The bulk-edge correspondence for continuous dislocated systems

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.