  We here introduce the concept of fragile topological dislocation modes, which
are localized only in a fraction of a topological phase, while otherwise leak
into the bulk continuum. We show that such dislocation modes are hosted in an
obstructed atomic topological phase in the two-dimensional Su-Schrieffer-Heeger
model, but only in a finite region with an indirect gap at high energy. These
dislocation modes are realized as chiral pairs at finite energies with
protection stemming from a combination of the chiral (unitary particle-hole)
and the point group (C$_{4v}$) symmetries, but only when the indirect gap is
open. In this regime, we corroborate the stability of the defect modes by
following their localization and also by explicitly adding a weak chemical
potential disorder. Our findings, therefore, should be consequential for the
experimental observation of such modes in designer topological crystals and
classical metamaterials.
