The bulk-edge correspondence for continuous dislocated systems

Drouot, Alexis

Annales de l'Institut Fourier, Tome 71 (2021) no. 3, pp. 1185-1239.

Nous étudions les aspects topologiques de modes propres d’une famille d’opérateurs ${\left\{𝒫\left(t\right)\right\}}_{t\in [0,2\pi ]}$, perturbations d’un opérateur $P_0$ par une dislocation : un potentiel avec un défaut de phase $t$ entre $-\infty$ and $+\infty$. Quand $t=\pi$ et la dislocation est adiabatique, Fefferman, Lee-Thorp and Weinstein ont montré que les dégénérescences coniques de $P_0$ bifurquent vers des états propres de $𝒫\left(t\right)$.

Nous montrons ici que ces états propres sont topologiquement protégés, même en dehors du régime adiabatique. Cela passe par une correspondance « bulk-edge » pour les systèmes de dislocations. Spécifiquement, nous démontrons que le nombre effectif d’états propres est égal à un nombre de Chern, calculé à partir du bulk. Nous montrons que ces nombres sont le degré d’une fonction définie à partir de la dégénérescence conique et de la dislocation. Finalement, nous illustrons la richesse du modèle à travers quelques exemples.

We study topological aspects of defect modes for a family of operators $𝒫\left(t\right)$, obtained as a Schrödinger operator $P_0$ perturbed by a phase defect $t$ between $-\infty$ and $+\infty$: a dislocation. When $t=\pi$ and the dislocation is adiabatic, Fefferman, Lee-Thorp and Weinstein showed that conical degeneracies of $P_0$ bifurcate to defect modes of $𝒫\left(\pi \right)$.

We show that these modes are topologically protected even outside the adiabatic regime. This relies on a bulk-edge correspondence for dislocations. Specifically, we show that the signed number of defect modes equals a Chern number computed from the bulk. We next derive an explicit formula for these indices in terms of the dislocation and of the conical point Bloch modes. We illustrate the depth of our result through a few examples.

Classification : 35P15, 35P25, 35Q40, 35Q41
Keywords: quantum matter, topological edge states, bulk-edge correspondence
Mot clés : matière quantinque, états de bord topologiques, correspondance bulk-edge

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