  Topological materials occupy the central stage in the modern condensed matter
physics because of their robust metallic edge or surface states protected by
the topological invariant, characterizing the electronic band structure in the
bulk. Higher-order topological (HOT) states extend this usual bulk-boundary
correspondence, so they host the modes localized at lower-dimensional
boundaries, such as corners and hinges. Here we theoretically demonstrate that
dislocations, ubiquitous defects in crystalline materials, can probe
higher-order topology, recently realized in various platforms. We uncover that
HOT insulators respond to dislocations through symmetry protected finite-energy
in-gap electronic modes, localized at the defect core, which originate from an
interplay between the orientation of the HOT mass domain wall and the Burgers
vector of the dislocation. As such, these modes become gapless only when the
Burgers vector points toward lower-dimensional gapless boundaries. Our findings
are consequential for the systematic probing of the extended bulk-boundary
correspondence in a broad range of HOT crystals, and photonic and phononic or
mechanical metamaterials through the bulk topological lattice defects.
