  Topological lattice defects, such as dislocations and grain boundaries (GBs),
are ubiquitously present in the bulk of quantum materials and externally
tunable in metamaterials. In terms of robust modes, localized near the defect
cores, they are instrumental in identifying topological crystals, featuring the
hallmark band inversion at a finite momentum (translationally active type).
Here we show that GB superlattices in both two-dimensional and
three-dimensional translationally active higher-order topological insulators
harbor a myriad of dispersive modes that are typically placed at finite
energies, but always well-separated from the bulk states. However, when the
Burgers vector of the constituting edge dislocations points toward the gapless
corners or hinges, both second-order and third-order topological insulators
accommodate self-organized emergent topological metals near the zero energy
(half-filling) in the GB mini Brillouin zone. We discuss possible material
platforms where our proposed scenarios can be realized through the
band-structure and defect engineering.
