  Nonzero weak topological indices are thought to be a necessary condition to
bind a single helical mode to lattice dislocations. In this work we show that
higher-order topological insulators (HOTIs) can, in fact, host a single helical
mode along screw or edge dislocations (including step edges) in the absence of
weak topological indices. When this occurs, the helical mode is necessarily
bound to a dislocation characterized by a fractional Burgers vector,
macroscopically detected by the existence of a stacking fault. The robustness
of a helical mode on a partial defect is demonstrated by an adiabatic
transformation that restores translation symmetry in the stacking fault. We
present two examples of HOTIs, one intrinsic and one extrinsic, that show
helical modes at partial dislocations. Since partial defects and stacking
faults are commonplace in bulk crystals, the existence of such helical modes
can measurably affect the expected conductivity in these materials.
