally and
computationally demonstrate that topological flat frequency bands can occur at
open edges of 2D planar metamaterials and at antiphase boundary seams of
ring-shaped or tubular metamaterials. Specifically, using mechanical systems
made of magnetically coupled spinners, we reveal that the presence of the edge
or seam bands that are flat in the entire projected reciprocal space follows
the predictions based on topological winding numbers. The edge-to-edge distance
sensitively controls the flatness of the edge bands and the localization of
excitations. The analogue of the fractional charge state is also observed.
Possible realizations of flat bands in a large class of metamaterials,
including photonic crystals and electronic metamaterials, are discussed.
