  We study topological aspects of defect modes for a family of operators
$\{\mathscr{P}(t)\}_{t \in [0,2\pi]}$ on $L^2(\mathbb{R})$. $\mathscr{P}(t)$ is
a periodic Schr\"odinger operator $P_0$ perturbed by a dislocated potential.
This potential is periodic on the left and on the right, but acquires a phase
defect $t$ from $-\infty$ relative to $+\infty$. When $t=\pi$ and the
dislocation is small and adiabatic, Fefferman, Lee-Thorp and Weinstein showed
in previous work that Dirac points of $P_0$ (degeneracies in the band spectrum
of $P_0$) bifurcate to defect modes of $\mathscr{P}(\pi)$.
  We show that these modes are topologically protected at the level of the
family $\{\mathscr{P}(t)\}_{t \in [0,2\pi]}$. This means that local
perturbations cannot remove these states for all values of $t$, even outside
the small adiabatic regime studied by Fefferman$-$Lee-Thorp$-$Weinstein. We
define two topological quantities: an edge index (the signed number of
eigenvalues crossing an energy gap as $t$ runs from $0$ to $2\pi$) and a bulk
index (the Chern number of a Bloch eigenbundle for the periodic operator near
$+\infty$). We prove that these indexes are equal to the same odd winding
number. This shows the bulk-edge correspondence for the family
$\{\mathscr{P}(t)\}_{t \in [0,2\pi]}$. We express this winding number in terms
of the asymptotic shape of the dislocation and of the Dirac point Bloch modes.
We illustrate the topological depth of our model via the computation of the
bulk/edge index on a few examples.
