Boundary conditions and violations of bulk-edge correspondence in a hydrodynamic model

Kurzfassung

Bulk-edge correspondence is a wide-ranging principle that applies to topological matter. According to the principle, the distinctive topological properties of matter, thought of as extending indefinitely in space, are equivalently reflected in the excitations running along its boundary. Indices encode those properties, and their values, when differing, are witness to a violation of that correspondence. We address such violations, as they occur in a hydrodynamic context. The model concerns a shallow layer of fluid in a rotating frame and provides a local description of waves propagating either across the oceans or along a coastline; it becomes topological when suitably modified at short distances. The edge index is sensitive to boundary conditions, hence exhibiting a violation. Here we present a classification of all (local, self-adjoint) boundary conditions. They come in four families, distinguished in part by the degree of their underlying differential operators. Generally, both the correspondence and its violation are typical. Across families though, the maximally possible amount of violation can vary with their degree. Several indices of interest are charted for all boundary conditions. A single spectral mechanism for the onset of violations is furthermore identified, and the role of a symmetry investigated.